KrisMinchev/finemath-4plus-tokenized · Datasets at Hugging Face

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https://nrich.maths.org/public/topic.php?code=125&cl=3&cldcmpid=574	1,596,728,764,000,000,000	text/html	crawl-data/CC-MAIN-2020-34/segments/1596439736972.79/warc/CC-MAIN-20200806151047-20200806181047-00522.warc.gz	370,684,246	8,169	# Resources tagged with: Sine, cosine, tangent Filter by: Content type: Age range: Challenge level: ### There are 35 results Broad Topics > Pythagoras and Trigonometry > Sine, cosine, tangent ##### Age 14 to 16 Challenge Level: The sides of a triangle are 25, 39 and 40 units of length. Find the diameter of the circumscribed circle. ### Circle Scaling ##### Age 14 to 16 Challenge Level: Describe how to construct three circles which have areas in the ratio 1:2:3. ### Where Is the Dot? ##### Age 14 to 16 Challenge Level: A dot starts at the point (1,0) and turns anticlockwise. Can you estimate the height of the dot after it has turned through 45 degrees? Can you calculate its height? ### Figure of Eight ##### Age 14 to 16 Challenge Level: On a nine-point pegboard a band is stretched over 4 pegs in a "figure of 8" arrangement. How many different "figure of 8" arrangements can be made ? ### Squ-areas ##### Age 14 to 16 Challenge Level: Three squares are drawn on the sides of a triangle ABC. Their areas are respectively 18 000, 20 000 and 26 000 square centimetres. If the outer vertices of the squares are joined, three more. . . . ### Sine and Cosine ##### Age 14 to 16 Challenge Level: The sine of an angle is equal to the cosine of its complement. Can you explain why and does this rule extend beyond angles of 90 degrees? ### Inscribed in a Circle ##### Age 14 to 16 Challenge Level: The area of a square inscribed in a circle with a unit radius is, satisfyingly, 2. What is the area of a regular hexagon inscribed in a circle with a unit radius? ### Circle Box ##### Age 14 to 16 Challenge Level: It is obvious that we can fit four circles of diameter 1 unit in a square of side 2 without overlapping. What is the smallest square into which we can fit 3 circles of diameter 1 unit? ### Round and Round ##### Age 14 to 16 Challenge Level: Prove that the shaded area of the semicircle is equal to the area of the inner circle. ### Dodecawhat ##### Age 14 to 16 Challenge Level: Follow instructions to fold sheets of A4 paper into pentagons and assemble them to form a dodecahedron. Calculate the error in the angle of the not perfectly regular pentagons you make. ### Six Discs ##### Age 14 to 16 Challenge Level: Six circular discs are packed in different-shaped boxes so that the discs touch their neighbours and the sides of the box. Can you put the boxes in order according to the areas of their bases? ### From All Corners ##### Age 14 to 16 Challenge Level: Straight lines are drawn from each corner of a square to the mid points of the opposite sides. Express the area of the octagon that is formed at the centre as a fraction of the area of the square. ### Raising the Roof ##### Age 14 to 16 Challenge Level: How far should the roof overhang to shade windows from the mid-day sun? ### Farhan's Poor Square ##### Age 14 to 16 Challenge Level: From the measurements and the clue given find the area of the square that is not covered by the triangle and the circle. ### Round and Round a Circle ##### Age 14 to 16 Challenge Level: Can you explain what is happening and account for the values being displayed? ### History of Trigonometry - Part 3 ##### Age 11 to 18 The third of three articles on the History of Trigonometry. ### A Scale for the Solar System ##### Age 14 to 16 Challenge Level: The Earth is further from the Sun than Venus, but how much further? Twice as far? Ten times? ### Lying and Cheating ##### Age 11 to 14 Challenge Level: Follow the instructions and you can take a rectangle, cut it into 4 pieces, discard two small triangles, put together the remaining two pieces and end up with a rectangle the same size. Try it! ### Cosines Rule ##### Age 14 to 16 Challenge Level: Three points A, B and C lie in this order on a line, and P is any point in the plane. Use the Cosine Rule to prove the following statement. ### At a Glance ##### Age 14 to 16 Challenge Level: The area of a regular pentagon looks about twice as a big as the pentangle star drawn within it. Is it? ### Sine and Cosine for Connected Angles ##### Age 14 to 16 Challenge Level: The length AM can be calculated using trigonometry in two different ways. Create this pair of equivalent calculations for different peg boards, notice a general result, and account for it. ### The History of Trigonometry- Part 1 ##### Age 11 to 18 The first of three articles on the History of Trigonometry. This takes us from the Egyptians to early work on trigonometry in China. ### History of Trigonometry - Part 2 ##### Age 11 to 18 The second of three articles on the History of Trigonometry. ### Making Maths: Clinometer ##### Age 11 to 14 Challenge Level: You can use a clinometer to measure the height of tall things that you can't possibly reach to the top of, Make a clinometer and use it to help you estimate the heights of tall objects. ### Moving Squares ##### Age 14 to 16 Challenge Level: How can you represent the curvature of a cylinder on a flat piece of paper? ### 8 Methods for Three by One ##### Age 14 to 18 Challenge Level: This problem in geometry has been solved in no less than EIGHT ways by a pair of students. How would you solve it? How many of their solutions can you follow? How are they the same or different?. . . . ### Orbiting Billiard Balls ##### Age 14 to 16 Challenge Level: What angle is needed for a ball to do a circuit of the billiard table and then pass through its original position? ### Screen Shot ##### Age 14 to 16 Challenge Level: A moveable screen slides along a mirrored corridor towards a centrally placed light source. A ray of light from that source is directed towards a wall of the corridor, which it strikes at 45 degrees. . . . ##### Age 11 to 18 Logo helps us to understand gradients of lines and why Muggles Magic is not magic but mathematics. See the problem Muggles magic. ### Coke Machine ##### Age 14 to 16 Challenge Level: The coke machine in college takes 50 pence pieces. It also takes a certain foreign coin of traditional design... ##### Age 14 to 16 Challenge Level: In this problem we are faced with an apparently easy area problem, but it has gone horribly wrong! What happened? ### Far Horizon ##### Age 14 to 16 Challenge Level: An observer is on top of a lighthouse. How far from the foot of the lighthouse is the horizon that the observer can see? ### Eight Ratios ##### Age 14 to 16 Challenge Level: Two perpendicular lines lie across each other and the end points are joined to form a quadrilateral. Eight ratios are defined, three are given but five need to be found. ### Trigonometric Protractor ##### Age 14 to 16 Challenge Level: An environment that simulates a protractor carrying a right- angled triangle of unit hypotenuse.	1,614	6,796	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	4.375	4	CC-MAIN-2020-34	latest	en	0.86502	[ 128000, 2, 16607, 38213, 449, 25, 328, 483, 11, 76359, 11, 69760, 271, 5750, 555, 25, 9059, 955, 512, 17166, 2134, 512, 63178, 2237, 1473, 14711, 2684, 527, 220, 1758, 3135, 271, 69424, 41994, 871, 5468, 96462, 65747, 323, 1183, 74981, 7133, 871, 328, 483, 11, 76359, 11, 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https://oeis.org/A127330	1,600,905,311,000,000,000	text/html	crawl-data/CC-MAIN-2020-40/segments/1600400212959.12/warc/CC-MAIN-20200923211300-20200924001300-00018.warc.gz	543,022,897	4,630	The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation. Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!) A127330 Begin with the empty sequence and a starting number s = 0. At step k (k >= 1) append the k consecutive numbers s to s+k-1 and change the starting number (for the next step) to 2s+2. 5 0, 2, 3, 6, 7, 8, 14, 15, 16, 17, 30, 31, 32, 33, 34, 62, 63, 64, 65, 66, 67, 126, 127, 128, 129, 130, 131, 132, 254, 255, 256, 257, 258, 259, 260, 261, 510, 511, 512, 513, 514, 515, 516, 517, 518, 1022, 1023, 1024, 1025, 1026, 1027, 1028, 1029, 1030, 1031, 2046 (list; table; graph; refs; listen; history; text; internal format) OFFSET 0,2 COMMENTS From a TV show. A129142 and A129143 are similar, slightly more natural, but for a puzzle perhaps too transparent sequences. Can be seen as a triangle (row by step) read by rows: T(n,k) = T(n-1,k) + 2^n for k < n and T(n,n) = T(n-1,n-1) + 2^n + 1. - Reinhard Zumkeller, Nov 16 2013 LINKS Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened EXAMPLE In step 1 starting number 0 is appended to the empty sequence and the next starting number is 20 + 2 = 2. In step 2 the two numbers 2, 3 are appended and the starting number is changed to 22 + 2 = 6. MATHEMATICA Table[ Range[2^k-2, 2^k+k-3], {k, 1, 11}] // Flatten (* Jean-François Alcover, Oct 07 2013, after Klaus Brockhaus ) Join[{0}, Flatten[With[{nn=10}, Range[#[[1]], Total[#]]&/@Thread[ {Accumulate[ 2^Range[nn]], Range[nn]}]]]] ( Harvey P. Dale, Nov 05 2017 ) PROG (PARI) {v=[]; s=0; for(k=1, 11, w=vector(k, j, j+s-1); s=2s+2; v=concat(v, w)); for(n=1, #v, print1(v[n], ", "))} \\ Klaus Brockhaus, Mar 31 2007 (MAGMA) &cat[ [2^k-2..2^k+k-3]: k in [1..11] ]; // Klaus Brockhaus, Mar 31 2007 (Haskell) a127330 n k = a127330_tabl !! n !! k a127330_row n = a127330_tabl !! n a127330_tabl = step 0 1 where step s k = [s .. s + k - 1] : step (2 * s + 2) (k + 1) -- Reinhard Zumkeller, Nov 16 2013 CROSSREFS Cf. A129142, A129143. Cf. A000918 (left edge), A145071 (right edge). Sequence in context: A166458 A189013 A242940 * A035346 A030164 A066646 Adjacent sequences: A127327 A127328 A127329 * A127331 A127332 A127333 KEYWORD easy,nice,nonn,tabl AUTHOR Steven Cartier (steven.cartier(AT)rogers.com), Mar 30 2007 EXTENSIONS Edited and extended by Klaus Brockhaus, Mar 31 2007 Keyword tabl added and offset changed by Reinhard Zumkeller, Nov 16 2013 STATUS approved Lookup \| Welcome \| Wiki \| Register \| Music \| Plot 2 \| Demos \| Index \| Browse \| More \| WebCam Contribute new seq. or comment \| Format \| Style Sheet \| Transforms \| Superseeker \| Recent The OEIS Community \| Maintained by The OEIS Foundation Inc. Last modified September 23 19:41 EDT 2020. Contains 337315 sequences. 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http://classblogmeister.com/blog.php?blogger_id=352731&user_id=352731&show=all	1,368,928,929,000,000,000	text/html	crawl-data/CC-MAIN-2013-20/segments/1368696383156/warc/CC-MAIN-20130516092623-00087-ip-10-60-113-184.ec2.internal.warc.gz	53,037,578	8,800	files/ kains -- Blogmeister We have three 6th grade Science classes and two 8th grade Science classes blogging here from the Pacific Northwest in Chimacum, WA! Sixth graders are learning a bit about Mt Saint Helens, environmental science through fresh water ecology, and physical science this year. Eighth graders are learning about life science this year. Please join us as we learn Science by exploring our world. Mr. G's Blog Mr. G's Class Facebook Page by kains teacher: Alfonso Gonzalez Blog Entries List 25, 50, all Article posted May 16, 2012 at 03:29 PM GMT • comment • Reads 1078 here are the blogs I visited (links) 1. http://danteawapuni.blogspot.co.nz/2012/02/wonder-beads.html 2.http://ryanawapuni.blogspot.co.nz/2012/02/this-is-me.html those are the blogs I visited here is some information about them 1.dante plays sports and reads 2.ryan likes superheros Article posted May 16, 2012 at 03:29 PM GMT • comment • Reads 1078 Article posted May 15, 2012 at 03:16 PM GMT • comment • Reads 790 A. I thought that the wide side on a block fo wood would be the side that would reaquier ( i know i spelled that wrong) the most force to pull because it has the most surface area but I found out that the side with the least surface ( the narrow side) took the most force to pull. What I learned about surface area, and weight of blocks I shall type in the next letters starting now. I learned that that if you have more weight on a smaller surface it will take more force to pull the block then if the block has the weight spread out and if you add more blocks the weight will add and it will be even more force needed to pull the block. B.the scale measures the fricton force witch ( yes I spelled that wrong to ( I think)) is caused by pulling on the scale with gravity pulling the block down. C.we would use almost no force to pull a block of wood across ice because it is very smooth so the force needed to pull the block will be about .1 newtons per block on the wide side of the block. Article posted May 15, 2012 at 03:16 PM GMT • comment • Reads 790 Article posted April 26, 2012 at 03:34 PM GMT • comment • Reads 867 You can put a rubber band on 2 sticks then you put it (the rubber band) around the cart then pull back then you let it go to slingshot it across the floor. A way to make it twice as powerful ( the rubber bands elastic force) is to double up the ruber band then reapeat the stuff before. The difference between mass and weight! well mass affects wieght because gravity affects mass makeing wieght Article posted April 26, 2012 at 03:34 PM GMT • comment • Reads 867 Article posted March 13, 2012 at 04:35 PM GMT • comment • Reads 1094 K: I know that without bateries we would not have cars or handheld gaming systems or anything like those things that need bateries. W: Some things that I want to learn about bateries are: 1. how to make one 2. how to see if I could make a way better one and 3. why there is acidic stuff in bateries L: I learned that batterys have strong acid in them to make energy because without that it would not work. I also learned that batterys have copper and zinc in them. Article posted March 13, 2012 at 04:35 PM GMT • comment • Reads 1094 Article posted March 13, 2012 at 04:20 PM GMT • comment (1) • Reads 1495 Hi I am a 6th grader going to chimacum middle school I live in a small house close to port townsend. If I want to I can walk to the beach next to my house. A few things I like are: riding my bike, playing video games and,talking to my friends. The subjects in school I like are science, p.e., and math. I am also interesed in space (yes that big empty thing thats all around us with other planets and stuff). Thats pretty much it so good bye. Article posted March 13, 2012 at 04:20 PM GMT • comment (1) • Reads 1495 Article posted March 5, 2012 at 04:34 PM GMT • comment • Reads 938 @font-face { font-family: "Times New Roman"; }@font-face { font-family: "Wingdings"; }@font-face { font-family: "Modern No. 20"; }@font-face { font-family: "ヒラギノ角ゴ Pro W3"; }p.MsoNormal, li.MsoNormal, div.MsoNormal { margin: 0in 0in 0.0001pt; font-size: 12pt; font-family: "Times New Roman"; }table.MsoNormalTable { font-size: 10pt; font-family: "Times New Roman"; }div.Section1 { page: Section1; } DO: what are these years average DO levels: 8.125. The oxygen in the air makes contact with water to make dissolved oxygen (do). Chimacum creek has been doing well even at a low of about 6 ppm. PH: the average is 6.3 ph for this year. 7 is pure water 1-6 is acidic 8-14 is alkaline. When base and acid come together it makes is a balance. Ph stands for positive hydrogen. Fish like 6.5-7. Turbitity: the average is in the 30’ of ntu’s. n.t.u stands for nephelometric turbitity units somebody might ask what turbitity is well turbitity is the difference between clear water and dirty water Nitrogen (this is my perrmater): nitrates are important because they help the soil and water and creatures and everything(about this).the average for this year in nitrates is 0.3 this is good because fish die in to high of nitrate and so will people mostly babys because to much nitrates can cause brown blood disease or blue baby syndrome(same thing) or Methogloblin (also the same thing). Ammonia: This years average ammonia is 1.6. this is good because it wont kill the fish. Ammonia is very dangeros chemical thing. Ammonia can be found in many cleaning supplies. Pure Ammonia is a very very smelly colorless liquid. Eutrophication is a thing that is very bad. A thing you do not want to do is mix ammonia and bleach ( it can kill you) Flow rate: you x length x width (in a river) to get cubic meters per second (if using that measurement) . flow rate can be important because if fish can not lay there eggs then they will die out. The average for flow rate is 1.5934 m3/s (??????) Temperature: water that is shallow takes shorter to warm up than deep. This years average water temp is 33.5 farenhite. The airs average is 37.5 farenhite. One of the highest temp max for fish is 97º farenhite where the fish will die L. Article posted March 5, 2012 at 04:34 PM GMT • comment • Reads 938 Article posted January 30, 2012 at 04:16 PM GMT • comment • Reads 1111 When I went to the place next to snow creek I was really exited to plant trees. when we started I found out that it was much harder to plant trees then I thought. We also found lots of worms and we put them next to the trees to help the soil. Article posted January 30, 2012 at 04:16 PM GMT • comment • Reads 1111 Article posted November 21, 2011 at 04:34 PM GMT • comment (2) • Reads 1855 this is the better video Click here. Article posted November 21, 2011 at 04:34 PM GMT • comment (2) • Reads 1855 Article posted November 7, 2011 at 04:22 PM GMT • comment (1) • Reads 1727 In some other blogs I showed pictures in this blog I will tell you which creek scored better yellow jacket or chimacum. Well my pictures would mean that chimacum scored 2 points because scuds are worth 2 points. but luckly other people found more bugs like mayflys and stoneflys that are worth more. chimacum scored 21 points total. And yellow jacket creek scored 25 points total. so that means yellow jacket creek is a little bit cleaner. it also means that we have potentially good water quality and yellow jacket creek has potentially exelent water quality. Article posted November 7, 2011 at 04:22 PM GMT • comment (1) • Reads 1727 Article posted November 7, 2009 at 06:00 AM GMT • comment • Reads 1639 this is a picture of a moving scud. this is another moving scud. these are worth 3 points from our out door education program which scored using how tolorent the bug was to pollution. there is 4 different groups 1.intolerent, 2. somewhat intolerent, 3.somewhat tolerent, 4. very tolerent this is another part of my other macro blog which also had a scud but it was dead. Article posted November 7, 2009 at 06:00 AM GMT • comment • Reads 1639 Article posted November 2, 2011 at 04:16 PM GMT • comment • Reads 1696 this is a photo of a dead scud i found in chimacum creek. thanks for reading and looking. Article posted November 2, 2011 at 04:16 PM GMT • comment • Reads 1696 Article posted October 20, 2011 at 04:35 PM GMT • comment (1) • Reads 1859 1.What is water pollution? well it's contamination of water bodies such as lakes, rivers, oceans, ground water, ect. 2.What are some sources of water pollution? Well 14 billion pounds of garbage is being dumped into the oceans every year. Another source is factories, refineries, waste treatment plants, ect. 3.What are the consequences of water pollution? Well it can be disastrous for both humans and the ecosystem .14,000 humans are suffering from waterborne diseases every day! Water pollution effects the chemical and physical properties of water afecting the life present inside the water. 4.What can we do to improve water quality. Well we can drive less because the pollution from the gas is geting into the water. Another way is to stop using chemicals like pesticide and herbicide because they get into our water polluting it. 5.What can you do to prevent water from being polluted? One thing to do is dispose of used oil, antifreeze, paints, ect. properly which means not in sewers or drains. Another thing to do is have your septic system [if you have one] inspected and pumped every three to five years. One last thing to do is purchase household detergents and cleaners that are low in phosphorous to reduce the pollution flowwing into our water. Article posted October 20, 2011 at 04:35 PM GMT • comment (1) • Reads 1859 Article posted October 3, 2011 at 04:27 PM GMT • comment (7) • Reads 2740 1. I Love sushi 2. I Like candy 3. I am a boy 4. My brain is abnormal 5. Most of my friends are smart 6. Some Are stupid(of my friends) 7. 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https://web2.0calc.com/questions/please-help-need-answer-asap	1,524,742,803,000,000,000	text/html	crawl-data/CC-MAIN-2018-17/segments/1524125948126.97/warc/CC-MAIN-20180426105552-20180426125552-00091.warc.gz	720,724,380	6,359	+0 0 1 286 1 +4 1.Use the figure to answer the questions. (a) Explain why the triangles are similar. (b) Find the value of y. Show your work 2. Danae is making a candle by pouring melted wax into a mold in the shape of a cone. The radius of the base of the cone is 2 in and the height of the cone is 8 in. To get the wax for the candle, Danae melts blocks of wax that are each 1 in by 1 in by 2 in. How many of the wax cubes will Danae need in order to make the candle? Show your work. StrongJet Jun 16, 2017 Sort: #1 +6954 +2 1. (a) Since ∠C = ∠S , these triangles are similar IF $$\frac{30}{45}=\frac{16}{24} \\~\\ \frac{30\,\div\,15}{45\,\div\,15}=\frac{16\,\div\,8}{24\,\div\,8} \\~\\ \frac23=\frac23\qquad\text{true}$$ So, these triangles are similar. (b) Since these triangles are similar... $$\frac{y}{14}=\frac{30}{16} \\~\\ y=\frac{30}{16}\,\,14 \\~\\ y=26.25$$ 2. volume of cone = (1/3) (pi * radius2) * (height) = (1/3) * (pi * 22) * (8) = (1/3) * pi * 4 * 8 = 32pi / 3 cubic inches volume of block = length * width * height = 1 * 1 * 2 = 2 cubic inches How many blocks fit into the cone? How many 2 cubic inches fit into 32pi / 3 cubic inches ? $$\frac{32\pi}{3}\,\div\,2 \,=\,\frac{32\pi}{3}\,*\,\frac12\,=\,\frac{32\pi}{6}\,\approx\,16.755$$ hectictar Jun 16, 2017 ### 11 Online Users We use cookies to personalise content and ads, to provide social media features and to analyse our traffic. We also share information about your use of our site with our social media, advertising and analytics partners. 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https://electronics.stackexchange.com/questions/700694/obtaining-transfer-function	1,721,502,088,000,000,000	text/html	crawl-data/CC-MAIN-2024-30/segments/1720763517515.18/warc/CC-MAIN-20240720174732-20240720204732-00811.warc.gz	196,265,329	39,326	# Obtaining transfer function In the above circuit j(t)=u(t)A, where u(t) is the step function. Z is a component/circuit system with no independent variables and all initial values zero. The goal is to find the transfer function Z(s). So I think I know the transfer function here is the laplace transform of the output divided by the laplace transform of the input. The output being u1(t) and input being j(t). u1(t) is also Zj(t) and the laplace transform Z(s)1/s. The laplace transform of the input is 1/s. So using the equation for the transfer function (Z(s)*1/s)/1/s you get Z(s), which does not seem right at all. What am I missing here? The goal is to find the transfer function Z(s). Z is a component/circuit system with no independent variables The transfer function is independent of the input so the fact that it is a current step is irrelevant. So the only information you have to express the transfer function is the output voltage and the input current. The ratio then is an impedance Z(s)=Z. This you have verified as: $$Z(s)=\frac{U1_{out}(s)}{J_{in}(s)}=Z$$ I think you are missing the fact that you have the right answer. Understand that the impedance of any passive element is the transfer function of the element where voltage is the output and current is the input. • Oh that's interesting. Makes sense now lol. Only thing that I don't understand is how you write the equation for the transfer function. I have never seen t(s) or n(s) before, so I'm kind of confused on that. Commented Feb 8 at 20:03 • Just some finger trouble with MathJax. The equation should make more sense now. @JohnnyB Commented Feb 8 at 20:06 • Oh lol. Thanks for the help @RussellH Commented Feb 8 at 22:11	426	1,713	{"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 1, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	3.5625	4	CC-MAIN-2024-30	latest	en	0.911571	[ 128000, 2, 57747, 2101, 8481, 734, 271, 644, 279, 3485, 16622, 503, 1175, 11992, 84, 1175, 8, 32, 11, 1405, 577, 1175, 8, 374, 279, 3094, 734, 13, 1901, 374, 264, 3777, 2971, 38368, 1887, 449, 912, 9678, 7482, 323, 682, 2926, 2819, 7315, 13, 578, 5915, 374, 311, 1505, 279, 8481, 734, 1901, 1161, 3677, 4516, 358, 1781, 358, 1440, 279, 8481, 734, 1618, 374, 279, 24301, 27634, 5276, 315, 279, 2612, 18255, 555, 279, 24301, 27634, 5276, 315, 279, 1988, 13, 578, 2612, 1694, 577, 16, 1175, 8, 323, 1988, 1694, 503, 1175, 570, 577, 16, 1175, 8, 374, 1101, 1901, 55145, 1175, 8, 323, 279, 24301, 27634, 5276, 1901, 1161, 4911, 16, 2754, 13, 578, 24301, 27634, 5276, 315, 279, 1988, 374, 220, 16, 2754, 13, 2100, 1701, 279, 24524, 369, 279, 8481, 734, 320, 57, 1161, 4911, 16, 2754, 5738, 16, 2754, 499, 636, 1901, 1161, 705, 902, 1587, 539, 2873, 1314, 520, 682, 13, 3639, 1097, 358, 7554, 1618, 1980, 791, 5915, 374, 311, 1505, 279, 8481, 734, 1901, 1161, 3677, 57, 374, 264, 3777, 2971, 38368, 1887, 449, 912, 9678, 7482, 271, 791, 8481, 734, 374, 9678, 315, 279, 1988, 779, 279, 2144, 430, 433, 374, 264, 1510, 3094, 374, 40815, 382, 4516, 279, 1193, 2038, 499, 617, 311, 3237, 279, 8481, 734, 374, 279, 2612, 22465, 323, 279, 1988, 1510, 13, 578, 11595, 1243, 374, 459, 91048, 1901, 1161, 11992, 57, 382, 2028, 499, 617, 24884, 439, 25, 27199, 57, 1161, 11992, 59, 38118, 90, 52, 16, 15511, 412, 26628, 82, 9317, 90, 41, 15511, 258, 26628, 82, 9317, 28, 57, 14415, 271, 40, 1781, 499, 527, 7554, 279, 2144, 430, 499, 617, 279, 1314, 4320, 13, 71994, 430, 279, 91048, 315, 904, 28979, 2449, 374, 279, 8481, 734, 315, 279, 2449, 1405, 22465, 374, 279, 2612, 323, 1510, 374, 279, 1988, 382, 6806, 8840, 430, 596, 7185, 13, 37970, 5647, 1457, 28509, 13, 8442, 3245, 430, 358, 1541, 956, 3619, 374, 1268, 499, 3350, 279, 24524, 369, 279, 8481, 734, 13, 358, 617, 2646, 3970, 259, 1161, 8, 477, 308, 1161, 8, 1603, 11, 779, 358, 2846, 3169, 315, 22568, 389, 430, 13, 12535, 291, 13806, 220, 23, 520, 220, 508, 25, 2839, 198, 6806, 4702, 1063, 14654, 12544, 449, 4242, 41, 710, 13, 578, 24524, 1288, 1304, 810, 5647, 1457, 13, 571, 88960, 33, 12535, 291, 13806, 220, 23, 520, 220, 508, 25, 2705, 198, 6806, 8840, 28509, 13, 11361, 369, 279, 1520, 571, 79070, 616, 39, 12535, 291, 13806, 220, 23, 520, 220, 1313, 25, 806, 128001 ]	[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ]
https://justaaa.com/chemistry/38496-when-solutions-of-silver-nitrate-and-potassium	1,701,981,433,000,000,000	text/html	crawl-data/CC-MAIN-2023-50/segments/1700679100686.78/warc/CC-MAIN-20231207185656-20231207215656-00482.warc.gz	372,822,489	9,397	Question When solutions of silver nitrate and potassium chloride are mixed, silver chloride precipitates out of solution... When solutions of silver nitrate and potassium chloride are mixed, silver chloride precipitates out of solution according to the equation AgNO3(aq)+KCl(aq)→AgCl(s)+KNO3(aq). A) What mass of silver chloride can be produced from 1.90 L of a 0.133 M solution of silver nitrate? B) The reaction described in Part A required 3.67 L of potassium chloride. What is the concentration of this potassium chloride solution? A) number of moles of AgNO3 = M(AgNO3)V(AgNO3) = 0.133 M 1.90 L = 0.2527 mol from reaction, mol of AgCl formed = mol of AgNO3 reacted = 0.2527 mol Molar mass of AgCl, MM = 1MM(Ag) + 1MM(Cl) = 1107.9 + 135.45 = 143.35 g/mol use: mass of AgCl, m = number of mol * molar mass = 0.2527 mol * 1.43410^2 g/mol = 36.22 g B) From reaction, mol of KCl reacted = mol of AgNO3 formed = 0.2527 mol Now use: mol of KCl = M(KCl)V(KCl) 0.2527 mol = M(KCl) * 3.67 L M(KCl) = 0.0689 M	352	1,041	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	3.609375	4	CC-MAIN-2023-50	latest	en	0.821464	[ 128000, 14924, 271, 4599, 10105, 315, 15310, 25719, 7853, 323, 62275, 82882, 527, 9709, 11, 15310, 82882, 36841, 82829, 704, 315, 6425, 2195, 4599, 10105, 315, 15310, 25719, 7853, 323, 62275, 82882, 527, 9709, 11, 15310, 82882, 36841, 82829, 704, 315, 6425, 4184, 311, 279, 24524, 4701, 9173, 18, 2948, 80, 7405, 42, 5176, 2948, 80, 8, 52118, 9219, 5176, 1161, 7405, 42, 9173, 18, 2948, 80, 3677, 32, 8, 3639, 3148, 315, 15310, 82882, 649, 387, 9124, 505, 220, 16, 13, 1954, 445, 315, 264, 220, 15, 13, 9423, 386, 6425, 315, 15310, 25719, 7853, 1980, 33, 8, 578, 13010, 7633, 304, 3744, 362, 2631, 220, 18, 13, 3080, 445, 315, 62275, 82882, 13, 3639, 374, 279, 20545, 315, 420, 62275, 82882, 6425, 1980, 32, 696, 4174, 315, 4647, 645, 315, 4701, 9173, 18, 284, 386, 4444, 70, 9173, 18, 4911, 53, 4444, 70, 9173, 18, 696, 28, 220, 15, 13, 9423, 386, 353, 220, 16, 13, 1954, 445, 271, 28, 220, 15, 13, 12326, 22, 22337, 271, 1527, 13010, 3638, 45444, 315, 4701, 5176, 14454, 284, 22337, 315, 4701, 9173, 18, 55841, 271, 28, 220, 15, 13, 12326, 22, 22337, 271, 44, 7569, 3148, 315, 4701, 5176, 3638, 8195, 284, 220, 16, 9, 8195, 4444, 70, 8, 489, 220, 16, 9, 8195, 44744, 696, 28, 220, 16, 9, 7699, 13, 24, 489, 220, 16, 9, 1758, 13, 1774, 271, 28, 220, 10290, 13, 1758, 342, 39971, 271, 817, 1473, 27428, 315, 4701, 5176, 3638, 76, 284, 1396, 315, 22337, 353, 296, 7569, 3148, 271, 28, 220, 15, 13, 12326, 22, 22337, 353, 220, 16, 13, 20165, 9, 605, 61, 17, 342, 39971, 271, 28, 220, 1927, 13, 1313, 342, 271, 33, 696, 3915, 13010, 3638, 45444, 315, 735, 5176, 55841, 284, 22337, 315, 4701, 9173, 18, 14454, 271, 28, 220, 15, 13, 12326, 22, 22337, 271, 7184, 1005, 1473, 45444, 315, 735, 5176, 284, 386, 17155, 5176, 4911, 53, 17155, 5176, 696, 15, 13, 12326, 22, 22337, 284, 386, 17155, 5176, 8, 353, 220, 18, 13, 3080, 445, 271, 44, 17155, 5176, 8, 284, 220, 15, 13, 26661, 24, 386, 128001 ]	[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ]
https://oeis.org/A006345	1,685,823,815,000,000,000	text/html	crawl-data/CC-MAIN-2023-23/segments/1685224649343.34/warc/CC-MAIN-20230603201228-20230603231228-00470.warc.gz	475,099,284	6,036	The OEIS is supported by the many generous donors to the OEIS Foundation. Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!) A006345 Linus sequence: a(n) "breaks the pattern" by avoiding the longest doubled suffix. (Formerly M0074) 9 1, 2, 1, 1, 2, 2, 1, 2, 1, 1, 2, 1, 2, 2, 1, 1, 2, 1, 1, 1, 2, 2, 1, 2, 1, 1, 2, 2, 1, 1, 1, 2, 1, 1, 2, 2, 1, 2, 1, 1, 2, 1, 2, 2, 1, 1, 2, 1, 1, 1, 2, 2, 1, 2, 1, 1, 2, 2, 1, 2, 2, 2, 1, 1, 2, 1, 2, 2, 1, 1, 2, 2, 2, 1, 2, 2, 1, 1, 2, 1, 2, 2, 1, 2, 1, 1, 2, 2, 1, 2, 2, 2, 1, 1, 2, 1, 2, 2, 1, 1, 2, 2, 2, 1, 2 (list; graph; refs; listen; history; text; internal format) OFFSET 1,2 COMMENTS To find a(n), consider either a 1 or a 2. For each, find the longest repeated suffix, that is, for each of a(n)=1,2, find the longest sequence s with the property that the sequence a(1),...,a(n) ends with ss. Use the digit that results in the shorter such suffix. a(1) = 1. The empty sequence of length 0 is the shortest possible suffix and is trivially doubled. Note that this doesn't result in exactly Linus's choices. - K. Ramsey, kramsey(AT)aol.com On average, it seems that (# of 1s up to n) - (# of 2s up to n) -> infinity as n -> infinity (as O(log n)?), while the asymptotic density of either 1s or 2s appears to be 1/2. - Daniel Forgues, Mar 01 2017 REFERENCES N. S. Hellerstein, Letter to N. J. A. Sloane (1978). N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS T. D. Noe, Robert Israel and Hugo van der Sanden, Table of n, a(n) for n = 1..50000 (1..1000 from T. D. Noe, 1001..20000 from Robert Israel) P. Balister, S. Kalikow, A. Sarkar, The Linus sequence, Preprint May 2007; Combinatorics, Probability and Computing, Volume 19, Issue 1 January 2010 , pp. 21-46.. N. Hellerstein, Letter to N. J. A. Sloane, 1978 N. Hellerstein, M. Gardner, & S. Kim, Correspondence related to the Linus and Sally sequences, 1977 N. J. A. Sloane, Illustration of initial terms Eric Weisstein's World of Mathematics, Linus Sequence. EXAMPLE After 1,2,1,1,2,2,1,2, if we put a 1, the suffix {2,1} repeats, but if we put a 2 the longer suffix {1,2,2} repeats, so the next term is 1. MAPLE LDS:= proc(L) local Cands, r, m; Cands:= {\$1..floor(nops(L)/2)}; r:= 0; for m from 1 while nops(Cands) > 0 do Cands:= select(c -> L[-m] = L[-c-m], Cands); if min(Cands) = m then r:= m; Cands:= subs(m=NULL, Cands); fi od; r end proc: A:= 1: for n from 2 to 10^3 do if LDS([A, 1]) < LDS([A, 2]) then A:= A, 1 else A:= A, 2 fi; od: seq(A[i], i=1..10^3); # Robert Israel, Jun 22 2015 MATHEMATICA a[1]=1; a[2]=2; suffix[lst_] := If[MatchQ[lst, {___, b__, b__}], lst /. {___, b__, b__} :> {b}, {}]; a[n_] := a[n] = Module[{aa, lg1, lg2}, aa = Array[a, n-1]; lg1 = suffix[Append[aa, 1]] // Length; lg2 = suffix[Append[aa, 2]] // Length; If[lg1 <= lg2, 1, 2]]; Table[a[n], {n, 1, 105}] (* Jean-François Alcover, Dec 11 2014 ) PROG (Perl) -le 'print\$_.=3/(.)(.)\1\$/-\$2for(\$_)x99' (Ton Hospel/Phil Carmody) [An example of Perl golfing: use as few (key)strokes as possible] (PARI) {a(n)=local(A, t); if(n<2, n>0, A=[1]; for(i=2, n, forstep(j=i\2-1, 0, -1, for(k=1, j, if(A[i-j-k-1]!=A[i-k], next(2))); t=j; break); A=concat(A, [3-A[i-t-1]])); A[n])} /* Michael Somos, May 04 2006 / Comment on calculating this sequence and A006346 with Perl, from Hugo van der Sanden, Jun 23 2015: (Start) The approach I used was to take advantage of Perl's regular expression capabilities, coupled with the realization that Perl can optimize patterns anchored to the start far better than those anchored to the end - reversing the string to allow that gave a speedup of several orders of magnitude: my \$string = ""; digit('1', 0); for (2 .. \$limit) { my(\$repeat, \$digit) = (\$string =~ m{ ^ (.) ([12]) \1 }x) or die; digit(\$digit eq '1' ? '2' : '1', length(\$repeat) + 1); } sub digit { my(\$digit, \$repeat) = @_; \$string = \$digit . \$string; # n A6345(n) A6346(n) printf "%s %s %s\n", length(\$string), \$digit, \$repeat; } This takes about 45s to calculate 50000 terms of both sequences. (End) CROSSREFS Cf. A006346, A094840, A157238 (A006345(n) - 1). Sequence in context: A202340 A049705 A060236 * A122497 A350330 A154402 Adjacent sequences: A006342 A006343 A006344 * A006346 A006347 A006348 KEYWORD nonn,easy,nice AUTHOR N. J. A. Sloane EXTENSIONS More terms from Naohiro Nomoto, May 21 2001 Additional comments from Mitch Harris, Dec 31 2003 STATUS approved Lookup \| Welcome \| Wiki \| Register \| Music \| Plot 2 \| Demos \| Index \| Browse \| More \| WebCam Contribute new seq. or comment \| Format \| Style Sheet \| Transforms \| Superseeker \| Recents The OEIS Community \| Maintained by The OEIS Foundation Inc. Last modified June 3 15:36 EDT 2023. Contains 363116 sequences. 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https://programmingpraxis.com/2017/03/28/rng147/	1,679,372,356,000,000,000	text/html	crawl-data/CC-MAIN-2023-14/segments/1679296943625.81/warc/CC-MAIN-20230321033306-20230321063306-00430.warc.gz	511,148,579	30,429	## RNG147 ### March 28, 2017 We have looked at random number generators in several previous exercises, but most of them returned integers. In today’s exercise we look at a simple random number generator that returns floating-point numbers. The generator is simple: Given a seed between zero and one, the next number in the sequence is the fractional portion of 147 times the seed. Your task is to implement the random number generator described above, and to assess its suitability. When you are finished, you are welcome to read or run a suggested solution, or to post your own solution or discuss the exercise in the comments below. Pages: 1 2 ### 4 Responses to “RNG147” 1. Rutger said In python 3 ```def rng147(seed): while True: yield seed seed = (147.0 * seed) % 1.0 r = rng147(0.1234) for v in range(10): print(next(r)) ``` 2. Graham said Your sine trick only moves the problem elsewhere; try seeding with pi/6 (arcsin of 1/2). Solution in Haskell, using the Stream package for explicitly infinite lists: ```module Main where import Data.Fixed (mod') import Data.Stream (Stream) import qualified Data.Stream as S rng147 :: Double -> Stream Double rng147 = S.iterate (\x -> (147 * x) `mod'` 1) main :: IO () main = mapM_ print . S.take 10 . rng147 \$ pi / 10 ``` 3. Globules said Here’s another Haskell version. With an integer argument, n, it prints the first n random numbers. With no arguments it converts the stream of random Doubles into a stream of 32-bit unsigned integers. The purpose is to feed that binary data to the dieharder program, which evaluates its randomness. We don’t make use of all the bits in the Double; it’s just an easy way of getting the data to dieharder in a form that it likes. ```{-# LANGUAGE ScopedTypeVariables #-} import Data.Word (Word32) import Foreign.Marshal.Alloc (alloca) import Foreign.Storable (poke, sizeOf) import System.Environment (getArgs) import System.IO (hPutBuf, stdout) -- A stream of "random" numbers. (We take the tail to exclude the seed from the -- values.) rng147 :: RealFrac b => b -> [b] rng147 = tail . iterate step where step x = let (_ :: Int, f) = properFraction (147 * x) in f main :: IO () main = do let rs = rng147 (0.1234567 :: Double) ns <- fmap (map read) getArgs :: IO [Int] case ns of -- Output an endless stream of 32-bit binary data for dieharder. [] -> put \$ map cvt rs -- Print the first n random values. [n] -> mapM_ print \$ take n rs -- Invalid argument(s). _ -> putStrLn "Usage: rng147 [n]" -- Convert a value to an 32-bit unsigned integer, possibly throwing away the -- least significant bits of the argument. cvt :: RealFrac a => a -> Word32 cvt x = round \$ fromIntegral (maxBound :: Word32) * x -- Output the unsigned 32-bit values as binary data. put :: [Word32] -> IO () put is = let n = sizeOf (0 :: Word32) in alloca (\p -> void (mapM_ (\i -> poke p i >> hPutBuf stdout p n) is)) ``` ```\$ ./rng147 10 0.14813489999999874 0.7758302999998143 4.705409997271204e-2 0.9169526959886696 0.7920463103344275 0.430807619160845 0.3287200166442119 0.32184244669915074 0.310839664775159 0.6934307219483742 ``` Here’s the result of feeding the output to dieharder. I won’t try to interpret the results, here. (The -a argument tells it to run all the tests it has; -g 200 says that stdin is a series of 32-bit unsigned integers.) The summary is that 82 tests passed, 27 failed and 5 were weak. ```\$ ./rng147 \| dieharder -a -g 200 #=============================================================================# # dieharder version 3.31.1 Copyright 2003 Robert G. Brown # #=============================================================================# rng_name \|rands/second\| Seed \| stdin_input_raw\| 1.01e+07 \|2776686247\| #=============================================================================# test_name \|ntup\| tsamples \|psamples\| p-value \|Assessment #=============================================================================# diehard_birthdays\| 0\| 100\| 100\|0.64044752\| PASSED diehard_operm5\| 0\| 1000000\| 100\|0.00000000\| FAILED diehard_rank_32x32\| 0\| 40000\| 100\|0.45366134\| PASSED diehard_rank_6x8\| 0\| 100000\| 100\|0.97658733\| PASSED diehard_bitstream\| 0\| 2097152\| 100\|0.72508491\| PASSED diehard_opso\| 0\| 2097152\| 100\|0.00000000\| FAILED diehard_oqso\| 0\| 2097152\| 100\|0.00000000\| FAILED diehard_dna\| 0\| 2097152\| 100\|0.00000000\| FAILED diehard_count_1s_str\| 0\| 256000\| 100\|0.00000000\| FAILED diehard_count_1s_byt\| 0\| 256000\| 100\|0.00000000\| FAILED diehard_parking_lot\| 0\| 12000\| 100\|0.00000000\| FAILED diehard_2dsphere\| 2\| 8000\| 100\|0.00000000\| FAILED diehard_3dsphere\| 3\| 4000\| 100\|0.00000000\| FAILED diehard_squeeze\| 0\| 100000\| 100\|0.00000000\| FAILED diehard_sums\| 0\| 100\| 100\|0.06709806\| PASSED diehard_runs\| 0\| 100000\| 100\|0.00000000\| FAILED diehard_runs\| 0\| 100000\| 100\|0.00000000\| FAILED diehard_craps\| 0\| 200000\| 100\|0.13428702\| PASSED diehard_craps\| 0\| 200000\| 100\|0.00096831\| WEAK marsaglia_tsang_gcd\| 0\| 10000000\| 100\|0.00000000\| FAILED marsaglia_tsang_gcd\| 0\| 10000000\| 100\|0.00000000\| FAILED sts_monobit\| 1\| 100000\| 100\|0.03994929\| PASSED sts_runs\| 2\| 100000\| 100\|0.22979976\| PASSED sts_serial\| 1\| 100000\| 100\|0.98471576\| PASSED sts_serial\| 2\| 100000\| 100\|0.31624292\| PASSED sts_serial\| 3\| 100000\| 100\|0.13274265\| PASSED sts_serial\| 3\| 100000\| 100\|0.06274970\| PASSED sts_serial\| 4\| 100000\| 100\|0.10906298\| PASSED sts_serial\| 4\| 100000\| 100\|0.71359846\| PASSED sts_serial\| 5\| 100000\| 100\|0.00790811\| PASSED sts_serial\| 5\| 100000\| 100\|0.07231695\| PASSED sts_serial\| 6\| 100000\| 100\|0.17659272\| PASSED sts_serial\| 6\| 100000\| 100\|0.18513720\| PASSED sts_serial\| 7\| 100000\| 100\|0.04315090\| PASSED sts_serial\| 7\| 100000\| 100\|0.45634930\| PASSED sts_serial\| 8\| 100000\| 100\|0.14826946\| PASSED sts_serial\| 8\| 100000\| 100\|0.42520026\| PASSED sts_serial\| 9\| 100000\| 100\|0.71305937\| PASSED sts_serial\| 9\| 100000\| 100\|0.20822407\| PASSED sts_serial\| 10\| 100000\| 100\|0.97154428\| PASSED sts_serial\| 10\| 100000\| 100\|0.69812982\| PASSED sts_serial\| 11\| 100000\| 100\|0.53454147\| PASSED sts_serial\| 11\| 100000\| 100\|0.08444014\| PASSED sts_serial\| 12\| 100000\| 100\|0.22453592\| PASSED sts_serial\| 12\| 100000\| 100\|0.72236320\| PASSED sts_serial\| 13\| 100000\| 100\|0.78366045\| PASSED sts_serial\| 13\| 100000\| 100\|0.37616667\| PASSED sts_serial\| 14\| 100000\| 100\|0.95764809\| PASSED sts_serial\| 14\| 100000\| 100\|0.14472891\| PASSED sts_serial\| 15\| 100000\| 100\|0.43901688\| PASSED sts_serial\| 15\| 100000\| 100\|0.76714556\| PASSED sts_serial\| 16\| 100000\| 100\|0.58587727\| PASSED sts_serial\| 16\| 100000\| 100\|0.67425978\| PASSED rgb_bitdist\| 1\| 100000\| 100\|0.05846675\| PASSED rgb_bitdist\| 2\| 100000\| 100\|0.00004623\| WEAK rgb_bitdist\| 3\| 100000\| 100\|0.66589418\| PASSED rgb_bitdist\| 4\| 100000\| 100\|0.00026565\| WEAK rgb_bitdist\| 5\| 100000\| 100\|0.07969442\| PASSED rgb_bitdist\| 6\| 100000\| 100\|0.17850326\| PASSED rgb_bitdist\| 7\| 100000\| 100\|0.25911992\| PASSED rgb_bitdist\| 8\| 100000\| 100\|0.00045592\| WEAK rgb_bitdist\| 9\| 100000\| 100\|0.00642740\| PASSED rgb_bitdist\| 10\| 100000\| 100\|0.27514527\| PASSED rgb_bitdist\| 11\| 100000\| 100\|0.91420132\| PASSED rgb_bitdist\| 12\| 100000\| 100\|0.09738292\| PASSED rgb_minimum_distance\| 2\| 10000\| 1000\|0.00000000\| FAILED rgb_minimum_distance\| 3\| 10000\| 1000\|0.00000000\| FAILED rgb_minimum_distance\| 4\| 10000\| 1000\|0.00000000\| FAILED rgb_minimum_distance\| 5\| 10000\| 1000\|0.01232138\| PASSED rgb_permutations\| 2\| 100000\| 100\|0.55323173\| PASSED rgb_permutations\| 3\| 100000\| 100\|0.00000000\| FAILED rgb_permutations\| 4\| 100000\| 100\|0.00000000\| FAILED rgb_permutations\| 5\| 100000\| 100\|0.00000000\| FAILED rgb_lagged_sum\| 0\| 1000000\| 100\|0.76792249\| PASSED rgb_lagged_sum\| 1\| 1000000\| 100\|0.59838659\| PASSED rgb_lagged_sum\| 2\| 1000000\| 100\|0.95607401\| PASSED rgb_lagged_sum\| 3\| 1000000\| 100\|0.28565715\| PASSED rgb_lagged_sum\| 4\| 1000000\| 100\|0.95085961\| PASSED rgb_lagged_sum\| 5\| 1000000\| 100\|0.45582064\| PASSED rgb_lagged_sum\| 6\| 1000000\| 100\|0.60565985\| PASSED rgb_lagged_sum\| 7\| 1000000\| 100\|0.04195406\| PASSED rgb_lagged_sum\| 8\| 1000000\| 100\|0.30831976\| PASSED rgb_lagged_sum\| 9\| 1000000\| 100\|0.36546234\| PASSED rgb_lagged_sum\| 10\| 1000000\| 100\|0.19936994\| PASSED rgb_lagged_sum\| 11\| 1000000\| 100\|0.07303233\| PASSED rgb_lagged_sum\| 12\| 1000000\| 100\|0.34645127\| PASSED rgb_lagged_sum\| 13\| 1000000\| 100\|0.29195838\| PASSED rgb_lagged_sum\| 14\| 1000000\| 100\|0.30094975\| PASSED rgb_lagged_sum\| 15\| 1000000\| 100\|0.99790005\| WEAK rgb_lagged_sum\| 16\| 1000000\| 100\|0.75196523\| PASSED rgb_lagged_sum\| 17\| 1000000\| 100\|0.38809626\| PASSED rgb_lagged_sum\| 18\| 1000000\| 100\|0.61180091\| PASSED rgb_lagged_sum\| 19\| 1000000\| 100\|0.19440479\| PASSED rgb_lagged_sum\| 20\| 1000000\| 100\|0.57917094\| PASSED rgb_lagged_sum\| 21\| 1000000\| 100\|0.83129601\| PASSED rgb_lagged_sum\| 22\| 1000000\| 100\|0.57292407\| PASSED rgb_lagged_sum\| 23\| 1000000\| 100\|0.10381929\| PASSED rgb_lagged_sum\| 24\| 1000000\| 100\|0.93651195\| PASSED rgb_lagged_sum\| 25\| 1000000\| 100\|0.50848729\| PASSED rgb_lagged_sum\| 26\| 1000000\| 100\|0.81756629\| PASSED rgb_lagged_sum\| 27\| 1000000\| 100\|0.29194323\| PASSED rgb_lagged_sum\| 28\| 1000000\| 100\|0.14934239\| PASSED rgb_lagged_sum\| 29\| 1000000\| 100\|0.96722837\| PASSED rgb_lagged_sum\| 30\| 1000000\| 100\|0.41045537\| PASSED rgb_lagged_sum\| 31\| 1000000\| 100\|0.33379469\| PASSED rgb_lagged_sum\| 32\| 1000000\| 100\|0.67528985\| PASSED rgb_kstest_test\| 0\| 10000\| 1000\|0.12422629\| PASSED dab_bytedistrib\| 0\| 51200000\| 1\|1.00000000\| FAILED dab_dct\| 256\| 50000\| 1\|0.00000000\| FAILED Preparing to run test 207. ntuple = 0 dab_filltree\| 32\| 15000000\| 1\|0.00000000\| FAILED dab_filltree\| 32\| 15000000\| 1\|0.00000000\| FAILED Preparing to run test 208. ntuple = 0 dab_filltree2\| 0\| 5000000\| 1\|0.00000000\| FAILED dab_filltree2\| 1\| 5000000\| 1\|0.00000000\| FAILED Preparing to run test 209. ntuple = 0 dab_monobit2\| 12\| 65000000\| 1\|1.00000000\| FAILED \$ ``` 4. […] built several random number generators: [1], [2], [3], [4], [5], [6], [7], [8], [9] (I didn’t realize it was so many until I went back and looked). 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https://myuniversalnk.com/the-chain-rule/	1,600,450,442,000,000,000	text/html	crawl-data/CC-MAIN-2020-40/segments/1600400188049.8/warc/CC-MAIN-20200918155203-20200918185203-00315.warc.gz	563,586,069	16,147	# The Chain Rule The Chain Rule – The topic for today’s article is the chain rule. So you all might have come across this particular equation or this particular term in your lower standards it is when you were in your Mathematics High School or somewhere. So but in deep learning is the most important concept when we have to deal with the backpropagation, So backpropagation is one such algorithm or technique where you try to backtrack to individual layers and you try to come to that particular unit or neuron in order to adjust the weights. So there you basically use this equation or this concept from mathematics. That is the chain rule. So we’ll be having a quick revision of what the chain rule is in this particular video. So say for example Simple you are given an equation like you have some exponential equation E raised to the sine of some function say x square. So now you have this particular equation and it’s been asked that you have to take the total derivative of with respect to X say, this is some function say Y and Y is equal to is given like this and you want to take the derivative of this particular function. So how would you basically do this? So for in order to perform this, you need to have to learn the equation so the derivatives of certain functions like implicit functions explicit function derivative of many things so that you come across when you do certain types of equations or when you have certain kinds of activation functions, especially their this particular equation comes into the picture. So now how would you take the derivative? So whenever if you want to differentiate y with respect to X now, you want to check whether where the X comes in this particular equation. So now this works in the format of Of derivative of an outer function multiplied by derivative of inner function So it is a kind of nested functions. So these are essentially called as composite functions. That is function inside function. So how do you take the derivative so for the derivative for E raised to X now this particular one particular equation that is you can consider it as e raised to t. So when you differentiate you first write it as e raised to sine x square. So for E raise to X or E raise to D. You have the same form. And then you need to take the derivative of that is d by dX of sine X Square. So till you reach the derivative of x squared e to differentiate it this so that is the each individual particular unit. You are trying to differentiate so that is you have e raised to the sine of x square into if you take the derivative of sine of x squared that becomes cosine of x square now, you should not stop that is you have to take the derivative with respect to x square also. So now this is pure in terms of X. So finally your derivative would be for this you have 2X. So now if you just observe you have each and everything as individual units, which are just multiplying. So these are essentially you’re seeing of different equations that you are multiplying. So essentially these are used for Composite functions so Whenever say you have a function like you are giving Y is equal to x square. So now say I’m representing this as Y. So we have taken x square as y now. We are taking Z is equal to sine y. So now this became sine of Y. So we are taking this sine of y as z and say we have W is equal to e raise to z. So now if we are asked to find the derivative, of course, we are considering here the partial derivative of with respect to say we want to take the derivative of w with respect to Z. Now in order to do this this W should have something that is expressed in terms of x square. So in order to differentiate this, there is some term there should be something in blue that is representative of X then only you can apply the chain rule else. You cannot go back and connect to that particular thing. So essentially when you take the derivative so first it will be dou W by dou z, so that is with respect to Z. Then you have dou z by dou Y and then you finally have dou y by dou X. So if you see just this particular equation this consists of this dou Z and dou Y so basically you can just assume like this cancels out, but in mathematically we cannot cancel this but just for Simplicity, you can consider this. So how you represent. Are you can find the chain of different equations that are coming across has given with the help of this chain rule now for the general format say this is represented as F of G of x. So if you want to take the derivative that is we have y Prime is equal to f’ G of X into the derivative of G of X. So that becomes a composite functions. Now this not only goes with this exponential functions. Like you have some functions. So 3x plus 1 raise to 7. So are you using the derivative? Also if you have some logarithmic functions like 5X Or if you have some square root functions like this, but we are familiar with these kinds of equations. Right? So these are nothing but your loss functions that you encounter. So if you have this y & y predicted, you may have some kind of equations where you have y log y cap plus 1 minus y into log 1 minus y cap so you can differentiate this and similarly. This is nothing but your x square plus y Square which is nothing but your L2 norm and so all these are basically convex functions. So this is the beauty of the chain rule comes into for differentiating in Neural Networks. So you can essentially take the derivative of this and you can backtrack and to update each and every particular individual unit. So well that was all regarding the chain rule in deep learning for backpropagation.	1,182	5,630	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	4.59375	5	CC-MAIN-2020-40	latest	en	0.953629	[ 128000, 2, 578, 29625, 18592, 271, 791, 29625, 18592, 1389, 578, 8712, 369, 3432, 753, 4652, 374, 279, 8957, 6037, 13, 2100, 499, 682, 2643, 617, 2586, 4028, 420, 4040, 24524, 477, 420, 4040, 4751, 304, 701, 4827, 10886, 433, 374, 994, 499, 1051, 304, 701, 50895, 5234, 6150, 477, 15038, 13, 2100, 719, 304, 5655, 6975, 374, 279, 1455, 3062, 7434, 994, 584, 617, 311, 3568, 449, 279, 1203, 2741, 28236, 11, 2100, 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https://www.onlinemath4all.com/one-step-equations-multiplication-and-division-worksheet.html	1,596,562,033,000,000,000	text/html	crawl-data/CC-MAIN-2020-34/segments/1596439735881.90/warc/CC-MAIN-20200804161521-20200804191521-00441.warc.gz	775,346,716	13,711	# ONE STEP EQUATIONS MULTIPLICATION AND DIVISION WORKSHEET Problem 1 : When we multiply a number by 4, we get 124. Find the number. Problem 2 : When we divide a number by 7, we get 14. Find the number. Problem 3 : Alex borrowed some money from Jose. After 3 years, Alex returned 2 times of borrowed money to Jose. If the returned money is \$226, how much money did Alex borrow from Jose ? Problem 4 : David has some money. He gave one fourth of the money to Lily. If Lily gets \$8 from David, how much money did David have initially ? Problem 5 : In a deposit, invested money will become 4 times itself in 5 years. If Rosy receives \$3280 after five years, how much money did Rosy invest ? Problem 6 : Jacob has some number of candies and Michael has 35 candies. If Michael has candies 5 times as Jacob, how many candies does Jacob have ? Problem 7 : Daniel had some hot dogs and he gave one third of the hot dogs to Alex. If Alex gets 8 hot dogs from Daniel, how many hot dogs did have initially ? Problem 8 : Between the hours of 10 P.M. and 6 A.M., the temperature decreases an average of of a degree per hour. How long, in hours and minutes, will it take for the temperature to decrease by 5 °F ? Problem 1 : When we multiply a number by 4, we get 124. Find the number. Solution : Let x be the number. Then, 4x = 124 Divide each side by 4. x = 31 So, the number is 31. Problem 2 : When we divide a number by 7, we get 14. Find the number. Solution : Let m be the number. Then, m/7 = 14 Multiply each side by 7. m = 98 So, the number is 98. Problem 3 : Alex borrowed some money from Jose. After 3 years, Alex returned 2 times of borrowed money to Jose. If the returned money is \$226, how much money did Alex borrow from Jose ? Solution : Let x be the borrowed money. Then, 2x = 226 Divide each side by 2. x = 113 So, the borrowed money is \$113. Problem 4 : David has some money. He gave one fourth of the money to Lily. If Lily gets \$8 from David, how much money did David have initially ? Solution : Let m be the money that David had initially. Then, m/4 = 32 Multiply each side by 4. m = 128 Problem 5 : In a deposit, invested money will become 4 times itself in 5 years. If Rosy receives \$3280 after five years, how much money did Rosy invest ? Solution : Let x be the money that Rosy invested. Then, 4x = 3280 Divide each side by 4. x = 820 So, Rosy invested \$820. Problem 6 : Jacob has some number of candies and Michael has 35 candies. If Michael has candies 5 times as Jacob, how many candies does Jacob have ? Solution : Let p be the number of candies that Jacob has. Then, 5p = 35 Divide each side by 5. p = 7 So, Jacob has 5 candies. Problem 7 : Daniel had some hot dogs and he gave one third of the hot dogs to Alex. If Alex gets 8 hot dogs from Daniel, how many hot dogs did have initially ? Solution : Let h be the number of hot dogs that Daniel had initially. Then, m/3 = 8 Multiply each side by 3. m = 24 So, Daniel had 24 hot dogs initially. Problem 8 : Between the hours of 10 P.M. and 6 A.M., the temperature decreases an average of of a degree per hour. How long, in hours and minutes, will it take for the temperature to decrease by 5 °F ? Solution : Let x represent the number of hours it takes for the temperature to decrease by 5 °F. Write an equation (-3/4)x = -5 3x/4 = -5 Solve the equation using an inverse operation. Multiply each side by - 4/3. x = 20/3 x = 6 2/3 It takes 6 2/3 hours. Convert 2/3 hours to minutes. 2/3 hours = (2/3) ⋅ 60 minutes 2/3 hours = 40 minutes So, it takes 6 hours and 40 minutes for the temperature to decrease by 5 °F. 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WORD PROBLEMS Word problems on simple equations Word problems on linear equations Algebra word problems Word problems on trains Area and perimeter word problems Word problems on direct variation and inverse variation Word problems on unit price Word problems on unit rate Word problems on comparing rates Converting customary units word problems Converting metric units word problems Word problems on simple interest Word problems on compound interest Word problems on types of angles Complementary and supplementary angles word problems Double facts word problems Trigonometry word problems Percentage word problems Profit and loss word problems Markup and markdown word problems Decimal word problems Word problems on fractions Word problems on mixed fractrions One step equation word problems Linear inequalities word problems Ratio and proportion word problems Time and work word problems Word problems on sets and venn diagrams Word problems on ages Pythagorean theorem word problems Percent of 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formed with non zero digits Sum of all three four digit numbers formed using 0, 1, 2, 3 Sum of all three four digit numbers formed using 1, 2, 5, 6	1,571	6,107	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	4.53125	5	CC-MAIN-2020-34	latest	en	0.91689	[ 128000, 2, 25002, 49456, 469, 5876, 22545, 83837, 19366, 3651, 47360, 25189, 32904, 50, 83576, 271, 32298, 220, 16, 14852, 4599, 584, 31370, 264, 1396, 555, 220, 19, 11, 584, 636, 220, 8874, 13, 7531, 279, 1396, 382, 32298, 220, 17, 14852, 4599, 584, 22497, 264, 1396, 555, 220, 22, 11, 584, 636, 220, 975, 13, 7531, 279, 1396, 382, 32298, 220, 18, 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https://socratic.org/questions/how-do-you-graph-x-2-2-y-1-2-32-1	1,606,603,478,000,000,000	text/html	crawl-data/CC-MAIN-2020-50/segments/1606141195929.39/warc/CC-MAIN-20201128214643-20201129004643-00536.warc.gz	490,729,186	5,969	# How do you graph (x+2)^2 + (y+1)^2 =32? Feb 5, 2016 Circle with the center $\left(- 2 , - 1\right)$ and the radius $4 \sqrt{2}$. #### Explanation: What you have here is a circle equation. The standard form of a circle equation is ${\left(x - {x}_{m}\right)}^{2} + {\left(y - {y}_{m}\right)}^{2} = {r}^{2}$ which describes a circle with the center point $\left({x}_{m} , {y}_{m}\right)$ and the radius $r$. In your case, ${x}_{m} = - 2$, ${y}_{m} = - 1$ and $r = \sqrt{32} = \sqrt{16 \cdot 2} = 4 \sqrt{2} \approx 5.66$ Thus, you can graph a circle with the center point $\left(- 2 , - 1\right)$ and the radius $\approx 5.66$: graph{(x+2)^2 + (y+1)^2 = 32 [-15.54, 16.49, -8.36, 7.66]}	275	697	{"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 10, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	4.46875	4	CC-MAIN-2020-50	latest	en	0.721715	[ 128000, 2, 2650, 656, 499, 4876, 320, 87, 10, 17, 30876, 17, 489, 320, 88, 10, 16, 30876, 17, 284, 843, 1980, 41691, 220, 20, 11, 220, 679, 21, 271, 26264, 449, 279, 4219, 59060, 2414, 4172, 220, 17, 1174, 482, 220, 16, 59, 1315, 15437, 323, 279, 10801, 400, 19, 1144, 27986, 90, 17, 32816, 382, 827, 72387, 1473, 3923, 499, 617, 1618, 374, 264, 12960, 24524, 382, 791, 5410, 1376, 315, 264, 12960, 24524, 374, 271, 2420, 59, 2414, 2120, 482, 314, 87, 52635, 76, 11281, 1315, 9317, 48922, 17, 92, 489, 29252, 2414, 7166, 482, 314, 88, 52635, 76, 11281, 1315, 9317, 48922, 17, 92, 284, 314, 81, 92, 48922, 17, 32816, 271, 8370, 16964, 264, 12960, 449, 279, 4219, 1486, 59060, 2414, 2358, 87, 52635, 76, 92, 1174, 314, 88, 52635, 76, 11281, 1315, 15437, 323, 279, 10801, 400, 81, 3, 382, 644, 701, 1162, 11, 3654, 87, 52635, 76, 92, 284, 482, 220, 17, 55976, 3654, 88, 52635, 76, 92, 284, 482, 220, 16, 3, 323, 271, 44060, 284, 1144, 27986, 90, 843, 92, 284, 1144, 27986, 90, 845, 1144, 51953, 220, 17, 92, 284, 220, 19, 1144, 27986, 90, 17, 92, 1144, 49153, 220, 20, 13, 2287, 67526, 45600, 11, 499, 649, 4876, 264, 12960, 449, 279, 4219, 1486, 59060, 2414, 4172, 220, 17, 1174, 482, 220, 16, 59, 1315, 15437, 323, 279, 10801, 59060, 49153, 220, 20, 13, 2287, 3, 1473, 4539, 97165, 87, 10, 17, 30876, 17, 489, 320, 88, 10, 16, 30876, 17, 284, 220, 843, 10261, 868, 13, 4370, 11, 220, 845, 13, 2491, 11, 482, 23, 13, 1927, 11, 220, 22, 13, 2287, 14316, 128001 ]	[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ]
https://www.kaysonseducation.co.in/questions/p-span-sty_42	1,657,109,960,000,000,000	text/html	crawl-data/CC-MAIN-2022-27/segments/1656104672585.89/warc/CC-MAIN-20220706121103-20220706151103-00798.warc.gz	876,321,211	12,896	A point source of light S is placed on the major optical axis of concave mirror at a distance of 60 cm. At what distance from the concave mirror should a flat mirror be placed for the rays to converge again at the point S having been reflected from the concave mirror and then from the flat one? Will the position of the point where the rays meet change if they are first reflected from the flat mirror? The radius of the concave mirror is 80 cm. : Kaysons Education # A Point Source Of Light S Is Placed On The Major Optical Axis Of Concave Mirror At A Distance Of 60 Cm. At What Distance From The Concave Mirror Should A Flat Mirror Be Placed For The Rays To Converge Again At The Point S Having Been Reflected From The Concave Mirror And Then From The Flat One? Will The Position Of The Point Where The Rays Meet Change If They Are First Reflected From The Flat Mirror? The Radius Of The Concave Mirror Is 80 Cm. #### Video lectures Access over 500+ hours of video lectures 24*7, covering complete syllabus for JEE preparation. #### Online Support Practice over 30000+ questions starting from basic level to JEE advance level. #### National Mock Tests Give tests to analyze your progress and evaluate where you stand in terms of your JEE preparation. #### Organized Learning Proper planning to complete syllabus is the key to get a decent rank in JEE. #### Test Series/Daily assignments Give tests to analyze your progress and evaluate where you stand in terms of your JEE preparation. ## Question ### Solution Correct option is 90 cm For concave mirror v = –120 cm Now, x = 120 – (x + 60) x = 30 cm So the distance is 60 + 30 = 90 cm. #### SIMILAR QUESTIONS Q1 A point object is placed at a distance of 30 cm from a convex mirror of focal length 30 cm. the image will form at Q2 Figure shows two rays A and B being reflected by a mirror and going as A` and B`. The mirror Q3 Figure shows three transparent media of refractive indices µ1, µ2 and µ3. A point object O is placed in the medium µ2. If the entire medium on the right of the spherical surface has refractive index µ1, the image forms at O`. If this entire medium has refractive index µ3, the image forms at O``. In the situation shown, Q4 An object of length 2.5 cm is placed at a distance of 1.5f from a concave mirror where f is the magnitude of the focal length of the mirror. The length of the object is perpendicular to the principal axis. Find the length of the image. Is the image erect or inverted? Q5 A convex mirror has its radius of curvature 20 cm. Find the position of the image of an object placed at a distance of 12 cm from the mirror. Q6 which mirror should a boy use, if he stands straight in front of a mirror at a distance of 30cm away from it and sees his erect image whose height is 1/6 th of his real height, is Q7 A hair dresser stands with her nose 20 cm in front of a plane mirror for what distance must she focus her eyes in order to see her nose in the mirror? Q8 An object of height 5 cm is placed in midway between a concave mirror of radius of curvature 30 cm and a convex mirror of radius of curvature 30 cm. The mirrors are placed opposite to each other and are 60 cm apart. The position of the image formed by reflection at convex mirror is. Q9 A convex lens of focal length 15 cm is placed in front of a convex mirror. Both are coaxial and the lens is 5 cm from the apex of the mirror. When an object is placed on the axis at a distance of 20 cm from the lens, if is found that image coincides with the object is 40 cm. Consider only two steps. Calculate the radius of curvature of mirror Q10 A small object is placed on the principal axis of a concave spherical mirror of radius 20 cm at a distance of 30 cm. By how much will the position and size of the image alter, when a parallel – sided slab of glass of thickness 6 cm and refractive index 1.5 is introduced between the centre of curvature and the object? 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http://www.foxbusiness.com/markets/2017/05/25/how-much-is-my-bond-worth.html	1,516,510,844,000,000,000	text/html	crawl-data/CC-MAIN-2018-05/segments/1516084890187.52/warc/CC-MAIN-20180121040927-20180121060927-00308.warc.gz	436,489,735	9,402	# How Much Is My Bond Worth? By A bond's current market value depends on its own interest rate, or coupon rate, along with its face value and the current market interest rate. There is a mathematical formula to calculate how much your bond is worth, but simply put, rising interest rates cause bond values to drop while falling interest rates cause bond values to rise. ## The basics of bond pricing When a bond is first issued, it is sold at a certain par value (also known as face value), which is typically \$1,000 but other amounts are possible. It also has a stated interest rate known as the coupon rate, which is the amount of interest the bond pays as a percentage of its par value. For example, a bond with a par value of \$1,000 and a coupon rate of 5% would pay \$50 in interest per year. Interest rate fluctuations can have a big impact on your bond's value. Image source: Getty Images. ## How interest rates affect bond values The market value of a bond depends on two factors -- the current market interest rate, and the bond's par value. In mathematical terms, the price of a bond is the sum of the present values of all of its future interest payments, plus its par value at maturity. If interest rates rise, the present value of future payments is less. I'll spare you the complex formula used to calculate this, but here's an overview of how this works in practice: For example, let's say that you buy a 30-year Treasury bond for \$1,000 with a coupon rate of 4%, so it pays \$40 per year. If the 30-year Treasury rate jumps to 5%, investors will now expect this yield from their Treasury bonds, or \$50 per \$1,000 invested. Since yours is only paying \$40, the market value must fall in order to make your bond's yield more attractive to new investors. At a 5% interest rate, a bond that pays \$40 per year would be worth \$800. However, keep in mind that the par value of the bond is \$1,000, so an investor would get back considerably more than they pay upon maturity. This adds to the bond's yield to maturity and needs to be taken into consideration. In reality, a 30-year Treasury bond with a 4% coupon rate in a 5% rate environment would be worth about \$845, with the extra \$45 to compensate for the future profit expected from the bond's higher par value. ## So, how much is your bond worth? With all of that in mind, here's a quick calculator that can help you determine your bond's value or predict what it would be worth if interest rates were to change. A couple of notes: • "Desired yield to maturity" refers to the theoretical interest rate you'd like to evaluate. See my example below the calculator. • If your bond is callable, it may affect the bond's value, based on the call date and market interest rate. In this case, the bond's yield to call needs to be considered as well. * Calculator is for estimation purposes only, and is not financial planning or advice. As with any tool, it is only as accurate as the assumptions it makes and the data it has, and should not be relied on as a substitute for a financial advisor or a tax professional. For example, let's say that I own a bond with a \$1,000 par value and a 5% coupon rate, with 12 years to maturity. We'll say that the current market interest rate for bonds with this maturity length is 4%. And I want to know what will happen to the value of my bond if the market interest rate spikes to 6%. Well, based on today's market rate, my bond is worth \$1,095, or \$95 more than its par value. However, if the market rate were to rise to 6%, my bond's market value would drop to just \$915. 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The Motley Fool has a disclosure policy.	947	4,209	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	3.796875	4	CC-MAIN-2018-05	latest	en	0.936572	[ 128000, 2, 2650, 24191, 2209, 3092, 24537, 37246, 1980, 1383, 271, 32, 11049, 596, 1510, 3157, 907, 14117, 389, 1202, 1866, 2802, 4478, 11, 477, 24759, 4478, 11, 3235, 449, 1202, 3663, 907, 323, 279, 1510, 3157, 2802, 4478, 13, 2684, 374, 264, 37072, 15150, 311, 11294, 1268, 1790, 701, 11049, 374, 5922, 11, 719, 5042, 2231, 11, 16448, 2802, 7969, 5353, 11049, 2819, 311, 6068, 1418, 16054, 2802, 7969, 5353, 11049, 2819, 311, 10205, 382, 567, 578, 32874, 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https://jeffreypost.dev/scattered-thoughts/probability%20distributions/modeling/seir/epidemiology/stochastic/hiv/aids/2021/05/07/stochastic_HIV_model.html	1,653,561,282,000,000,000	text/html	crawl-data/CC-MAIN-2022-21/segments/1652662604794.68/warc/CC-MAIN-20220526100301-20220526130301-00250.warc.gz	387,949,743	18,414	## Motivation for write-up This is the 4th part of a multi-part series blog post on modeling in epidemiology. The COVID-19 pandemic has brought a lot of attention to study of epidemiology and more specifically to the various mathematical models that are used to inform public health policies. Everyone has been trying to understand the growth or slowing of new cases and trying to predict the necessary sanitary resources. This blog post attempts to explain the foundations for some of the most used models and enlighten the reader on two key points. After introducing the concepts of compartmentalization and disease dynamics in the first blog post, the second part looked at a deterministic numerical solution for the SEIR model discussed, and the effects of the parameters $\beta$, $\sigma$, and $\gamma$ in parts 1 and 2. Part 3 made the argument that most models ignore individual-level disease dynamics in favor of averaging population-level $\sigma$ and $\gamma$ parameters and showed some big discrepancies between actual COVID-19 probability distributions for those parameters and those used in research. This 4th part is where I build a numerical SEIR model that takes into account these probability distributions in order to tweak the model as close to COVID-19 data as possible. ## Building a stochastic model As opposed to the deterministic model from Part 2, this model is going to focus on individual level disease dynamics to model the disease propagation. The basic idea of this model is to have a dataframe with the number of rows equal to the population size (each individual is a row) and two columns: • State column to describe the state of each individual (S, E, I, or R) • Day column to save the day of transition of the individual into that state However, the population-level rates of transmission still apply here i.e. a person goes from S → E following three points: 1. the number of contacts the person has per unit time (given by $r$) 2. the chance a given contact is with an I - infectious individual (the higher thenumber of I, the higher the chance) 3. the chance of an S contracting the disease from a contact with an I (given by $\rho$) This is done stochastically. Once a person becomes E, their progression is unique to them. This progression is calculated in advance for computational reason, but it allows to use the time ditributions we want. #collapse_hide !pip install plotly==4.14.3 import pandas as pd import numpy as np import math import plotly.graph_objects as go import plotly.express as px from scipy.stats import expon from scipy.stats import gamma from scipy.stats import weibull_min from numpy.random import default_rng rng = default_rng() # Let's build a numerical solution def seir_model(init, parms, days): S_0, E_0, I_0, R_0 = init Epd, Ipd, Rpd = [0], [0], [0] S, E, I, R = [S_0], [E_0], [I_0], [R_0] dt=0.1 t = np.linspace(0,days,int(days/dt)) sigma, beta, gam = parms for _ in t[1:]: next_S = S[-1] - betaS[-1]I[-1]dt Epd.append(betaS[-1]I[-1]dt) next_E = E[-1] + (betaS[-1]I[-1] - sigmaE[-1])dt Ipd.append(sigmaE[-1]dt) next_I = I[-1] + (sigmaE[-1] - gamI[-1])dt Rpd.append(gamI[-1]dt) next_R = R[-1] + (gamI[-1])dt S.append(next_S) E.append(next_E) I.append(next_I) R.append(next_R) return np.stack([S, E, I, R, Epd, Ipd, Rpd]).T Collecting plotly==4.14.3 \|████████████████████████████████\| 13.2MB 303kB/s Requirement already satisfied: retrying>=1.3.3 in /usr/local/lib/python3.7/dist-packages (from plotly==4.14.3) (1.3.3) Requirement already satisfied: six in /usr/local/lib/python3.7/dist-packages (from plotly==4.14.3) (1.15.0) Installing collected packages: plotly Found existing installation: plotly 4.4.1 Uninstalling plotly-4.4.1: Successfully uninstalled plotly-4.4.1 Successfully installed plotly-4.14.3 ### Creating the initial population dataframe Below is a function to create the initial population dataframe: • $p$ is the population number • $num_E$ is the number of people exposed on day 0 • $num_I$ is the number of infectious on day 0 • $num_R$ is the number of people recovered on day 0 #collapse_hide # Need this new function for model below: def make_df(p, num_I, num_R): df = pd.DataFrame(np.full((p,1), 'S').T[0], columns=['State']) df['Year'] = 0 df['Age'] = (np.random.random(p)35+15).astype(int) tochange=df.loc[rng.choice(p, size=num_I+num_R, replace=False),'State'].index df.loc[tochange[0:num_I],'State'] = 'I' df.loc[tochange[num_I:num_I+num_R],'State'] = 'R' return df ### Building the model #np.random.random(size=(p,days)) #np.log(4) j=11 over = 10 #10/np.cumsum(np.ones(100)) 0.05 + (0.3/np.exp((j+1-over)/10)) 0.29561922592339457 #collapse_hide def seir_model_stoch(beta, beta2, p, num_I, num_R, years, T_Infectious, ART, control): # Initialize population dataframe with data given by user df = make_df(p, num_I, num_R) # This variable is used to track daily value of beta if it varies over time xxbeta=np.array([],dtype=float) # Initialize the arrays to return # Below are numbers of S, I, R total S=np.array([],dtype=int) I=np.array([],dtype=int) R=np.array([],dtype=int) # Below are the daily additions in S, I, R Spd=np.array([],dtype=int) Ipd=np.array([],dtype=int) Rpd=np.array([],dtype=int) b=beta #b2=beta[0] b2=np.array([],dtype=float) b1=b # signal diminshing beta over = 0 # signal end of deaths due to ART art1 = 0 art2 = 0 # Stochastic model so use random values to decide on progression rand = np.random.random(size=(p,years)) # Depending if you want exponential, gamma, or Weibull distribution for T_Infectious # Uses distributions found on blog part 3 if T_Infectious == 'expon': ItoR = expon.rvs(loc=0,scale=10,size=p) elif T_Infectious == 'gamma': ItoR = gamma.rvs(4,loc=3,scale=2,size=p) else: ItoR = weibull_min.rvs(2.3, loc=2, scale=20.11, size=p) # Iterate over every day the simulation is run for j in range(0,years-1): # Record daily beta values xxbeta=np.append(xxbeta, b[j]) # First we get the index of the individuals that will change state today: # Random number tells you which 'S' have been exposed on this day #StoE_index = df.loc[(df.State == 'S') & (rand[:,j] < b[j]len(np.where(df.State=='I')[0])/p)].index if ART < 2: StoI_index = df.loc[(df.State == 'S') & (df.Age < 49) & (rand[:,j] < b[j]len(np.where(df.State=='I')[0])/(len(np.where(df.State=='I')[0])+len(np.where(df.State=='S')[0])))].index StoS_index = df.loc[(df.State == 'S') & (df.Age < 49) & (rand[:,j] < b[j]len(np.where(df.State=='I')[0])/(len(np.where(df.State=='I')[0])+len(np.where(df.State=='S')[0])))].index elif ART == 2: if art2 == 0: StoI_index = df.loc[(df.State == 'S') & (df.Age < 49) & (rand[:,j] < b[j]len(np.where(df.State=='I')[0])/(len(np.where(df.State=='I')[0])+len(np.where(df.State=='S')[0])))].index StoS_index = df.loc[(df.State == 'S') & (df.Age < 49) & (rand[:,j] < b[j]len(np.where(df.State=='I')[0])/(len(np.where(df.State=='I')[0])+len(np.where(df.State=='S')[0])))].index elif art2 == 1: StoI_index = df.loc[(df.State == 'S') & (df.Age > 55)].index StoS_index = df.loc[(df.State == 'S') & (df.Age < 49)].index StoRem_index = df.loc[(df.State == 'S') & (df.Age == 49)].index # For each row, if a person has been a certain number of days in E, they will go to I # This follows EtoI variable which is either exponential or gamma distributed according to above #EtoI_index = df.loc[(df.State == 'E') & (j-df.Day >= EtoI)].index # Similaraly as above # For each row, if a person has been a certain number of days in I, they will go to R # This follows EtoI variable which is either exponential or gamma distributed according to above ItoRem_index = df.loc[(df.State == 'I') & (df.Age == 49)].index if ART == 0: #don't use ART ItoR_index = df.loc[(df.State == 'I') & (j-df.Year >= ItoR) & (df.Age < 49)].index ItoI_index = df.loc[(df.State == 'I') & (j-df.Year < ItoR) & (df.Age < 49)].index elif ART > 0: if art2 == 0: ItoR_index = df.loc[(df.State == 'I') & (j-df.Year >= ItoR) & (df.Age < 49)].index ItoI_index = df.loc[(df.State == 'I') & (j-df.Year < ItoR) & (df.Age < 49)].index elif art2 ==1: ItoR_index = df.loc[(df.State == 'I') & (df.Age > 49)].index ItoI_index = df.loc[(df.State == 'I') & (df.Age < 49)].index RtoRem_index = df.loc[(df.State == 'R') & (df.Age == 49)].index RtoR_index = df.loc[(df.State == 'R') & (df.Age < 49)].index # Use indexes collected above to populate per day values #Epd = np.append(Epd,len(StoE_index)) #Ipd = np.append(Ipd,len(EtoI_index)) Ipd = np.append(Ipd,len(StoI_index)) Rpd = np.append(Rpd,len(ItoR_index)) # Now we use the indexes collected above randomly to change the actual population dataframe to the new states df.iloc[ItoRem_index] = ['S', j, 15] df.loc[ItoR_index, ['State','Year']] = ['S', j] df.loc[ItoR_index, 'Age'] = df.loc[ItoR_index, 'Age'] + 1 df.loc[ItoI_index, 'Age'] = df.loc[ItoI_index, 'Age'] + 1 df.iloc[StoRem_index] = ['S', j, 15] df.loc[StoI_index, ['State','Year']] = ['I', j] df.loc[StoI_index, 'Age'] = df.loc[StoI_index, 'Age'] + 1 df.loc[StoS_index, 'Age'] = df.loc[StoS_index, 'Age'] + 1 df.iloc[RtoRem_index] = ['S', j, 15] df.loc[RtoR_index, 'Age'] = df.loc[RtoR_index, 'Age'] + 1 # Append the S, I, and R arrays S=np.append(S,len(np.where(df.State=='S')[0])) I=np.append(I,len(np.where(df.State=='I')[0])) R=np.append(R,len(np.where(df.State=='R')[0])) # Code below for control measures to reduce beta values if control == 1: if (I[-1]/p > 0.015): art1 = 1 if over == 0: over = j if art1 == 1: if j > over + 15: #if Ipd[-2] > Ipd[-1]: art2 = 1 if over != 0: #b = beta2+(b1/np.exp((j+3-over)/15)) b = beta2+(b1/np.exp((j+1-over)/10)) if control == 2: if (I[-1]/p > 0.3): art1 = 1 if over == 0: over = j #print(over) if art1 == 1: if j > over + 15: #if Ipd[-2] > Ipd[-1]: art2 = 1 if over != 0: #b = beta2+(b1/np.exp((j+3-over)/15)) b = beta2+(b1/np.exp((j+1-over)/10)) xxbeta2 = ((S[j-1]+I[j-1])/I[j-1])Ipd[j]/S[j-1] #xxbeta2 = 0.5 #print(xxbeta2) b2 = np.append(b2, xxbeta2) #Epd[0]+=num_E Ipd[0]+=num_I Rpd[0]+=num_R #return S,E,I,R, Epd, Ipd, Rpd, xxbeta return S, I, R, Spd, Ipd, Rpd, xxbeta, b2, over ## Testing the model #collapse_hide # Define parameters for stochastic model days = 200 p = 10000 num_E = 0 num_I = 1 num_R = 0 beta_stoch = 0.3np.ones(days) beta_stoch2 = 0.05 # Run 3 stochastic simulations results_stoch1 = seir_model_stoch(beta_stoch,beta_stoch2, p, num_I, num_R, years, 'gamma', 0, 1) results_stoch2 = seir_model_stoch(beta_stoch, beta_stoch2, p, num_I, num_R, years, 'gamma', 0, 1) results_stoch3 = seir_model_stoch(beta_stoch, beta_stoch2, p, num_I, num_R, years, 'gamma', 0, 2) results_stoch4 = seir_model_stoch(beta_stoch, beta_stoch2, p, num_I, num_R, years, 'gamma', 0, 2) results_stoch1[8] 26 #collapse_hide fig = go.Figure(data=[ go.Scatter(name='Beta_stoch1', x=np.arange(len(results_stoch1[0])), y=results_stoch1[6], line={'dash':'dot','color':'yellow'}, legendgroup="stoch1"), go.Scatter(name='Beta_meas1', x=np.arange(len(results_stoch1[0])), y=results_stoch1[7], line={'dash':'dot','color':'yellow'}, legendgroup="stoch1"), go.Scatter(name='I_stoch1', x=np.arange(len(results_stoch1[0])), y=results_stoch1[1]/p, line={'dash':'dot', 'color':'red'}, legendgroup="stoch1"), go.Bar(name='Ip_stoch1', x=np.arange(len(results_stoch1[0])), y=results_stoch1[4]10/p, legendgroup="stoch1"), go.Scatter(name='R_stoch1', x=np.arange(len(results_stoch1[0])), y=results_stoch1[2]/p, line={'dash':'dot', 'color':'green'}, legendgroup="stoch1"), go.Scatter(name='Beta_stoch2', x=np.arange(len(results_stoch2[0])), y=results_stoch2[6], line={'dash':'dot','color':'yellow'}, legendgroup="stoch2"), go.Scatter(name='Beta_meas2', x=np.arange(len(results_stoch2[0])), y=results_stoch2[7], line={'dash':'dot','color':'yellow'}, legendgroup="stoch2"), go.Scatter(name='I_stoch2', x=np.arange(len(results_stoch2[0])), y=results_stoch2[1]/p, line={'dash':'dot', 'color':'red'}, legendgroup="stoch2"), go.Bar(name='Ip_stoch2', x=np.arange(len(results_stoch2[0])), y=results_stoch2[4]10/p, legendgroup="stoch2"), go.Scatter(name='R_stoch2', x=np.arange(len(results_stoch2[0])), y=results_stoch2[2]/p, line={'dash':'dot', 'color':'green'}, legendgroup="stoch2"), go.Scatter(name='Beta_stoch3', x=np.arange(len(results_stoch3[0])), y=results_stoch3[6], line={'dash':'dot', 'color':'yellow'}, legendgroup="stoch3"), go.Scatter(name='Beta_meas3', x=np.arange(len(results_stoch3[0])), y=results_stoch3[7], line={'dash':'dot','color':'yellow'}, legendgroup="stoch3"), go.Scatter(name='I_stoch3', x=np.arange(len(results_stoch3[0])), y=results_stoch3[1]/p, line={'dash':'dot', 'color':'red'}, legendgroup="stoch3"), go.Bar(name='Ip_stoch3', x=np.arange(len(results_stoch3[0])), y=results_stoch3[4]10/p, legendgroup="stoch3"), go.Scatter(name='R_stoch3', x=np.arange(len(results_stoch3[0])), y=results_stoch3[2]/p, line={'dash':'dot', 'color':'green'}, legendgroup="stoch3"), go.Scatter(name='Beta_stoch4', x=np.arange(len(results_stoch4[0])), y=results_stoch4[6], line={'dash':'dot', 'color':'yellow'}, legendgroup="stoch4"), go.Scatter(name='Beta_meas4', x=np.arange(len(results_stoch4[0])), y=results_stoch4[7], line={'dash':'dot','color':'yellow'}, legendgroup="stoch4"), go.Scatter(name='I_stoch4', x=np.arange(len(results_stoch4[0])), y=results_stoch4[1]/p, line={'dash':'dot', 'color':'red'}, legendgroup="stoch4"), go.Bar(name='Ip_stoch4', x=np.arange(len(results_stoch4[0])), y=results_stoch4[4]*10/p, legendgroup="stoch4"), go.Scatter(name='R_stoch4', x=np.arange(len(results_stoch4[0])), y=results_stoch4[2]/p, line={'dash':'dot', 'color':'green'}, legendgroup="stoch4") ]) fig.update_layout( xaxis_title = 'Day', yaxis_title = 'Proportion of population', title={ 'text':r'$\text{Effect of stochasticity on Deterministic SEIR model}$', 'x':0.5, 'xanchor':'center' } ) fig.show()	4,472	13,783	{"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 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https://math-master.org/general/a-rectangular-swimming-pool-has-a-length-of-14-feet-a-width-of-26-feet-and-a-depth-of-5-feet-round-answers-to-the-nearest-hundredth-as-needed-a-how-many-cubic-feet-of-water-can-the-pool-hold	1,723,113,786,000,000,000	text/html	crawl-data/CC-MAIN-2024-33/segments/1722640726723.42/warc/CC-MAIN-20240808093647-20240808123647-00302.warc.gz	300,102,749	50,985	Question # A rectangular swimming pool has a length of 14 feet, a width of 26 feet and a depth of 5 feet. Round answers to the nearest hundredth as needed. $a$ How many cubic feet of water can the pool hold? cubic feet $b$ The manufacturer suggests filling the pool to 95% capacity. How many cubic feet of water is this? cubic feet 220 likes 1102 views ## Answer to a math question A rectangular swimming pool has a length of 14 feet, a width of 26 feet and a depth of 5 feet. Round answers to the nearest hundredth as needed. $a$ How many cubic feet of water can the pool hold? cubic feet $b$ The manufacturer suggests filling the pool to 95% capacity. How many cubic feet of water is this? cubic feet Eliseo 4.6 $a$ To find the volume of the rectangular swimming pool, we need to multiply its length, width, and depth. The volume of a rectangular prism can be found using the formula: \text{{Volume}} = \text{{length}} \times \text{{width}} \times \text{{depth}} Given: Length $L$ = 14 feet Width $W$ = 26 feet Depth $D$ = 5 feet Substituting the values into the formula: \text{{Volume}} = 14 \times 26 \times 5 Calculating this: \text{{Volume}} = 1820 \text{{ cubic feet}} Therefore, the pool can hold 1820 cubic feet of water. $b$ To find the amount of water to fill the pool to 95% capacity, we need to multiply the volume of the pool by 95%. The amount of water is given by: \text{{Amount of water}} = \text{{Volume of pool}} \times \left$\frac{{95}}{{100}} \right$ Substituting the value of the volume of the pool: \text{{Amount of water}} = 1820 \times \left$\frac{{95}}{{100}} \right$ Calculating this: \text{{Amount of water}} = 1729 \text{{ cubic feet}} Therefore, filling the pool to 95% capacity would require 1729 cubic feet of water. Frequently asked questions $FAQs$ Question: What is the value of f$x$ = 3x² - 4x + 2 when x = 5?	516	1,867	{"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	4.46875	4	CC-MAIN-2024-33	latest	en	0.867533	[ 128000, 14924, 271, 2, 362, 52524, 24269, 7463, 706, 264, 3160, 315, 220, 975, 7693, 11, 264, 2430, 315, 220, 1627, 7693, 323, 264, 8149, 315, 220, 20, 7693, 13, 17535, 11503, 311, 279, 24379, 7895, 339, 439, 4460, 13, 400, 64, 3, 2650, 1690, 41999, 7693, 315, 3090, 649, 279, 7463, 3412, 30, 41999, 7693, 400, 65, 3, 578, 14290, 13533, 21973, 279, 7463, 311, 220, 2721, 4, 8824, 13, 2650, 1690, 41999, 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http://www.docstoc.com/docs/122874662/Section-14-1-Practice-Problems	1,369,121,794,000,000,000	text/html	crawl-data/CC-MAIN-2013-20/segments/1368699776315/warc/CC-MAIN-20130516102256-00042-ip-10-60-113-184.ec2.internal.warc.gz	429,904,180	15,776	# Section 14 1 Practice Problems Shared by: Categories Tags - Stats views: 65 posted: 6/16/2012 language: pages: 3 Document Sample ``` Section 14.1 Practice Problems 1. A teacher asked her 8 introductory statistics students to record the total amount of time they spent studying for a particular test. The amounts of study time x (in hours) and the resulting test grades y are given below. x 2 1 1.5 0.5 1 3 0 2 y 92 81 84 68 85 96 48 74 a. Make a scatterplot of the data. b. Use your TI-83 to obtain the equation of the least-squares regression line and the correlation. c. Explain in words what the slope  of the true regression line says about d. What is the estimate of  from the data? What is your estimate of the intercept  of the true regression line? e. Use your calculator to calculate the residuals. Report the sum of the residuals and the sum of the squares of the residuals. Then use these results to estimate the standard deviation  in the regression model.  SE b = s f. The standard error of the slope SEb is defined as  (x – x) 2 Calculate SEb. g. Suppose we want to find out if the number of hours studied helps predict grade awarded on this statistics test. Formulate null and alternative hypotheses about the slope of the true regression line. State a two-sided alternative. h. Determine the test statistic, the degrees of freedom, and the P-value of t against the alternative. 2. Ideal proportions Once upon a time, a class like yours made measurements of their arm span and height. They entered their results into a Minitab worksheet, requested least squares regression of height on arm span (both in inches) and obtained the following output: Predictor Coef Stdev t-ratio p Constant 11.547 5.600 2.06 0.056 Arm span 0.84042 0.08091 10.39 0.000 s = 1.613 R-sq = 87.1% R-sq(adj) = 86.3% A residual plot for the data looks like this: o o o o o o o o 0.0 o o o o o o RESI1 o o o o -3.0 64.0 68.0 72.0 76.0 armspan a. Determine the equation of the least squares regression line from the printout. b. In your opinion, is the least squares line an appropriate model for the data? Would you be willing to predict a student’s height, knowing that his arm span is 76 inches? Explain. Then do it – use this model to predict the height of a student whose arm span is 76 inches. c. Estimate the parameters , and . d. Construct a 95% confidence interval for the true slope of the regression line. e. Would you reject the null hypothesis at the 1% significance level? Explain briefly. f. Write your conclusions in plain language. g. 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https://study.com/academy/lesson/what-is-an-absolute-value.html	1,542,485,705,000,000,000	text/html	crawl-data/CC-MAIN-2018-47/segments/1542039743732.41/warc/CC-MAIN-20181117185331-20181117211331-00281.warc.gz	735,962,289	49,916	# What is an Absolute Value? An error occurred trying to load this video. Try refreshing the page, or contact customer support. Coming up next: How to Evaluate Absolute Value Expressions ### You're on a roll. Keep up the good work! Replay Your next lesson will play in 10 seconds • 0:10 Distance from Zero • 2:16 Absolute Value Notation • 2:55 Solving Equations Want to watch this again later? Timeline Autoplay Autoplay #### Recommended Lessons and Courses for You Lesson Transcript Instructor: Jeff Calareso Jeff teaches high school English, math and other subjects. He has a master's degree in writing and literature. When we're talking and comparing numbers, we often don't care whether its positive or negative, just how big it is. This is often called the magnitude of a number and we find it by taking the absolute value. Learn all about it here! ## The Distance from Zero One February, during Super Bowl XLV, which was the Packers vs. the Steelers, I made a bet with a friend over how much money it costs to get a 30-second commercial on during one of the commercial breaks. Now I'm a huge Packers fan and he was a Steelers fan, so we decided that the loser would have to wear the opposing team's jersey for the rest of the game. I thought it would cost around \$1,250,000 and my friend was convinced that it cost \$5,500,000. I wasn't sure if I was exactly right or not, but I was pretty confident that he was way off, so I decided to make the bet. Well, a few internet searches later, we discovered that a 30-second commercial that year cost \$3,000,000. So who won? Well, a simple calculation of \$3,000,000 minus \$1,250,000 gave us the answer that I was \$1,750,000 off the actual answer. Doing the same for him, \$3,000,000 minus \$5,500,000 told us that he was \$2,500,000 off. This means that I was definitely closer, so I won and my Packer pride stayed intact. But a closer inspection into this reveals a situation that actually comes up a lot in mathematics. What my friend and I cared about was purely how far away our guess was from the actual answer. We didn't care about whether we were over or under, or whether the number we did after the little subtraction problem was positive or negative. All that we cared about was the magnitude of the answer, or how large the number was regardless of the sign in front of it. In mathematics, this operation of taking the number and only looking at the size of it and ignoring the sign, is called the absolute value. We often use absolute values when talking about distance because there is no such thing as negative distance. For example, just because the store you're going to is a block behind you doesn't mean it's negative one blocks away. It's just one block, right? ## Notation of Absolute Values Just like how the winner of our bet was the one whose guess was less far away from the answer, the absolute value of any number can be thought of as how far away that number is from zero. For example, the absolute value 7 (written as \|7\|)? Still 7. It's 7 units away from zero. The absolute value of -7 (written as \|-7\|)? 7, because it's still 7 units away from zero. To unlock this lesson you must be a Study.com Member. ### Register for a free trial Are you a student or a teacher? #### See for yourself why 30 million people use Study.com ##### Become a Study.com member and start learning now. Back What teachers are saying about Study.com ### Earning College Credit Did you know… We have over 160 college courses that prepare you to earn credit by exam that is accepted by over 1,500 colleges and universities. 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http://mathhelpforum.com/advanced-math-topics/225208-inverse-3x3-matrix.html	1,527,152,591,000,000,000	text/html	crawl-data/CC-MAIN-2018-22/segments/1526794866107.79/warc/CC-MAIN-20180524073324-20180524093324-00464.warc.gz	198,138,479	10,880	# Thread: Inverse of 3X3 Matrix 1. ## Inverse of 3X3 Matrix Any hints? Got everything down with the "cofactor method" for finding the inverse of a 3X3 Matrix, except this part: 2. ## Re: Inverse of 3X3 Matrix First of all, you have calculated the matrix of minors wrong, it should be: $\displaystyle \begin{bmatrix}2&2&2\\-2&3&3\\0&-10&0 \end{bmatrix}$. Now to create the adjugate (or adjoint) matrix, we multiply each ij-entry by $\displaystyle (-1)^{i+j}$, that is, alternate signs, starting with postive in the upper-left corner, to get: $\displaystyle \begin{bmatrix}2&-2&2\\2&3&-3\\0&10&0 \end{bmatrix}$. We will want the transpose of this matrix, which is: $\displaystyle \begin{bmatrix}2&2&0\\-2&3&10\\2&-3&0 \end{bmatrix}$. We need to now calculate the determinant of the original matrix, to get a suitable scalar factor: The easiest way is to expand by minors along the second column, since it has only one non-zero entry. Since this is in the 3,2 position, we take the negative of this determinant (since 3+2 = 5 is odd), the cofactor of the other 2 subdeterminants will be 0, so we don't have to calculate the matrix of the other minors (although we've already done it, so meh...). that is, the determinant of the original matrix is: $\displaystyle \det(A) = (-1)\ast 1 \ast \begin{vmatrix}3&2\\2&-2 \end{vmatrix} = (-1)(-10) = 10$. So our scaling factor will be 1/10. This gives the inverse: $\displaystyle A^{-1} = \frac{1}{\det(A)}\text{adj}(A) = \begin{bmatrix} \frac{1}{5}&\frac{1}{5}&0\\ -\frac{1}{5}&\frac{3}{10}&1\\ \frac{1}{5}&-\frac{3}{10}&0 \end{bmatrix}$ 3. ## Re: Inverse of 3X3 Matrix Ok, corrected now. 4. ## Re: Inverse of 3X3 Matrix Ok here are the two charts. I have one last question on the 2nd one. As far as understanding the question in the first chart, I understand it. Wherever the two lines meet (in a cross-out pattern), that place is where the determinant goes for the new "matrix of determinant answers", so to speak. 5. ## Re: Inverse of 3X3 Matrix Go entry-by-entry. Draw one horizontal and one vertical line that intersect at the entry you are on. 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https://goldenssport.com/how-do-you-find-a-unit-vector-that-is-orthogonal-to-both-u-1-0-v-0-1/	1,722,826,541,000,000,000	text/html	crawl-data/CC-MAIN-2024-33/segments/1722640427760.16/warc/CC-MAIN-20240805021234-20240805051234-00296.warc.gz	221,743,599	13,259	Home > blog > How Do You Find A Unit Vector That Is Orthogonal To Both U = 1, 0, V = 0, 1, ? # How Do You Find A Unit Vector That Is Orthogonal To Both U = 1, 0, V = 0, 1, ? We can formalize this result into a theorem regarding orthogonal vectors. Find the net force Express the answer using standard unit vectors. Find a vector of magnitude that points in the opposite direction than vector where and Express the answer in component form. Assume the quarterback and the receiver are in the same place as in the previous example. Therefore, two vectors are parallel if they have the same or opposite directions. We will see the first application of this in the next chapter. Let’s start off by supposing that we wanted the rate of change of $$f$$ at a particular point, say $$\left( , \right)$$. Let’s also suppose that both $$x$$ and $$y$$ are increasing and that, in this case, $$x$$ is increasing twice as fast as $$y$$ is increasing. So, as $$y$$ increases one unit of measure $$x$$ will increase two units of measure. To this point we’ve only looked at the two partial derivatives $$\left( \right)$$ and $$\left( \right)$$. Its magnitude is obtained by multiplying their magnitudes by the sine of the angle between them. The direction of the vector product can be determined by the corkscrew right-hand rule. The vector product of two either parallel or antiparallel vectors vanishes. The magnitude of the vector product is largest for orthogonal vectors. The distributive property is applied frequently when vectors are expressed in their component forms, in terms of unit vectors of Cartesian axes. The corkscrew right-hand rule is a common mnemonic used to determine the direction of the vector product. Determine the requested vectors and express each of them a. Vector uu is a unit vector that forms an angle of 60°60° with the positive x-axis. Determine all three-dimensional vectors u orthogonal to vector Express the answer by using standard unit vectors. The boat’s motor generates a force in one direction, and the current of the river generates a force in another direction. We must take both the magnitude and direction of each force into account if we want to know where the boat will go. Each arrow has the same length and direction. A closely related concept is the idea of parallel vectors. When finding the cross product, in practice, we can use either or , depending on which one of them seems to be less complex computationally. One way to make sure if the final result is correct is to use them both. Is a well-known indicator that they are perpendicular. Where $$\theta$$ is the angle between the gradient and $$\vec u$$. You appear to be on a device with a “narrow” screen width (i.e. you are probably on a mobile phone). Due to the nature of the mathematics on this site it is best views in landscape mode. For the following exercises, find the center and radius of the sphere with an equation in general form that is given. Write the equation of the plane passing through point that is parallel to the xz-plane. Write the equation of the plane passing through point that is parallel to the xy-plane. what is the maximum frame size for ethernet ii frames on a vlan? To add vectors in three dimensions, we follow the same procedures we learned for two dimensions. Think about what happens if you plot this equation in two dimensions in the xz-plane. Describe the set of points in three dimensional space that satisfies and graph the surface. /r/cheatatmathhomework is FREE math homework help sub. Asking for or offering payment will result in a permanent ban. The triangle looks to be equilateral and equiangular. 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https://thvinhtuy.edu.vn/simple-double-integration-of-square-wave-question-8dicu0se/	1,721,582,698,000,000,000	text/html	crawl-data/CC-MAIN-2024-30/segments/1720763517747.98/warc/CC-MAIN-20240721152016-20240721182016-00581.warc.gz	499,799,718	11,378	Simple double integration of square wave question Finding Fourier coefficients for square wave Finding Fourier coefficients for square wave • #1 864 17 Hi, Simple question, sort of: I see that according to the internet the mathematical description of a triangular wave is rather complex, so I’ll try to stay as far away from that as I can, because I’m a bit rusty. I understand that if you integrate a square wave you get a triangular wave on the x-axis. But If you integrate that triangular wave you get something resembling a sineusoid. (something about it being the first harmonic of the triangular wave I believe) My questions are: How close is that to a sine wave? What does it look like graphically? (I don’t have matlab) Secondly, are there different slopes of triangular waves, including asymmetrical triangular waves, which are closer to a sine wave? I am so rusty on Fourier transforms / series, it’s not funny. But regarding this: http://mathworld.wolfram.com/FourierSeriesTriangleWave.html I see the asymmetrical figure, which is kind of what I have in mind, but I’m not sure what the red sine wave is indicating. Cheers Simple question, sort of: I see that according to the internet the mathematical description of a triangular wave is rather complex, so I’ll try to stay as far away from that as I can, because I’m a bit rusty. I understand that if you integrate a square wave you get a triangular wave on the x-axis. But If you integrate that triangular wave you get something resembling a sineusoid. (something about it being the first harmonic of the triangular wave I believe) My questions are: How close is that to a sine wave? What does it look like graphically? (I don’t have matlab) Secondly, are there different slopes of triangular waves, including asymmetrical triangular waves, which are closer to a sine wave? I am so rusty on Fourier transforms / series, it’s not funny. But regarding this: http://mathworld.wolfram.com/FourierSeriesTriangleWave.html I see the asymmetrical figure, which is kind of what I have in mind, but I’m not sure what the red sine wave is indicating. Cheers You are watching: Simple double integration of square wave question. Info created by THVinhTuy selection and synthesis along with other related topics. 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https://brainly.in/question/59714	1,484,929,328,000,000,000	text/html	crawl-data/CC-MAIN-2017-04/segments/1484560280835.60/warc/CC-MAIN-20170116095120-00032-ip-10-171-10-70.ec2.internal.warc.gz	808,236,735	9,838	# A number has two digits whose sum is 8. If 18 is added to the number, its digits are reversed. What is that number.Help me please. 2 2014-12-03T20:36:29+05:30 Let the no be 10x + y given, x + y = 8 ------------ (1) 10x +y + 18 = 10y + x 9x - 9y + 18 =0 x - y +2 = 0 x - y = -2 ----- (2) from (1) & (2) x + y = 8 x - y = -2 ----------------- 2x = 6 x = 3 sub x = 3 in (1) 3 + y = 8 y = 5 number 10x + y = 10(3) + 5 = 30+5 = 35 required number is 35. 2014-12-03T20:37:03+05:30 The number is 62. 6+2=8 +18= 26 hence the number is reversed	249	540	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	4.03125	4	CC-MAIN-2017-04	latest	en	0.720026	[ 128000, 2, 362, 1396, 706, 1403, 19016, 6832, 2694, 374, 220, 23, 13, 1442, 220, 972, 374, 3779, 311, 279, 1396, 11, 1202, 19016, 527, 28537, 13, 3639, 374, 430, 1396, 71982, 757, 4587, 382, 17, 271, 679, 19, 12, 717, 12, 2839, 51, 508, 25, 1927, 25, 1682, 10, 2304, 25, 966, 198, 10267, 279, 912, 387, 220, 605, 87, 489, 379, 198, 43068, 11, 865, 489, 379, 284, 220, 23, 33658, 320, 16, 340, 605, 87, 489, 88, 489, 220, 972, 284, 220, 605, 88, 489, 865, 198, 24, 87, 482, 220, 24, 88, 489, 220, 972, 284, 15, 198, 87, 482, 379, 489, 17, 284, 220, 15, 198, 87, 482, 379, 284, 482, 17, 35803, 320, 17, 340, 1527, 320, 16, 8, 612, 320, 17, 340, 87, 489, 379, 284, 220, 23, 198, 87, 482, 379, 284, 482, 17, 198, 776, 7058, 17, 87, 284, 220, 21, 198, 87, 284, 220, 18, 198, 2008, 865, 284, 220, 18, 304, 320, 16, 340, 18, 489, 379, 284, 220, 23, 198, 88, 284, 220, 20, 198, 4174, 220, 605, 87, 489, 379, 284, 220, 605, 7, 18, 8, 489, 220, 20, 198, 28, 220, 966, 10, 20, 198, 28, 220, 1758, 198, 6413, 1396, 374, 220, 1758, 382, 679, 19, 12, 717, 12, 2839, 51, 508, 25, 1806, 25, 2839, 10, 2304, 25, 966, 198, 791, 1396, 374, 220, 5538, 627, 21, 10, 17, 28, 23, 489, 972, 28, 220, 1627, 198, 71, 768, 279, 1396, 374, 28537, 128001 ]	[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ]
https://math.stackexchange.com/questions/545474/a-weird-infinity-problem?noredirect=1	1,581,956,745,000,000,000	text/html	crawl-data/CC-MAIN-2020-10/segments/1581875142603.80/warc/CC-MAIN-20200217145609-20200217175609-00207.warc.gz	469,192,616	27,137	A weird infinity problem. [duplicate] A weird infinity problem. I saw this on youtube but could not understand it: Let us add 1 + 2 + 4 + 8 + 16 + ... up to infinity x=(1+2+4+8+...) = 1(1+2+4+8+...) = (2-1)(1+2+4+8+...) = (2+4+8+16+...)-(1+2+4+8+...) = -1 x=-1 Now that is weird since we are adding up to infinity towards the positive side yet we end up with a negative number. Somebody please explain me why. • As long as only sums are involved, everything is fine. But to substract $+\infty$ from $+\infty$ (which you do as soon as a minus sign appears) can lead to chaos. This has been explained several times on the site. – Did Oct 30 '13 at 12:09 • can u provide me a link? – Shaurya Gupta Oct 30 '13 at 12:18 • – Gerry Myerson Oct 30 '13 at 12:34	256	755	{"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	3.5	4	CC-MAIN-2020-10	latest	en	0.88874	[ 128000, 32, 16682, 56010, 3575, 13, 510, 64926, 2595, 32, 16682, 56010, 3575, 13, 358, 5602, 420, 389, 28277, 719, 1436, 539, 3619, 433, 512, 10267, 603, 923, 220, 16, 489, 220, 17, 489, 220, 19, 489, 220, 23, 489, 220, 845, 489, 2564, 709, 311, 56010, 198, 87, 4640, 16, 10, 17, 10, 19, 10, 23, 10, 32318, 28, 220, 16, 7, 16, 10, 17, 10, 19, 10, 23, 10, 33674, 284, 320, 17, 12, 16, 2432, 16, 10, 17, 10, 19, 10, 23, 10, 33674, 284, 320, 17, 10, 19, 10, 23, 10, 845, 10, 1131, 52456, 16, 10, 17, 10, 19, 10, 23, 10, 33674, 284, 482, 16, 198, 87, 11065, 16, 271, 7184, 430, 374, 16682, 2533, 584, 527, 7999, 709, 311, 56010, 7119, 279, 6928, 3185, 3686, 584, 842, 709, 449, 264, 8389, 1396, 13, 94192, 4587, 10552, 757, 3249, 382, 6806, 1666, 1317, 439, 1193, 37498, 527, 6532, 11, 4395, 374, 7060, 13, 2030, 311, 1207, 4437, 400, 42815, 258, 38058, 3, 505, 400, 42815, 258, 38058, 3, 320, 8370, 499, 656, 439, 5246, 439, 264, 28382, 1879, 8111, 8, 649, 3063, 311, 28013, 13, 1115, 706, 1027, 11497, 3892, 3115, 389, 279, 2816, 13, 1389, 4194, 7131, 5020, 220, 966, 364, 1032, 520, 220, 717, 25, 2545, 198, 6806, 649, 577, 3493, 757, 264, 2723, 30, 1389, 4194, 63416, 3431, 64, 74798, 5020, 220, 966, 364, 1032, 520, 220, 717, 25, 972, 198, 6806, 1389, 121890, 5515, 3092, 1293, 5020, 220, 966, 364, 1032, 520, 220, 717, 25, 1958, 128001 ]	[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ]
http://www.ask.com/question/what-is-2-3-cups-in-oz	1,409,623,346,000,000,000	text/html	crawl-data/CC-MAIN-2014-35/segments/1409535921318.10/warc/CC-MAIN-20140909054337-00452-ip-10-180-136-8.ec2.internal.warc.gz	659,571,781	16,480	# What is 2/3 cup in ounces? There are 5 1/3 ounces in 2/3 of a cup. One cup is the equivalent of 8 ounces, and 2/3 of 8 is 5 1/3. Because 1 tablespoon equals 1/2 ounce, 5 1/3 tablespoons equal 1/3 of a cup. This also means that 10 2/3 tablespoons equal 2/3 of a cup. Since there are 3 teaspoons in 1 tablespoon, 2/3 of a cup contains 32 teaspoons and because 2 cups equal 1 pint, 2/3 of a cup is the equivalent of 1/3 of a pint. The measurement of 2/3 cup is also equal to 1/6 of a quart and 1/24 of a gallon. Reference:	184	524	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	3.609375	4	CC-MAIN-2014-35	latest	en	0.930447	[ 128000, 2, 3639, 374, 220, 17, 14, 18, 10747, 304, 49138, 1980, 3947, 527, 220, 20, 220, 16, 14, 18, 49138, 304, 220, 17, 14, 18, 315, 264, 10747, 13, 3861, 10747, 374, 279, 13890, 315, 220, 23, 49138, 11, 323, 220, 17, 14, 18, 315, 220, 23, 374, 220, 20, 220, 16, 14, 18, 382, 18433, 220, 16, 62611, 17239, 220, 16, 14, 17, 54808, 11, 220, 20, 220, 16, 14, 18, 56588, 6273, 220, 16, 14, 18, 315, 264, 10747, 13, 1115, 1101, 3445, 430, 220, 605, 220, 17, 14, 18, 56588, 6273, 220, 17, 14, 18, 315, 264, 10747, 13, 8876, 1070, 527, 220, 18, 93200, 304, 220, 16, 62611, 11, 220, 17, 14, 18, 315, 264, 10747, 5727, 220, 843, 93200, 323, 1606, 220, 17, 26446, 6273, 220, 16, 46746, 11, 220, 17, 14, 18, 315, 264, 10747, 374, 279, 13890, 315, 220, 16, 14, 18, 315, 264, 46746, 13, 578, 19179, 315, 220, 17, 14, 18, 10747, 374, 1101, 6273, 311, 220, 16, 14, 21, 315, 264, 41376, 323, 220, 16, 14, 1187, 315, 264, 50680, 382, 9032, 25, 128001 ]	[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ]
https://www.objectivistliving.com/topic/14133-is-using-someones-reason-against-them-fraud/page/2/	1,669,729,757,000,000,000	text/html	crawl-data/CC-MAIN-2022-49/segments/1669446710698.62/warc/CC-MAIN-20221129132340-20221129162340-00283.warc.gz	991,235,999	37,132	# Is Using Someone's Reason Against Them Fraud? ## Recommended Posts Why isn't it reasonable to suppose that they could have predicted it? My 9-year-old son could have predicted it, and he's only just now learning long division. If a third grader's basic math skills could work that out, I don't see why I couldn't expect the same of Bob and Alice. I think you don't give poor Bob and Alice nearly enough credit for being thinking folks. I explained why in my response to selene. But I'll explain why in more detail here: At the beginning of the auction, Alice bids \$1, because she thinks that there is a 50% chance that Bob will drop out and she will get \$19. There is also a 50% chance that Bob will bid \$2 and win the auction, in which case she loses \$1. Her expected gains for bidding on the first round is 0.519 - 0.51 = \$9, while her expected gains for not bidding at all is \$0. Thus, she should bid \$1 on the first round. Similarly for Bob. Since Alice has already bid \$1, he needs to bid \$2 to win. If he does, his expected gains are 0.518 - 0.52 = \$8, which is more than if he doesn't, i.e. \$0. This goes on and on until the bid is at \$10. At this point, Alice reasons, well if Bob bets more than \$10, then his expected gains would be negative, so he won't bid more than \$10, and Alice can certainly win the auction by betting \$11 and getting \$9. Bob, of course, is thinking the same thing and so he tries to bet more than \$10. Once the bid reaches \$20, the same reasoning applies as in the first paragraph, except in this case each player is trying to minimize his losses. But by that point, Carl has already won. That doesn't explain anything except that you think Bob and Alice are lacking in basic math skills. If you can do the math (and if a third-grader can do the math), why would I not expect Bob and Alice to be able to do the math? • Replies 208 • Created #### Popular Days Any endeavor which depends entirely on 'knowing' the content of another person's consciousness is irrational, so the premise of your game is irrational as are the actions of the 'players'. ##### Share on other sites Thing is, you are buying into the notion that people should be treated as beings that require protection both from things that they cannot predict and from things that they can predict but don't. Objectivism treats people as 1) beings that are entirely capable of protecting themselves, and 2) beings that can survive (learn, recover, thrive) even when they have failed to protect themselves. For the sake of argument, go ahead and tell us what specifically it is that you think Carl has done wrong and what you would suggest to do about it. Again, you're misunderstanding the situation. The auction was not something that simply happened to Alice and Bob, like a natural disaster. It was all Carl's doing. Unlike a natural disaster or some kind of accident, there is someone who can be held responsible for their actions. I can't tell you exactly what Carl has done wrong since, as I said, it has not yet been identified as a kind of wrong. I think Carl should be tried and punished if found guilty, just like with any other crime. That doesn't explain anything except that you think Bob and Alice are lacking in basic math skills. If you can do the math (and if a third-grader can do the math), why would I not expect Bob and Alice to be able to do the math? As I've shown, Bob and Alice already did do the math. They did it perfectly. And yet, they were still screwed. ##### Share on other sites Any endeavor which depends entirely on 'knowing' the content of another person's consciousness is irrational, so the premise of your game is irrational as are the actions of the 'players'. This is absolutely ridiculous. If it is irrational to try to predict what someone else will do, then any attempt to deal with a reality with more than one person in it would be irrational. ##### Share on other sites Alice and Bob can be held responsible for their actions. We can agree to disagree. ##### Share on other sites Alice and Bob can be held responsible for their actions. We can agree to disagree. If you believe that someone can act against their own interests by being rational, then you need to check your premises. Reason cannot fail, unless someone tries to use force or fraud or this third thing, apparently. ##### Share on other sites Any endeavor which depends entirely on 'knowing' the content of another person's consciousness is irrational, so the premise of your game is irrational as are the actions of the 'players'. This is absolutely ridiculous. If it is irrational to try to predict what someone else will do, then any attempt to deal with a reality with more than one person in it would be irrational. Nope, your OP described the participants as acting "in their rational self-interest all the way". Do you understand rational self-interest? It is not rational when the outcome turns completely on another's unpredictable acts. It's irrational and unself-interested to guess and gamble on anything significant, it's instead mystical whimsy. Notwithstanding those mathematical calculations, which prove - what? So the game can be dismissed out of hand as ridiculous. ##### Share on other sites Nope, your OP described the participants as acting "in their rational self-interest all the way". Do you understand rational self-interest? It is not rational when the outcome turns completely on another's unpredictable acts. It's irrational to guess and gamble on anything significant, it's instead mystical whimsy. Notwithstanding those mathematical calculations, which prove - what? So the game can be dismissed out of hand as ridiculous. In what sense is it "not rational"? When you do something rationally, that means that you've done the best you can given your goals and information. Sometimes, you have no information about something, but reality still demands that you make a decision. ##### Share on other sites Here's my issue with that line of reasoning. If it is not fraud, then Alice and Bob were being entirely rational. Non sequitur. You seem to be laboring under the mistaken opinion that any decision absent fraud must, by definition, be "rational." Not so. "Rational" doesn't mean "in the absence of fraud." J ##### Share on other sites Nope, your OP described the participants as acting "in their rational self-interest all the way". Do you understand rational self-interest? It is not rational when the outcome turns completely on another's unpredictable acts. It's irrational to guess and gamble on anything significant, it's instead mystical whimsy. Notwithstanding those mathematical calculations, which prove - what? So the game can be dismissed out of hand as ridiculous. In what sense is it "not rational"? When you do something rationally, that means that you've done the best you can given your goals and information. Sometimes, you have no information about something, but reality still demands that you make a decision. SoMad: Traders trade value for value where both sides feel they're getting the greater value. It is irrational to suppose someone would not pursue their self interest and trade a greater value for a lesser one (i.e. \$20 for \$1). The persons who thought they could accomplish this deserved the lesson. Everyone pursues, rightfully, their self interest. ##### Share on other sites As some of you have said, I don't think that Carl did anything fraudulent, in the technical sense. However, I disagree that Alice and Bob's choice to participate was rational because they were entertained. Are you claiming that any and all forms of entertainment that people choose must be rational?!!! First of all, they were not entertained. I'd say they were the exact opposite of entertained. We may have been entertained, Carl may have been very very entertained, but Alice and Bob certainly weren't. Alice and Bob appeared to be entertaining themselves with the act of competing with each other. If not, then your hypothetical seems to be something constructed out of imaginary psychology and behavior that has no relevance to reality. Secondly, they were not expecting to purchase entertainment in any case. They were expecting to buy money. Yeah, but then time continued on, and they discovered an opportunity for entertainment, and took it. Their "not expecting it" earlier has no bearing. If they knew that they were going to be "entertained" they may have considered the price too high, which would mean that Carl's plan would fail. Think about it this way. Who would go to a store where every item was some unknown price that you couldn't know until you agreed to buy the item? No one, obviously. Yeah, so why invent fictional people who would do what real people wouldn't? As for gambling, gambling is irrational. If you are a rational person, then you would know you can't beat the odds in a casino. Wrong. If you're a rational person, you know that you can beat the odds, but that doing so is unlikely. J ##### Share on other sites Here's my issue with that line of reasoning. If it is not fraud, then Alice and Bob were being entirely rational. Non sequitur. You seem to be laboring under the mistaken opinion that any decision absent fraud must, by definition, be "rational." Not so. "Rational" doesn't mean "in the absence of fraud." J Sorry, in that post I meant to say that Bob and Alice were capable of acting in their rational self-interest. If this was fraud, then you're right they could still be rational even if they could not act in their rational self-interest. ##### Share on other sites SoMad: Traders trade value for value where both sides feel they're getting the greater value. It is irrational to suppose someone would not pursue their self interest and trade a greater value for a lesser one (i.e. \$20 for \$1). The persons who thought they could accomplish this deserved the lesson. Everyone pursues, rightfully, their self interest. False. Immoral people certainly don't. ##### Share on other sites Not false. Explain yourself. ##### Share on other sites As some of you have said, I don't think that Carl did anything fraudulent, in the technical sense. However, I disagree that Alice and Bob's choice to participate was rational because they were entertained. Are you claiming that any and all forms of entertainment that people choose must be rational?!!! The context of that quote is that some people were arguing that Alice and Bob were acting rationally because they may have been entertained by the outcome. But I was pointing out that just because one is entertained by the outcome, that does not mean that their behavior was rational. Alice and Bob appeared to be entertaining themselves with the act of competing with each other. If not, then your hypothetical seems to be something constructed out of imaginary psychology and behavior that has no relevance to reality. Non-sequitur. It's entirely possible that someone might do something even if they were not entertained by the act. Yeah, but then time continued on, and they discovered an opportunity for entertainment, and took it. Their "not expecting it" earlier has no bearing. It does matter because if you don't get the product you expected to buy and have paid for, then you have been defrauded. Yeah, so why invent fictional people who would do what real people wouldn't? That case only considers what would happen if they knew the outcome. The whole point is that they didn't. Wrong. If you're a rational person, you know that you can beat the odds, but that doing so is unlikely. You know what I mean. ##### Share on other sites Not false. Explain yourself. Morality is acting in your own rational-self interest. Immoral people, by definition, don't do this. ##### Share on other sites I think it is time to send Doll Head to the Doll Hospital and do some serious interventions: Do some testing... write some meds.... and keep her under observation... ##### Share on other sites And what did Bob and Alice do with Ted and Carol? ...inquiring kinksters need to know... ##### Share on other sites Not false. Explain yourself. Morality is acting in your own rational-self interest. Immoral people, by definition, don't do this. I did not say rational self interest. Value is subjective (Mises). People pursue what they perceive is their self interest in trying to trade what to them is a lessor value for a greater one. In this case it would be hard to argue that a lesser amount of money is subjectively of more value than a greater one (unless Carl was nuts or a member of some strange cult). The trader principle is trading value for value, it is understood by both sides that a trade will take place when both parties perceive gain. That is a rational exchange. Therefore Alice and Bob were irrational to suppose they could gain while Carl lost. By your definition they were acting immorally and deserved what they got. ##### Share on other sites I did not say rational self interest. Value is subjective (Mises). People pursue what they perceive is their self interest in trying to trade what to them is a lessor value for a greater one. In this case it would be hard to argue that a lesser amount of money is subjectively of more value than a greater one (unless Carl was nuts or a member of some strange cult). The trader principle is trading value for value, it is understood by both sides that a trade will take place when both parties perceive gain. That is a rational exchange. Therefore Alice and Bob were irrational to suppose they could gain while Carl lost. By your definition they were acting immorally and deserved what they got. Is this objectively true? Because if value is subjective, then Alice and Bob cannot say that Carl is acting irrationally just because his values don't conform to their own. ##### Share on other sites I think that the kind of thing that Carl pulled is called a "gambit": A device, action, or opening remark, typically one entailing a degree of risk, that is calculated to gain an advantage. ##### Share on other sites I did not say rational self interest. Value is subjective (Mises). People pursue what they perceive is their self interest in trying to trade what to them is a lessor value for a greater one. In this case it would be hard to argue that a lesser amount of money is subjectively of more value than a greater one (unless Carl was nuts or a member of some strange cult). The trader principle is trading value for value, it is understood by both sides that a trade will take place when both parties perceive gain. That is a rational exchange. Therefore Alice and Bob were irrational to suppose they could gain while Carl lost. By your definition they were acting immorally and deserved what they got. Is this objectively true? Because if value is subjective, then Alice and Bob cannot say that Carl is acting irrationally just because his values don't conform to their own. SoMad: I believe so. If you haven't read Mises he is such a pleasure to read. The exemplar reasoning human being. When I first read Mises many years ago I filled a notebook with words and their definitions. I reread many times before not needing (mostly) my notebook. I recommend him for years of enjoyment and enlightenment if you'd like to put off your deathwish for awhile. I started this habit reading Ayn Rand when I read the word "epistemology" for the first time and I was so delighted when I looked it up and found out what it meant. I found a home. Carl is running a con but it is so obvious and the price is cheap the lesson does Alice and Bob a service. ##### Share on other sites I started this habit reading Ayn Rand when I read the word "epistemology" for the first time and I was so delighted when I looked it up and found out what it meant. I found a home. Very nicely stated. James Joyce was wrong on that because Ayn showed me that same home and I can live in it each and every day that I conscously choose too. Thanks A... ##### Share on other sites Carl is running a con but it is so obvious and the price is cheap the lesson does Alice and Bob a service. Carl is running a con, but the question is, is he doing anything wrong? Do there exist non-coercive and non-fraudulent unethical actions? ##### Share on other sites Carl is running a con but it is so obvious and the price is cheap the lesson does Alice and Bob a service. Carl is running a con, but the question is, is he doing anything wrong? Do there exist non-coercive and non-fraudulent unethical actions? Carl is perpetrating a fraud but an obvious one with not serious consequences. Are you familiar with the term hormesis? A low level toxin can have a positive effect on an organism. Carl's con game does not rise to the level of being prosecuted for a crime. 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https://www.gamedev.net/forums/topic/474905-how-many-vertices-can-result-from-clipping-a-a-triangle-to-a-view-frustum/	1,542,436,050,000,000,000	text/html	crawl-data/CC-MAIN-2018-47/segments/1542039743294.62/warc/CC-MAIN-20181117061450-20181117083450-00451.warc.gz	870,437,133	27,728	# How many vertices can result from clipping a a triangle to a view frustum. This topic is 3997 days old which is more than the 365 day threshold we allow for new replies. Please post a new topic. ## Recommended Posts If I have a triangle and a frustum how many vertices can result from clipping that triangle to the frustum. I can only come up with 7 points. Is this really the maximum? I think that a triangle can only pass through more than 4 planes of a frustum if it passes exactly through the corners in which case it only creates one additional vertex anyways. ##### Share on other sites A triangle (or plane) can pass through all 6 planes of the frustum...but I only see 6 vertices being created there ##### Share on other sites In the worst case, 64 , assuming the frustum has 6 planes and the triangle and its resulting part intersect the other 6-1 planes in turn. This is a rare case, but it could happen, More general cases depend on triangle positions , generally the formula is P= ( original triangle points - clipped points ) intersected frustum planes ##### Share on other sites The triangle could potentially intersect all of the six frustum planes. You would have to test one plane after the other. For each plane two new vertices could be generated if an intersection occurs, so in the end it would be possible that you would have to generate 6*2=12 new vertices in total of which not all have to actually be used by the resulting, clipped polygon. ##### Share on other sites Seven is the maximum for clipping to the X and Y planes, but you can get one more each from clipping to the near and far planes as well. Think of it this way - each plane can turn a single vertex of your (already partially clipped) polygon into an edge - so it takes away one vertex and adds two. So you start with a triangle = 3 vertices, and each of the 6 planes +/-X, +/-Y, +/-Z can add a single vertex each. Total = 9. 1. 1 Rutin 38 2. 2 3. 3 4. 4 5. 5 • 11 • 9 • 12 • 14 • 9 • ### Forum Statistics • Total Topics 633350 • Total Posts 3011470 • ### Who's Online (See full list) There are no registered users currently online ×	528	2,141	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	4.0625	4	CC-MAIN-2018-47	latest	en	0.923758	[ 128000, 2, 2650, 1690, 17672, 649, 1121, 505, 62956, 264, 264, 22217, 311, 264, 1684, 1448, 76408, 382, 2028, 8712, 374, 220, 18572, 22, 2919, 2362, 902, 374, 810, 1109, 279, 220, 12676, 1938, 12447, 584, 2187, 369, 502, 31737, 13, 5321, 1772, 264, 502, 8712, 382, 567, 51762, 15781, 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https://www.greenemath.com/Prealgebra/4/RoundingNumbersTest3.html	1,585,954,704,000,000,000	text/html	crawl-data/CC-MAIN-2020-16/segments/1585370518767.60/warc/CC-MAIN-20200403220847-20200404010847-00372.warc.gz	971,139,814	9,835	Sections: # Rounding Numbers Test #3 About: Additional Resources: In this section, we learn about rounding numbers. Specifically, we will focus on how to round with whole numbers. Often, an exact value is not needed. Let’s suppose we are looking at cars. We might find that a model X car costs $29,755.78, whereas a model G car costs$40,226.38. To make matters simple and easy to remember, we may just say the model X car costs about $30,000 and the model G car costs about$40,000. These values of $30,000 and$40,000 represent approximations. An approximation is an estimate or value that is close, but not exactly the same as the original. Rounding a whole number is a way to give an approximation. To round a whole number, first we must identify how precise we want our approximation to be. This stems from giving a round off place. Once this is known, we begin our procedure: 1. Identify the digit in the round off place 2. Move one digit right: • If this digit is 4 or less, we keep the digit in the round off place as is and change every digit that follows into a zero • If this digit is 5 or greater, we add one to the digit in the round off place, and change every digit that follows into a zero. Example 1: Round 326 to the nearest ten 326 - The 2 is the digit in the round off place 326 - The 6 is one digit to the right of the 2, this digit is in the category of 5 or greater. For this scenario we add one to the digit in the round off place (2). So 2 + 1 = 3. The digit in the tens place will be changed into a 3 and the digit that follows is changed into a 0. 326 Rounded to the Nearest Ten is 330 + Show More +	410	1,628	{"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	4.65625	5	CC-MAIN-2020-16	latest	en	0.909684	[ 128000, 39222, 1473, 2, 432, 13900, 35813, 3475, 674, 18, 271, 10714, 512, 30119, 16607, 1473, 644, 420, 3857, 11, 584, 4048, 922, 52662, 5219, 13, 45863, 11, 584, 690, 5357, 389, 1268, 311, 4883, 449, 4459, 5219, 13, 36016, 11, 459, 4839, 907, 374, 539, 4460, 13, 6914, 753, 23289, 584, 527, 3411, 520, 9515, 13, 1226, 2643, 1505, 430, 264, 1646, 1630, 1841, 7194, 400, 1682, 11, 23532, 13, 2495, 11, 20444, 264, 1646, 480, 1841, 7194, 3, 1272, 11, 14057, 13, 1987, 13, 2057, 1304, 13146, 4382, 323, 4228, 311, 6227, 11, 584, 1253, 1120, 2019, 279, 1646, 1630, 1841, 7194, 922, 400, 966, 11, 931, 323, 279, 1646, 480, 1841, 7194, 922, 3, 1272, 11, 931, 13, 4314, 2819, 315, 400, 966, 11, 931, 323, 3, 1272, 11, 931, 4097, 10049, 97476, 13, 1556, 57304, 374, 459, 16430, 477, 907, 430, 374, 3345, 11, 719, 539, 7041, 279, 1890, 439, 279, 4113, 13, 432, 13900, 264, 4459, 1396, 374, 264, 1648, 311, 3041, 459, 57304, 13, 2057, 4883, 264, 4459, 1396, 11, 1176, 584, 2011, 10765, 1268, 24473, 584, 1390, 1057, 57304, 311, 387, 13, 1115, 44814, 505, 7231, 264, 4883, 1022, 2035, 13, 9843, 420, 374, 3967, 11, 584, 3240, 1057, 10537, 512, 16, 13, 65647, 279, 16099, 304, 279, 4883, 1022, 2035, 198, 17, 13, 14903, 832, 16099, 1314, 512, 6806, 1442, 420, 16099, 374, 220, 19, 477, 2753, 11, 584, 2567, 279, 16099, 304, 279, 4883, 1022, 2035, 439, 374, 323, 2349, 1475, 16099, 430, 11263, 1139, 264, 7315, 198, 6806, 1442, 420, 16099, 374, 220, 20, 477, 7191, 11, 584, 923, 832, 311, 279, 16099, 304, 279, 4883, 1022, 2035, 11, 323, 2349, 1475, 16099, 430, 11263, 1139, 264, 7315, 382, 13617, 220, 16, 25, 17535, 220, 17470, 311, 279, 24379, 5899, 198, 17470, 482, 578, 220, 17, 374, 279, 16099, 304, 279, 4883, 1022, 2035, 198, 17470, 482, 578, 220, 21, 374, 832, 16099, 311, 279, 1314, 315, 279, 220, 17, 11, 420, 16099, 374, 304, 279, 5699, 315, 220, 20, 477, 7191, 13, 1789, 420, 15398, 584, 923, 832, 311, 279, 16099, 304, 279, 4883, 1022, 2035, 320, 17, 570, 2100, 220, 17, 489, 220, 16, 284, 220, 18, 13, 578, 16099, 304, 279, 22781, 2035, 690, 387, 5614, 1139, 264, 220, 18, 323, 279, 16099, 430, 11263, 374, 5614, 1139, 264, 220, 15, 627, 17470, 58550, 311, 279, 4275, 15795, 18165, 374, 220, 10568, 271, 10, 7073, 4497, 489, 128001 ]	[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ]
https://www.calculatoratoz.com/en/power-on-exponential-of-temperature-time-relation-calculator/Calc-30978	1,642,684,664,000,000,000	text/html	crawl-data/CC-MAIN-2022-05/segments/1642320301863.7/warc/CC-MAIN-20220120130236-20220120160236-00063.warc.gz	752,944,915	40,474	Credits Shri Govindram Seksaria Institute of Technology and Science (SGSITS), Indore Ravi Khiyani has created this Calculator and 100+ more calculators! National Institute Of Technology (NIT), Hamirpur Anshika Arya has verified this Calculator and 1600+ more calculators! Power on exponential of temperature-time relation Solution STEP 0: Pre-Calculation Summary Formula Used constantt = -(Convection heat transfer coefficientSurface AreaTime elapsed)/(DensityVolumeSpecific Heat Capacity) a = -(hSAt)/(ρVc) This formula uses 6 Variables Variables Used Convection heat transfer coefficient - Convection heat transfer coefficient is the rate of heat transfer between a solid surface and a fluid per unit surface area per unit kellvin. (Measured in Watt per Meter² per K) Surface Area - The Surface Area of a three-dimensional shape is the sum of all of the surface areas of each of the sides. (Measured in Square Meter) Time elapsed - Time elapsed after a particular task is started. (Measured in Second) Density - Density is the degree of compactness of a substance. (Measured in Kilogram per Meter³) Volume - Volume is the amount of space that a substance or object occupies or that is enclosed within a container. (Measured in Cubic Meter) Specific Heat Capacity - Specific Heat Capacity is the heat required to raise the temperature of the unit mass of a given substance by a given amount. (Measured in Kilojoule per Kilogram per K) STEP 1: Convert Input(s) to Base Unit Convection heat transfer coefficient: 1 Watt per Meter² per K --> 1 Watt per Meter² per K No Conversion Required Surface Area: 50 Square Meter --> 50 Square Meter No Conversion Required Time elapsed: 1 Second --> 1 Second No Conversion Required Density: 5.51 Kilogram per Meter³ --> 5.51 Kilogram per Meter³ No Conversion Required Volume: 63 Cubic Meter --> 63 Cubic Meter No Conversion Required Specific Heat Capacity: 4.184 Kilojoule per Kilogram per K --> 4184 Joule per Kilogram per K (Check conversion here) STEP 2: Evaluate Formula Substituting Input Values in Formula a = -(hSAt)/(ρVc) --> -(1501)/(5.51634184) Evaluating ... ... a = -3.44259695413343E-05 STEP 3: Convert Result to Output's Unit -3.44259695413343E-05 --> No Conversion Required -3.44259695413343E-05 <-- Constant (Calculation completed in 00.016 seconds) < 10+ Transient Heat Conduction Calculators Instantaneous heat transfer rate heat_rate = Convection heat transfer coefficientSurface Area(Initial Temperature-Fluid temperature)(exp(-(Convection heat transfer coefficientSurface AreaTime elapsed)/(DensityVolumeSpecific Heat Capacity))) Go Temperature after given time elapsed temperature = ((Initial Temperature-Fluid temperature)(exp(-(Convection heat transfer coefficientSurface AreaTime elapsed)/(DensityVolumeSpecific Heat Capacity))))+Fluid temperature Go Total heat transfer during a time interval heat_transfer_KJ = DensitySpecific heatVolume(Initial Temperature-Fluid temperature)(1-(exp(-(Biot numberFourier Number)))) Go Ratio of temperature difference for given time elapsed temperature_ratio = exp(-(Convection heat transfer coefficientSurface AreaTime elapsed)/(DensityVolumeSpecific Heat Capacity)) Go Power on exponential of temperature-time relation constantt = -(Convection heat transfer coefficientSurface AreaTime elapsed)/(DensityVolumeSpecific Heat Capacity) Go Product of Biot and Fourier Number in terms of system properties constantt = (Convection heat transfer coefficientSurface AreaTime elapsed)/(DensityVolumeSpecific Heat Capacity) Go Time Constant in unsteady state heat transfer tau_mi = (DensitySpecific Heat CapacityVolume)/(Convection heat transfer coefficientSurface Area) Go Thermal Capacitance thermal_capacitance = DensitySpecific Heat CapacityVolume Go Ratio of temperature difference for given time elapsed in terms of Biot and Fourier Number temperature_ratio = exp(-(Biot numberFourier Number)) Go Power on exponential of temperature-time relation in terms of Biot and Fourier Number constantt = -(Biot numberFourier Number) Go Power on exponential of temperature-time relation Formula constantt = -(Convection heat transfer coefficientSurface AreaTime elapsed)/(DensityVolumeSpecific Heat Capacity) a = -(hSAt)/(ρVc) What is Temperature-Time relation? The temperature-time relationship of unsteady-state heat transfer helps to determine the rate of heat transfer that has been conducted in the lumped system in a given time period. How to Calculate Power on exponential of temperature-time relation? Power on exponential of temperature-time relation calculator uses constantt = -(Convection heat transfer coefficientSurface AreaTime elapsed)/(DensityVolumeSpecific Heat Capacity) to calculate the Constant, Power on exponential of temperature-time relation formula calculates the value of power on exponential in the equation which makes the further calculation of lumped system easy. Constant is denoted by a symbol. How to calculate Power on exponential of temperature-time relation using this online calculator? To use this online calculator for Power on exponential of temperature-time relation, enter Convection heat transfer coefficient (h), Surface Area (SA), Time elapsed (t), Density (ρ), Volume (V) & Specific Heat Capacity (c) and hit the calculate button. Here is how the Power on exponential of temperature-time relation calculation can be explained with given input values -> -3.443E-5 = -(1501)/(5.51634184). FAQ What is Power on exponential of temperature-time relation? Power on exponential of temperature-time relation formula calculates the value of power on exponential in the equation which makes the further calculation of lumped system easy and is represented as a = -(hSAt)/(ρVc) or constantt = -(Convection heat transfer coefficientSurface AreaTime elapsed)/(DensityVolumeSpecific Heat Capacity). Convection heat transfer coefficient is the rate of heat transfer between a solid surface and a fluid per unit surface area per unit kellvin, The Surface Area of a three-dimensional shape is the sum of all of the surface areas of each of the sides, Time elapsed after a particular task is started, Density is the degree of compactness of a substance, Volume is the amount of space that a substance or object occupies or that is enclosed within a container & Specific Heat Capacity is the heat required to raise the temperature of the unit mass of a given substance by a given amount. How to calculate Power on exponential of temperature-time relation? Power on exponential of temperature-time relation formula calculates the value of power on exponential in the equation which makes the further calculation of lumped system easy is calculated using constantt = -(Convection heat transfer coefficientSurface AreaTime elapsed)/(DensityVolumeSpecific Heat Capacity). To calculate Power on exponential of temperature-time relation, you need Convection heat transfer coefficient (h), Surface Area (SA), Time elapsed (t), Density (ρ), Volume (V) & Specific Heat Capacity (c). With our tool, you need to enter the respective value for Convection heat transfer coefficient, Surface Area, Time elapsed, Density, Volume & Specific Heat Capacity and hit the calculate button. You can also select the units (if any) for Input(s) and the Output as well. How many ways are there to calculate Constant? In this formula, Constant uses Convection heat transfer coefficient, Surface Area, Time elapsed, Density, Volume & Specific Heat Capacity. We can use 10 other way(s) to calculate the same, which is/are as follows - • heat_rate = Convection heat transfer coefficientSurface Area(Initial Temperature-Fluid temperature)(exp(-(Convection heat transfer coefficientSurface AreaTime elapsed)/(DensityVolumeSpecific Heat Capacity))) • heat_transfer_KJ = DensitySpecific heatVolume(Initial Temperature-Fluid temperature)(1-(exp(-(Biot numberFourier Number)))) • tau_mi = (DensitySpecific Heat CapacityVolume)/(Convection heat transfer coefficientSurface Area) • constantt = -(Convection heat transfer coefficientSurface AreaTime elapsed)/(DensityVolumeSpecific Heat Capacity) • constantt = -(Biot numberFourier Number) • temperature_ratio = exp(-(Convection heat transfer coefficientSurface AreaTime elapsed)/(DensityVolumeSpecific Heat Capacity)) • temperature_ratio = exp(-(Biot numberFourier Number)) • temperature = ((Initial Temperature-Fluid temperature)(exp(-(Convection heat transfer coefficientSurface AreaTime elapsed)/(DensityVolumeSpecific Heat Capacity))))+Fluid temperature • constantt = (Convection heat transfer coefficientSurface AreaTime elapsed)/(DensityVolumeSpecific Heat Capacity) • thermal_capacitance = DensitySpecific Heat CapacityVolume Let Others Know	1,906	8,804	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	3.671875	4	CC-MAIN-2022-05	longest	en	0.772789	[ 128000, 70439, 271, 2059, 462, 25428, 485, 2453, 77494, 10649, 10181, 315, 12053, 323, 10170, 320, 7939, 50, 14813, 705, 2314, 461, 198, 49, 6321, 108856, 88, 5676, 706, 3549, 420, 37128, 323, 220, 1041, 10, 810, 5935, 3046, 4999, 31912, 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https://socratic.org/questions/how-do-you-rationalize-the-denominator-and-simplify-sqrt2-sqrt5	1,623,851,407,000,000,000	text/html	crawl-data/CC-MAIN-2021-25/segments/1623487623942.48/warc/CC-MAIN-20210616124819-20210616154819-00118.warc.gz	478,427,664	5,770	# How do you rationalize the denominator and simplify sqrt2/sqrt5? Apr 4, 2016 $\frac{\sqrt{10}}{5}$ #### Explanation: Multiply by 1 and you do not change the inherent value. 1 can come in many forms. Foe example: $1 = \frac{\sqrt{5}}{\sqrt{5}}$ Multiply by 1 but in the form of $1 = \frac{\sqrt{5}}{\sqrt{5}}$ giving $\frac{\sqrt{2}}{\sqrt{5}} \times \frac{\sqrt{5}}{\sqrt{5}}$ $= \frac{\sqrt{2} \sqrt{5}}{5}$ $= \frac{\sqrt{2 \times 5}}{5}$ $\frac{\sqrt{10}}{5}$	179	473	{"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 7, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	4.3125	4	CC-MAIN-2021-25	latest	en	0.689015	[ 128000, 2, 2650, 656, 499, 25442, 553, 279, 48012, 323, 40821, 18430, 17, 2754, 8303, 20, 1980, 21383, 220, 19, 11, 220, 679, 21, 271, 59836, 38118, 36802, 27986, 90, 605, 3500, 90, 20, 32816, 271, 827, 72387, 1473, 96255, 555, 220, 16, 323, 499, 656, 539, 2349, 279, 38088, 907, 13, 220, 16, 649, 2586, 304, 1690, 7739, 13, 435, 4748, 3187, 25, 400, 16, 284, 1144, 38118, 36802, 27986, 90, 20, 3500, 36802, 27986, 90, 20, 3500, 67526, 96255, 555, 220, 16, 719, 304, 279, 1376, 315, 400, 16, 284, 1144, 38118, 36802, 27986, 90, 20, 3500, 36802, 27986, 90, 20, 3500, 3, 7231, 271, 59836, 38118, 36802, 27986, 90, 17, 3500, 36802, 27986, 90, 20, 3500, 1144, 15487, 1144, 38118, 36802, 27986, 90, 20, 3500, 36802, 27986, 90, 20, 3500, 67526, 3, 28, 1144, 38118, 36802, 27986, 90, 17, 92, 1144, 27986, 90, 20, 3500, 90, 20, 32816, 271, 3, 28, 1144, 38118, 36802, 27986, 90, 17, 1144, 15487, 220, 20, 3500, 90, 20, 32816, 271, 59836, 38118, 36802, 27986, 90, 605, 3500, 90, 20, 32816, 128001 ]	[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ]
http://www.derkeiler.com/Newsgroups/sci.crypt/2009-10/msg00225.html	1,493,473,028,000,000,000	text/html	crawl-data/CC-MAIN-2017-17/segments/1492917123491.79/warc/CC-MAIN-20170423031203-00419-ip-10-145-167-34.ec2.internal.warc.gz	495,273,319	4,194	# Re: how to encrypt the 10-digit values into encrypted 10-digit values? On 26 Sep., 10:08, makissy <maki...@xxxxxxxxx> wrote: Hi All, I have an assignment about increasing the security of the system by encrypting the identification numbers in database. how to encrypt the 10-digit values into encrypted 10-digit values? i don't have any idea yet. Can anyone give me any idea. pls.. requirements • Each plain text 10-digit value with digits in the interval [0..9] should be encrypted to a 10-digit number with digits in the interval [0..9]. • Each encrypted value of 10 digits should be possible to decrypt back to a plain text 10-digit value. Thanks for any guidance. Hello, I would do it like this (this method is easy on a piece of paper): 1) split the number into 2 rows of 5 digits 2) shift the digits in the first row 1 place the the left, and 2 places to the left in the second row - appending the leftmost digits to the end 3) mod10 add each column's digits 4) mod10 subtract each column's calculated value from that column's digits 5) repeat steps 2-4 (at least once) It is very fast and very easy, and has the benefit over just simply to the number - that it adds diffusion, a very important property in encryption (changing the data in the smallest amount results in changes in the entire cipher). Let me show an example: Number to be encrypted: 4685197643 Secret key (that you have created somehow): 5216940876 1) split number into 2 rows 46851 97643 2) shift places to the left 68514 64397 3) mod10 add each column's digits 68514 64397 -------- 22801 (modulo 10 addition of each column's digits) 4) mod10 subtract each column's calculated value from that column's digits 46713 42596 Let me explain this step. First column: 6 6 -- 2 and becomes: 4 4 because: 6-2=4, 6-2=4 Let's take the 3rd column: 5 3 -- 8 becomes: 7 5 because: 5-8 = -3. We cant have a negative number, so we add 10 - and it then becomes 7. And the last digit: 3-8 = -5. -5 + 10 = 5 So as you can see it is pretty easy. 5) repeat steps 2-4 (at least once) 46713 42596 ....shifted: 67134 59642 6) mod10 add each column's digits 67134 59642 -------- 16776 7) mod10 subtract each column's calculated value from that column's digits 51468 43976 5146843976 (straightened out) 5216940876 (key) ----------------- 0352783742 = cipher (modulo 10 addition of the number and the key) And the encryption is done. To decrypt, just do everything in reverse: 0352783742 (cipher) 5216940876 (key) ----------------- 5146843976 (modulo 10 subtraction) 51468 43976 -------- .... 94334 mod10 subtracted 67134 59642 .... unshift 46713 42596 .... repeat these steps once more 46713 42596 ----- .... 88209 mod10 subtracted 68514 64397 .... unshift 46851 97643 .... straighten 4685197643 And it's decrypted! If you can't decrypt it, you made an error somewhere (errors happen quite easily, so you should always either: doublecheck the calculations or try to decrypt it). .	847	2,983	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	3.78125	4	CC-MAIN-2017-17	latest	en	0.813492	[ 128000, 2, 1050, 25, 1268, 311, 30725, 279, 220, 605, 49442, 2819, 1139, 25461, 220, 605, 49442, 2819, 1980, 1966, 220, 1627, 17907, 2637, 220, 605, 25, 2318, 11, 52016, 61315, 366, 76, 14966, 1131, 31, 45202, 87, 29, 6267, 512, 13347, 2052, 345, 40, 617, 459, 16720, 922, 7859, 279, 4868, 315, 279, 198, 9125, 555, 30725, 287, 279, 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https://www.mindmeister.com/1342932036/computational-thinking	1,571,163,936,000,000,000	text/html	crawl-data/CC-MAIN-2019-43/segments/1570986660231.30/warc/CC-MAIN-20191015182235-20191015205735-00313.warc.gz	942,828,178	14,113	# Computational Thinking Computational Thinking by Laura Schonfeld Get Started. It's Free Computational Thinking ## 2. Problem: Laura wants to go see band, but can’t take off work and only have \$375 of funds. ### 2.1. Decomposition: Laura needs to determine all the locations and times that band is playing where she would not need to take off work. Laura needs to figure out the cost of plane tickets, concert tickets, and accommodations for all of those times and dates. She will need to add up all the costs and determine which locations would be within her budget. Then, she would make a decision of where to see band from the options she had left within her budget. 2.1.1. Rationale: This is an example of the computational step, decomposition. Decomposition is breaking down a problem into smaller parts (“Google computational thinking for educators”, n.d.) Because the problem included determining where band was playing on days she could see them without taking off work, calculating multiple different costs and then adding up those costs to determine the total costs, each part of the process was separated to make it easier to solve the whole problem of wanting to go to the concert on a budget. 2.1.2. Pattern Recognition- Laura researches the tour dates to see what days and places band plays. She learns that band only plays on Saturdays (the only day when she could see them without taking off work) this year in Chicago, New York, and Red Rocks. She begins researching flights to these locations and determines that the cost for flying in on Fridays is \$100 for both Red Rocks and New York while the cost for flying to Chicago is \$275. She also finds that flying in on Saturdays is \$200 for Red Rocks and \$175 for both Chicago and New York. Next, she must research the costs of accommodations in each of those places. The cost for accommodations in Red Rocks is \$75 a night. The cost for accommodations for New York is \$150 a night. There is no cost for accommodations in Chicago because she can stay with her friend Lauren. Next, Laura must research the costs of each concert ticket in their respective cities. She discovers that Chicago has the most expensive ticket at \$125 followed by \$75 for Red Rocks and \$60 for New York. 2.1.2.1. Rationale: This is an example of the next step, pattern recognition. In this step, the computational thinker must look at the data and find patterns (“Google computational thinking for educators”, n.d.). Here, Laura has looked at all of the data including tour schedules, airplane tickets, and accommodations to determine the pattern of lowest cost options as well as options that fit within her time constraint. 2.1.2.2. Algorithm Design- Laura must write down and calculate all of the possible combinations of cost by adding up the cost for flying into New York on Friday and Saturday, the accommodations cost for staying both one and two nights depending on what day she flies in, and the cost of the ticket. She must also add up the cost for flying to Chicago on both Friday and Saturday as well as the cost of the ticket, but she does not need to add anything for accommodations since she would stay with her friend. Laura must add up the cost for flying to Red Rocks on both Friday and Saturday, the cost of accommodations for staying both one and two nights, as well as the cost for the concert ticket. She learns that it would be within her budget to see band at Red Rocks if she flys in on either Friday or Saturday or Chicago if she flys in on Saturday. She decides to buy a ticket to see band on Saturday in Chicago, books her Saturday plane ticket, and has a blast! 2.1.2.2.1. Rationale: In this step of computational thinking, algorithm design, the thinker must come up with a step by step process that will solve this problem and others like it (“Google computational thinking for educators”, n.d.). Because Laura needed to determine the total cost of each trip, she considered each different scenario (Friday New York, Friday Red Rocks, Friday Chicago, Saturday New York, Saturday Red Rocks, Saturday Chicago), calculated the costs, and compared the costs to her budget of \$375 to determine the possible options. From her possible options, she makes a decision on which show to attend. This is an example of computational thinking because of the steps that have just been described. These same steps could also be used to figure out attending future concerts and could thus be applied to similar problems. 2.1.2.2.2. Abstraction- Laura decides that other concert going lovers would benefit from being able to streamline this process using technology. She decides she could create an app using her friend Walton’s computer science skills that automatically scours the internet for concert tickets for a specific band, and combines it with the cost of plane tickets and accommodation all in one place! The user simply inputs their budget as well as the artists they are interested in seeing. They can also input their hometowns or places where they would not need accommodations. The app quickly searches possible combinations that would allow the user to see the artist within their budget. She becomes rich off of the success of her app and she never has to worry about not being able to spend money on seeing concerts ever again.	1,089	5,320	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	3.96875	4	CC-MAIN-2019-43	latest	en	0.963767	[ 128000, 2, 93028, 53389, 271, 59122, 1697, 53389, 555, 30928, 5124, 263, 31202, 271, 1991, 36912, 13, 1102, 596, 3658, 198, 59122, 1697, 53389, 271, 567, 220, 17, 13, 22854, 25, 30928, 6944, 311, 733, 1518, 7200, 11, 719, 649, 1431, 1935, 1022, 990, 323, 1193, 617, 33982, 12935, 315, 10736, 382, 14711, 220, 17, 13, 16, 13, 97478, 3571, 25, 30928, 3966, 311, 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http://www.radford.edu/~ibarland/Public/Humor/timeIsMoney	1,369,271,001,000,000,000	text/plain	crawl-data/CC-MAIN-2013-20/segments/1368702718570/warc/CC-MAIN-20130516111158-00058-ip-10-60-113-184.ec2.internal.warc.gz	672,450,577	1,714	Date: Thu, 4 Jun 1998 11:18:21 -0700 Dilbert's Salary Law: Engineers Vs. Managers -------------------------------------------- Engineers and scientists will never make as much money as business managers. Now we have a mathematical proof that explains why this is true: Postulate 1: Knowledge is Power. Postulate 2: Time is Money. As every engineer knows, Work ---------- = Power Time Since Knowledge = Power, and Time = Money, we have: Work ---------- = Knowledge Money Solving for Money, we get: Work ---------- = Money Knowledge Thus, as Knowledge approaches zero, Money approaches infinity regardless of the amount of Work done. Conclusion: The Less you Know, the More you Make. Note: It has been speculated that the reason why Bill Gates dropped out of Harvard's math program was because he stumbled upon this proof as an undergraduate, and dedicated the rest of his career to the pursuit of ignorance. [1997.jul.30] After applying some simple algebra to some trite phrases and cliches a new understanding can be reached of the secret to wealth and success. > > Here it goes. > Knowledge is Power > Time is Money and as every engineer knows, Power is Work over Time. > > So, substituting algebraic equations for these time worn bits of > wisdom, we get: > > K = P (1) > T = M (2) > P = W/T (3) > > Now, do a few simple substitutions: > > Put W/T in for P in equation (1), which yields: > K = W/T (4) > > Put M in for T into equation (4), which yields: > > K = W/M (5) > Now we've got something. Expanding back into English, we get: > > Knowledge equals Work over Money. > > What this MEANS is that: > > 1. The More You Know, the More Work You Do, and > 2. The More You Know, the Less Money You Make. > > Solving for Money, we get: > > M = W/K (6) > > Money equals Work Over Knowledge. > > From equation (6) we see that Money approaches infinity as Knowledge approaches 0, regardless of the Work done. > > What THIS MEANS is: > > The More you Make, the Less you Know. > > Solving for Work, we get > > W = M K (7) > > Work equals Money times Knowledge > > From equation (7) we see that Work approaches 0 as Knowledge > approaches 0. > > What THIS MEANS is: > The stupid rich do little or no work. > > Working out the socioeconomic implications of this breakthrough is > left as an exercise for the reader > > >------------------------------------------------------------------+ > Hard work has a future payoff. Laziness pays off now.	592	2,435	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	3.828125	4	CC-MAIN-2013-20	latest	en	0.909113	[ 128000, 1956, 25, 36992, 11, 220, 19, 12044, 220, 2550, 23, 220, 806, 25, 972, 25, 1691, 482, 17819, 15, 53867, 9339, 596, 42858, 7658, 25, 49796, 44082, 13, 62534, 20308, 5272, 49796, 323, 14248, 690, 2646, 1304, 439, 1790, 3300, 439, 2626, 20258, 13, 4800, 584, 617, 264, 37072, 11311, 430, 15100, 3249, 420, 374, 837, 25, 3962, 6468, 220, 16, 25, 33025, 374, 7572, 13, 3962, 6468, 220, 17, 25, 4212, 374, 18099, 13, 1666, 1475, 24490, 8964, 11, 5664, 44889, 284, 7572, 4212, 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http://www.numbersaplenty.com/15761	1,603,408,631,000,000,000	text/html	crawl-data/CC-MAIN-2020-45/segments/1603107880401.35/warc/CC-MAIN-20201022225046-20201023015046-00459.warc.gz	161,812,402	3,365	Search a number 15761 is a prime number BaseRepresentation bin11110110010001 3210121202 43312101 51001021 6200545 763644 oct36621 923552 1015761 1110929 129155 137235 145a5b 154a0b hex3d91 15761 has 2 divisors, whose sum is σ = 15762. Its totient is φ = 15760. The previous prime is 15749. The next prime is 15767. The reversal of 15761 is 16751. Subtracting from 15761 its product of digits (210), we obtain a palindrome (15551). Subtracting 15761 from its reverse (16751), we obtain a triangular number (990 = T44). It is a strong prime. It can be written as a sum of positive squares in only one way, i.e., 14161 + 1600 = 119^2 + 40^2 . It is a cyclic number. It is not a de Polignac number, because 15761 - 210 = 14737 is a prime. It is a Chen prime. It is equal to p1838 and since 15761 and 1838 have the same sum of digits, it is a Honaker prime. It is a congruent number. It is not a weakly prime, because it can be changed into another prime (15767) by changing a digit. It is a polite number, since it can be written as a sum of consecutive naturals, namely, 7880 + 7881. It is an arithmetic number, because the mean of its divisors is an integer number (7881). 215761 is an apocalyptic number. It is an amenable number. 15761 is a deficient number, since it is larger than the sum of its proper divisors (1). 15761 is an equidigital number, since it uses as much as digits as its factorization. 15761 is an evil number, because the sum of its binary digits is even. The product of its digits is 210, while the sum is 20. The square root of 15761 is about 125.5428213798. The cubic root of 15761 is about 25.0723239017. 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https://www.cs.odu.edu/~toida/nerzic/content/set/intr_to_set.html	1,632,352,137,000,000,000	text/html	crawl-data/CC-MAIN-2021-39/segments/1631780057403.84/warc/CC-MAIN-20210922223752-20210923013752-00388.warc.gz	702,067,012	2,059	## Introduction to Set Theory ### Subjects to be Learned • set • set membership --- "belong to" ### Contents The concept of set is fundamental to mathematics and computer science. Everything mathematical starts with sets. For example, relationships between two objects are represented as a set of ordered pairs of objects, the concept of ordered pair is defined using sets, natural numbers, which are the basis of other numbers, are also defined using sets, the concept of function, being a special type of relation, is based on sets, and graphs and digraphs consisting of lines and points are described as an ordered pair of sets. Though the concept of set is fundamental to mathematics, it is not defined rigorously here. Instead we rely on everyone's notion of "set" as a collection of objects or a container of objects. In that sense "set" is an undefined concept here. Similarly we say an object "belongs to " or "is a member of" a set without rigorously defining what it means. "An object(element) x belongs to a set A" is symbolically represented by "x A" . It is also assumed that sets have certain (obvious) properties usually asssociated with a collection of objects such as the union of sets exists, for any pair of sets there is a set that contains them etc. This approach to set theory is called "naive set theory " as opposed to more rigorous "axiomatic set theory". It was first developed by the German mathematician Georg Cantor at the end of the 19th century. For more on naive and axiomatic set theories click here which is not required for this course. Though the naive set theory is not rigorous, it is simpler and practically all the results we need can be derived within the naive set theory. Thus we shall be following this naive set theory in this course. Next -- Representation of Set Back to Schedule	394	1,837	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	3.90625	4	CC-MAIN-2021-39	latest	en	0.964476	[ 128000, 567, 29438, 311, 2638, 31535, 271, 14711, 65818, 311, 387, 90496, 271, 6806, 743, 198, 6806, 743, 16250, 12730, 330, 9978, 647, 311, 1875, 14711, 36962, 271, 791, 7434, 315, 743, 374, 16188, 311, 38696, 323, 6500, 8198, 13, 20696, 37072, 8638, 449, 7437, 13, 1789, 3187, 11, 12135, 1990, 1403, 6302, 527, 15609, 439, 264, 743, 315, 11713, 13840, 315, 6302, 11, 279, 7434, 315, 11713, 6857, 374, 4613, 1701, 7437, 11, 5933, 5219, 11, 902, 527, 279, 8197, 315, 1023, 5219, 11, 527, 1101, 4613, 1701, 7437, 11, 279, 7434, 315, 734, 11, 1694, 264, 3361, 955, 315, 12976, 11, 374, 3196, 389, 7437, 11, 323, 40099, 323, 4170, 1976, 82, 31706, 315, 5238, 323, 3585, 527, 7633, 439, 459, 11713, 6857, 315, 7437, 382, 27831, 279, 7434, 315, 743, 374, 16188, 311, 38696, 11, 433, 374, 539, 4613, 78477, 7162, 1618, 13, 12361, 584, 17631, 389, 5127, 596, 23035, 315, 330, 751, 1, 439, 264, 4526, 315, 6302, 477, 264, 5593, 315, 6302, 13, 763, 430, 5647, 330, 751, 1, 374, 459, 5732, 7434, 1618, 13, 35339, 584, 2019, 459, 1665, 330, 81997, 311, 330, 477, 330, 285, 264, 4562, 315, 1, 264, 743, 2085, 78477, 7162, 27409, 1148, 433, 3445, 13, 330, 2127, 1665, 11747, 8, 865, 17623, 311, 264, 743, 362, 1, 220, 4194, 285, 7891, 2740, 15609, 555, 4194, 330, 87, 362, 1, 662, 1102, 374, 1101, 19655, 430, 7437, 617, 3738, 320, 677, 2528, 8, 6012, 6118, 439, 784, 2168, 660, 449, 264, 4526, 315, 6302, 1778, 439, 279, 11552, 315, 7437, 6866, 11, 369, 904, 6857, 315, 7437, 1070, 374, 264, 743, 430, 5727, 1124, 5099, 382, 2028, 5603, 311, 743, 10334, 374, 2663, 330, 3458, 535, 743, 10334, 330, 439, 16475, 311, 810, 47999, 330, 710, 72, 13795, 743, 10334, 3343, 4194, 1102, 574, 1176, 8040, 555, 279, 6063, 21651, 1122, 13629, 42931, 269, 520, 279, 842, 315, 279, 220, 777, 339, 9478, 13, 1789, 810, 389, 50765, 323, 3944, 72, 13795, 743, 26018, 4299, 1618, 902, 374, 539, 2631, 369, 420, 3388, 13, 18056, 279, 50765, 743, 10334, 374, 539, 47999, 11, 433, 374, 35388, 323, 32367, 682, 279, 3135, 584, 1205, 649, 387, 14592, 2949, 279, 50765, 743, 10334, 13, 14636, 584, 4985, 387, 2768, 420, 50765, 743, 10334, 304, 420, 3388, 382, 5971, 1198, 79146, 315, 2638, 271, 3792, 311, 24416, 128001 ]	[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ]
https://algebra-class-ecourse.com/question/please-help-i-give-brainliest-sal-s-sandwich-shop-sells-wraps-and-sandwiches-as-part-of-its-lunc-11535266-59/	1,638,978,456,000,000,000	text/html	crawl-data/CC-MAIN-2021-49/segments/1637964363515.28/warc/CC-MAIN-20211208144647-20211208174647-00142.warc.gz	160,840,237	12,274	## PLEASE HELP!!! I GIVE BRAINLIEST!!!! Sal’s Sandwich Shop sells wraps and sandwiches as part of its lunch specials The pro Question PLEASE HELP!!! I GIVE BRAINLIEST!!!! Sal’s Sandwich Shop sells wraps and sandwiches as part of its lunch specials The profit on every sandwich is \$2 and The profit on every wrap is \$3. Sal made a profit of \$1,470 from lunch specials last month. The equation 2x + 3y = 1,470 represents Sal’s profits last month, where x is the number of sandwich lunch specials sold and y is the number of wrap lunch specials sold. 1. Change the equation into slope-intercept form. Identify the slope and y-intercept of the equation. Be sure to show all of your work below: 0 ## Answers ( No ) 1. 2x+3y =1470 slope intercept form y=mx+b 2x+3y =1470 subtract 2x from each side 3y = -2x+1470 divide by 3 y = -2/3 x +1470/3 y = -2/3x +490 slope = -2/3 y intercept = 490	262	903	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	4	4	CC-MAIN-2021-49	latest	en	0.876421	[ 128000, 567, 54233, 56571, 12340, 358, 480, 6674, 74863, 691, 19046, 5288, 17523, 8375, 753, 67836, 14355, 31878, 40809, 323, 57758, 439, 961, 315, 1202, 16163, 60874, 578, 463, 271, 14924, 271, 81663, 56571, 12340, 358, 480, 6674, 74863, 691, 19046, 5288, 17523, 198, 17691, 753, 67836, 14355, 31878, 40809, 323, 57758, 439, 961, 315, 1202, 16163, 60874, 271, 791, 11626, 389, 1475, 28974, 374, 33982, 17, 323, 198, 791, 11626, 389, 1475, 15411, 374, 33982, 18, 382, 17691, 1903, 264, 11626, 315, 33982, 16, 11, 17711, 505, 16163, 60874, 1566, 2305, 13, 578, 24524, 220, 17, 87, 489, 220, 18, 88, 284, 220, 16, 11, 17711, 11105, 8375, 753, 22613, 1566, 2305, 11, 1405, 865, 374, 279, 1396, 315, 28974, 16163, 60874, 6216, 323, 379, 374, 279, 1396, 315, 15411, 16163, 60874, 6216, 382, 16, 13, 10604, 279, 24524, 1139, 31332, 45994, 1512, 1376, 13, 65647, 279, 31332, 323, 379, 45994, 1512, 315, 279, 24524, 13, 2893, 2771, 311, 1501, 682, 315, 701, 990, 3770, 1473, 15, 271, 567, 38343, 320, 2360, 5235, 16, 13, 220, 17, 87, 10, 18, 88, 284, 10288, 15, 271, 97612, 29739, 1376, 271, 88, 28, 18577, 36193, 271, 17, 87, 10, 18, 88, 284, 10288, 15, 271, 60542, 220, 17, 87, 505, 1855, 3185, 271, 18, 88, 284, 482, 17, 87, 10, 10288, 15, 271, 60494, 555, 220, 18, 271, 88, 284, 482, 17, 14, 18, 865, 489, 10288, 15, 14, 18, 271, 88, 284, 482, 17, 14, 18, 87, 489, 18518, 271, 97612, 284, 482, 17, 14, 18, 271, 88, 29739, 284, 220, 18518, 128001 ]	[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ]
https://notebook.community/Radcliffe/project-euler/Euler%20060%20-%20Prime%20pair%20sets	1,642,530,976,000,000,000	text/html	crawl-data/CC-MAIN-2022-05/segments/1642320300997.67/warc/CC-MAIN-20220118182855-20220118212855-00261.warc.gz	487,727,562	5,637	# Euler Problem 60 The primes 3, 7, 109, and 673, are quite remarkable. By taking any two primes and concatenating them in any order the result will always be prime. For example, taking 7 and 109, both 7109 and 1097 are prime. The sum of these four primes, 792, represents the lowest sum for a set of four primes with this property. Find the lowest sum for a set of five primes for which any two primes concatenate to produce another prime. `````` In [2]: from time import time start = time() from sympy import primerange, isprime primes = list(primerange(2, 50000)) N = len(primes) minsum = 1e100 def concat(p, q): return int(f'{p}{q}') def edge(p, q): if (p,q) in E: return E[p,q] cliques = [([], 0)] for p in primes: E = {} if p >= minsum: break new_cliques = [] for clique, weight in cliques: if all(edge(p,q) for q in clique): new_clique = clique + [p] if len(new_clique) == 5: minsum = min(minsum, weight + p) new_cliques.append((new_clique, weight + p)) cliques.extend(new_cliques) print(minsum) print("Time: %.2f seconds" % (time() - start)) `````` `````` 26033 Time: 293.43 seconds `````` `````` In [ ]: ``````	352	1,137	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	3.59375	4	CC-MAIN-2022-05	latest	en	0.684046	[ 128000, 2, 81118, 22854, 220, 1399, 271, 791, 50533, 220, 18, 11, 220, 22, 11, 220, 7743, 11, 323, 220, 24938, 11, 527, 5115, 23649, 13, 3296, 4737, 904, 1403, 50533, 323, 41072, 1113, 1124, 304, 904, 2015, 279, 1121, 690, 2744, 387, 10461, 13, 1789, 3187, 11, 4737, 220, 22, 323, 220, 7743, 11, 2225, 220, 19027, 24, 323, 220, 7743, 22, 527, 10461, 13, 578, 2694, 315, 1521, 3116, 50533, 11, 220, 24763, 11, 11105, 279, 15821, 2694, 369, 264, 743, 315, 3116, 50533, 449, 420, 3424, 382, 10086, 279, 15821, 2694, 369, 264, 743, 315, 4330, 50533, 369, 902, 904, 1403, 50533, 78884, 311, 8356, 2500, 10461, 382, 14196, 14196, 14196, 271, 644, 4194, 58, 17, 69662, 1527, 892, 1179, 892, 198, 2527, 284, 892, 2892, 1527, 22176, 88, 1179, 27909, 853, 11, 374, 33438, 271, 652, 1769, 284, 1160, 26022, 3212, 853, 7, 17, 11, 220, 2636, 410, 1192, 45, 284, 2479, 26022, 1769, 340, 1083, 1264, 284, 220, 16, 68, 1041, 271, 755, 34820, 1319, 11, 2874, 997, 693, 528, 968, 25097, 79, 15523, 80, 75484, 755, 6964, 1319, 11, 2874, 997, 333, 320, 79, 36280, 8, 304, 469, 512, 693, 469, 11661, 36280, 2595, 566, 8467, 284, 510, 41156, 220, 15, 28871, 2000, 281, 304, 50533, 512, 36, 284, 5731, 333, 281, 2669, 1332, 1264, 512, 9137, 198, 943, 6937, 8467, 284, 4260, 2000, 81479, 11, 4785, 304, 22059, 14295, 512, 333, 682, 40889, 1319, 36280, 8, 369, 2874, 304, 81479, 997, 943, 6937, 2428, 284, 81479, 489, 510, 79, 933, 333, 2479, 1792, 6937, 2428, 8, 624, 220, 20, 512, 1083, 1264, 284, 1332, 14478, 1264, 11, 4785, 489, 281, 340, 943, 6937, 8467, 2102, 1209, 943, 6937, 2428, 11, 4785, 489, 281, 1192, 566, 8467, 16209, 1792, 6937, 8467, 696, 1374, 14478, 1264, 340, 1374, 446, 1489, 25, 19032, 17, 69, 6622, 1, 220, 1034, 320, 1712, 368, 482, 1212, 4489, 14196, 74694, 4077, 14196, 14196, 14196, 271, 11387, 1644, 198, 1489, 25, 220, 17313, 13, 3391, 6622, 271, 14196, 74694, 4077, 14196, 14196, 14196, 271, 644, 4194, 58, 4194, 69662, 14196, 14196, 14196, 128001 ]	[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ]
https://tutorial.math.lamar.edu/Solutions/CalcI/TrigEquations_CalcII/Prob7.aspx	1,701,399,848,000,000,000	text/html	crawl-data/CC-MAIN-2023-50/segments/1700679100264.9/warc/CC-MAIN-20231201021234-20231201051234-00052.warc.gz	669,653,221	16,936	Paul's Online Notes Home / Calculus I / Review / Trig Equations with Calculators, Part II Show Mobile Notice Show All Notes Hide All Notes Mobile Notice You appear to be on a device with a "narrow" screen width (i.e. you are probably on a mobile phone). Due to the nature of the mathematics on this site it is best views in landscape mode. If your device is not in landscape mode many of the equations will run off the side of your device (should be able to scroll to see them) and some of the menu items will be cut off due to the narrow screen width. ### Section 1.6 : Solving Trig Equations with Calculators, Part II 7. Find all the solutions to $$4{\csc ^2}\left( {1 - t} \right) + 6 = 25\csc \left( {1 - t} \right)$$. Use at least 4 decimal places in your work. Show All Steps Hide All Steps Hint : Factor the equation and using basic algebraic properties get two equations that can be dealt with using known techniques. If you’re not sure how to factor this think about how you would factor $$4{x^2} - 25x + 6 = 0$$. Start Solution This equation may look very different from anything that we’ve ever been asked to factor, however it is something that we can factor. First think about factoring the following, $4{x^2} + 6 = 25x\,\,\,\,\,\,\,\,\,\,\, \to \,\,\,\,\,\,\,\,\,\,4{x^2} - 25x + 6 = \left( {4x - 1} \right)\left( {x - 6} \right) = 0$ If we can factor this algebraic equation then we can factor the given equation in exactly the same manner. \begin{align}4{\csc ^2}\left( {1 - t} \right) + 6 & = 25\csc \left( {1 - t} \right)\\ 4{\csc ^2}\left( {1 - t} \right) - 25\csc \left( {1 - t} \right) + 6 & = 0\\ \left( {4\csc \left( {1 - t} \right) - 1} \right)\left( {\csc \left( {1 - t} \right) - 6} \right) & = 0\end{align} Now, we have a product of two factors that equals zero and so by basic algebraic properties we know that we must have, $4\csc \left( {1 - t} \right) - 1 = 0\hspace{0.25in}{\rm{OR}}\hspace{0.25in}\csc \left( {1 - t} \right) - 6 = 0$ Hint : Solve each of these two equations to attain all the solutions to the original equation. Show Step 2 Each of these equations are similar to equations solved in the previous section and earlier in this section. Therefore, we will be assuming that you can recall the solution process for each and we will not be putting in as many details. If you are unsure of the process you should go back to the previous section and work some of the problems there before proceeding with the solution to this problem. We’ll start with the first equation, isolate the cosecant and convert to an equation in terms of sine for easier solving. Doing this gives, $\csc \left( {1 - t} \right) = \frac{1}{4}\hspace{0.25in}\hspace{0.25in} \to \hspace{0.25in}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\sin \left( {1 - t} \right) = 4 > 1$ We now know that there are now solutions to the first equation because we know $$- 1 \le \sin \theta \le 1$$. Now, let’s solve the second equation. $\csc \left( {1 - t} \right) = 6\hspace{0.25in}\hspace{0.25in} \to \hspace{0.25in}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\sin \left( {1 - t} \right) = \frac{1}{6}$ Using our calculator we get, $1 - t = {\sin ^{ - 1}}\left( {\frac{1}{6}} \right) = 0.1674$ A quick glance at a unit circle shows us that the second angle in the range $$\left[ {0,2\pi } \right]$$ is $$\pi - 0.1674 = 2.9742$$. All the solutions to the second equation are then, \begin{align}1 - t & = 0.1674 + 2\pi n & \hspace{0.25in}{\rm{OR}}\hspace{0.25in}1 - t & = 2.9742 + 2\pi n & \hspace{0.25in} n = 0, \pm 1, \pm 2, \ldots \\t & = 0.8326 - 2\pi n & \hspace{0.25in}{\rm{OR}}\hspace{0.55in}t & = - 1.9742 - 2\pi n & \hspace{0.25in} n = 0, \pm 1, \pm 2, \ldots \end{align} Because we had not solutions to the first equation all the solutions to the original equation are then, $\require{bbox} \bbox[2pt,border:1px solid black]{{t = 0.8326 - 2\pi n \hspace{0.25in} {\rm{OR}} \hspace{0.25in} t = - 1.9742 - 2\pi n \hspace{0.25in} n = 0, \pm 1, \pm 2, \ldots }}$ Do get too excited about the fact that we only got solutions from one of the two equations we got after factoring. This will happen on occasion and so we need to be ready for these cases when they happen. If an interval had been given we would next proceed with plugging in values of $$n$$ to determine which solutions fall in that interval. Since we were not given an interval this is as far as we can go. Note that depending upon the amount of decimals you used here your answers may vary slightly from these due to round off error. Any differences should be slight and only appear around the 4th decimal place or so however.	1,499	4,593	{"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	4.8125	5	CC-MAIN-2023-50	latest	en	0.834584	[ 128000, 26368, 596, 8267, 18559, 198, 7778, 611, 32459, 355, 358, 611, 10506, 611, 1183, 343, 11964, 811, 449, 32459, 3046, 11, 3744, 8105, 198, 7968, 13716, 25773, 7073, 2052, 18559, 4194, 22434, 2052, 18559, 198, 18876, 25773, 198, 2675, 5101, 311, 387, 389, 264, 3756, 449, 264, 330, 77, 6172, 1, 4264, 2430, 320, 72, 1770, 13, 499, 527, 4762, 389, 264, 6505, 4641, 570, 24586, 311, 279, 7138, 315, 279, 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http://mathhelpforum.com/calculus/210003-integral-1-infiinity-ln-x-x-2-equals-1-how-please-help-print.html	1,526,854,398,000,000,000	text/html	crawl-data/CC-MAIN-2018-22/segments/1526794863689.50/warc/CC-MAIN-20180520205455-20180520225455-00474.warc.gz	189,027,199	3,520	• Dec 17th 2012, 04:28 PM skinsdomination09 integral from 1 to infiinity of ln x / (x^2) equals 1. How?! (IMPROPER INTEGRATION) I integrated the function to become ln x / (x^2) u = lnx du = 1/x dv = 1/x^2 v = -1/x UV - int VDU lnx * -1/x - - int of 1/x^2 becomes -1/x so - lnx/x - lnx (or close to this) • Dec 17th 2012, 04:58 PM Plato Re: integral from 1 to infiinity of ln x / (x^2) equals 1. How?! (IMPROPER INTEGRATIO Quote: Originally Posted by skinsdomination09 I integrated the function to become ln x / (x^2) u = lnx du = 1/x dv = 1/x^2 v = -1/x UV - int VDU lnx * -1/x - - int of 1/x^2 becomes -1/x so - lnx/x - lnx (or close to this) Look at this webpage. • Dec 17th 2012, 05:08 PM skeeter $\displaystyle \int_1^\infty \frac{\ln{x}}{x^2} \, dx$ $\displaystyle u = \ln{x}$ ... $\displaystyle du = \frac{1}{x} \, dx$ $\displaystyle dv = \frac{1}{x^2} dx$ ... $\displaystyle v = -\frac{1}{x}$ $\displaystyle \lim_{b \to \infty} \int_1^b \frac{\ln{x}}{x^2} \, dx = \lim_{b \to \infty} \left[-\frac{\ln{x}}{x}\right]_1^b - \int_1^b -\frac{1}{x^2} \, dx$ $\displaystyle \lim_{b \to \infty} \left[-\frac{\ln{x}}{x}\right]_1^b - \left[\frac{1}{x} \right]_1^b$ $\displaystyle \lim_{b \to \infty} \left[-\frac{\ln{b}}{b}\right] - \left[\frac{1}{b} - 1 \right]_1^b = 1$ • Dec 17th 2012, 05:09 PM skinsdomination09 Re: integral from 1 to infiinity of ln x / (x^2) equals 1. How?! (IMPROPER INTEGRATIO Quote: Originally Posted by Plato delete • Dec 17th 2012, 05:13 PM skinsdomination09 Quote: Originally Posted by skeeter $\displaystyle \int_1^\infty \frac{\ln{x}}{x^2} \, dx$ $\displaystyle u = \ln{x}$ ... $\displaystyle du = \frac{1}{x} \, dx$ $\displaystyle dv = \frac{1}{x^2} dx$ ... $\displaystyle v = -\frac{1}{x}$ $\displaystyle \lim_{b \to \infty} \int_1^b \frac{\ln{x}}{x^2} \, dx = \lim_{b \to \infty} \left[-\frac{\ln{x}}{x}\right]_1^b - \int_1^b -\frac{1}{x^2} \, dx$ $\displaystyle \lim_{b \to \infty} \left[-\frac{\ln{x}}{x}\right]_1^b - \left[\frac{1}{x} \right]_1^b$ $\displaystyle \lim_{b \to \infty} \left[-\frac{\ln{b}}{b}\right] - \left[\frac{1}{b} - 1 \right]_1^b = 1$ Makes more sense, Thanks But wouldn't the ln b / b (since it's approaching infinity) be indetirminate so it would be divergant. My original mistake was that I integrated the second half twice by accident, but I still don't get how you can just "ignore" ln b /b since ln(infinity) doesn't equal 0 or anything. Oh... is it bc the top is growing slower than the bottom so its just a really small number over a big one and thus goes to zero? Simple Alg. + early Calc skills holding me back(Angry) • Dec 17th 2012, 05:17 PM skeeter L'Hopital ... $\displaystyle \displaystyle \lim_{b \to \infty} \frac{\ln{b}}{b} = \lim_{b \to \infty} \frac{1}{b} = 0$ • Dec 17th 2012, 05:19 PM skinsdomination09 $\displaystyle \displaystyle \lim_{b \to \infty} \frac{\ln{b}}{b} = \lim_{b \to \infty} \frac{1}{b} = 0$	1,182	2,914	{"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	3.96875	4	CC-MAIN-2018-22	latest	en	0.51854	[ 128000, 6806, 3799, 220, 1114, 339, 220, 679, 17, 11, 220, 2371, 25, 1591, 5975, 198, 50418, 5717, 2617, 2545, 198, 82135, 505, 220, 16, 311, 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https://golylyparize.regardbouddhiste.com/excercise-14-problems-47568pe.html	1,606,288,650,000,000,000	text/html	crawl-data/CC-MAIN-2020-50/segments/1606141181482.18/warc/CC-MAIN-20201125071137-20201125101137-00336.warc.gz	324,944,317	3,709	# Excercise 14 problems For each exercise, a link to a possible solution is provided. Each solution includes a discussion of how a programmer might approach the problem and interesting points raised by the problem or its solution, as well as complete source code of the solution. Trigometric Ratios We can use Pythagoras' Theorem see above to calculate the length of an unknown side in a right-angled triangle when we are given information about the lengths of 2 other sides. However, if we are given information about an angle other than the 90 degrees and one side and need to calculate the length of another side then we use the trig ratios. This first tutorial takes you through naming the sides of a triangle and an introduction to the trig ratios. ## Best Types of Exercise for Asthma - Health This tutorial is suitable for students in Year 10 or We can use trig ratios to calulate the length of a side on a triangle if we know the length of one side and the size of one angle. You will need a scientific calculator for this tutorial This tutorial is suitable for students in Year 11 or Calculating angles using the trigometric ratios. OK, so probably saw this coming. If you know the length of the two sides of a right-angle triangle then you can caculate the size of the other angles inside the triangle. We're really getting to know stuff about right-angle triangles.Physical Geography Laboratory Manual for McKnightâ€™s Physical Geography: A Landscape Appreciation, Eleventh Edition offers a comprehensive set of lab exercises to accompany any physical geography class. The manual is organized to meet your needs, providing the flexibility to pick and choose a. Student Resources For more information on how to order these items, Chapter Exercise Exercise Exercise Exercise Problem A. Working Papers Plus for Select Exercises and Problems Chapters ISBN: Shop ProForm online. ProForm is a world leader in home fitness equipment. ## Solutions from 0 to 116, not including 113 which is probably not findable by the rules given. Shop professional-grade treadmills, training cycles, and ellipticals! Example Exercise Dilution of a Solution Concentrated hydrochloric acid is available commercially as a 12 M solution. What is the molarity of an HCl solution. Math []. 1. Write a function to calculate if a number is prime. Return 1 if it is prime and 0 if it is not a prime. 2. Write a function to determine the number of prime numbers below n. C programming Exercises, Practice, Solution: C is a general-purpose, imperative computer programming language, supporting structured programming, lexical variable scope and recursion, while a static type system prevents many unintended operations. 14 Leg Exercises for Men - Elite Men's Guide	554	2,755	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	3.671875	4	CC-MAIN-2020-50	latest	en	0.896775	[ 128000, 2, 39995, 18865, 220, 975, 5435, 271, 2520, 1855, 10368, 11, 264, 2723, 311, 264, 3284, 6425, 374, 3984, 13, 9062, 6425, 5764, 264, 10430, 315, 1268, 264, 48888, 2643, 5603, 279, 3575, 323, 7185, 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https://www.jiskha.com/display.cgi?id=1335584620	1,503,524,852,000,000,000	text/html	crawl-data/CC-MAIN-2017-34/segments/1502886124563.93/warc/CC-MAIN-20170823210143-20170823230143-00494.warc.gz	915,507,180	4,117	# algebra posted by . The circular path of cars on a ferris wheel can be modeled with the equation x^2-14x+y^2-150y=-49, measured in feet. What is the maximum height above ground of the riders. • algebra - write the circle equation in standard form by completing the square x^2 - 14x + 48 + y^2 - 150y + 5625 = -49 + 49 + 5625 (x-7)^2 + (y - 75)^2 = 75^2 so the centre is at (7, 75) and the radius is 75 after making a sketch you should be able to answer the question ## Similar Questions 1. ### trig A ferris wheel with a diameter of 100 feet rotates at a constant rate of 4 revolutions per minute. Let the center of the ferris wheel be at the origin. 1. Each of the ferris wheel's cars travels around a cirlce. a) Write an equation … 2. ### math a ferris wheel has the diameter of 240 feet and the bottom of the ferris wheel is 9 feet above the ground. find the equation of the wheel if the origin is placed on the ground directly below the center of the wheel 3. ### College Algebra A ferris wheel has a diameter of 320 feet and the bottom of the Ferris wheel is 9 feet above the ground. Find the equation of the wheel if the origin is placed on the ground directly below the center of the wheel. 4. ### Trig As you ride a ferris wheel, your distance from the ground varies sinusoidally with time. When the last passenger boards the ferris wheel and the ride starts moving, let your position be modeled by the diagram provided. Let t be the … 5. ### MATH A Ferris wheel completes one revolution every 90 s. The cars reach a maximum of 55 m above the ground and a minimum of 5 m above the ground. The height, h, in metres, above the ground can be modeled using a sine function, where t represents … 6. ### math A Ferris wheel has a maximum height of 245 feet and a wheel diameter of 230 feet. Find an equation for the wheel if the center of the wheel is on the y-axis and y represents the height above the ground. equation is ? 7. ### math trig A Ferris wheel with a diameter of 37 meters rotates at a rate of 4 minutes per revolution. Riders board the Ferris wheel 4 meters above the ground at the bottom of the wheel. A couple boards the Ferris wheel and the ride starts. Write … 8. ### math The height of a person above the ground on a Ferris wheel is given by the function h(t)=9 sin {pi/20 (t- pi/2))+ 12 where h(t) is the height in meters of person t seconds after getting on a) how long does it take the Ferris wheel to … 9. ### Math-Trigonometry Can someone help me with this problem. -A Ferris wheel has a radius of 37.8 feet. The bottom of the Ferris wheel sits 0.7 feet above the ground. You board the Ferris wheel at the 6 o'clock position and rotate counter-clockwise. A)Define … 10. ### precalculus A Ferris wheel has a radius of 40 feet. The bottom of the Ferris wheel sits 0.5 feet above the ground. You board the Ferris wheel at the 6 o'clock position and rotate counter-clockwise. Define a function,f that gives your height above … More Similar Questions	775	2,995	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	3.703125	4	CC-MAIN-2017-34	latest	en	0.871122	[ 128000, 2, 47976, 271, 44182, 555, 6905, 791, 28029, 1853, 315, 9515, 389, 264, 58139, 285, 13587, 649, 387, 62653, 449, 279, 24524, 865, 61, 17, 12, 975, 87, 44110, 61, 17, 12, 3965, 88, 11065, 2491, 11, 17303, 304, 7693, 13, 3639, 374, 279, 7340, 2673, 3485, 5015, 315, 279, 30803, 382, 6806, 47976, 22742, 5040, 279, 12960, 24524, 304, 5410, 1376, 555, 27666, 279, 9518, 271, 87, 61, 17, 482, 220, 975, 87, 489, 220, 2166, 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https://www.casinoguardian.co.uk/roulette/dalembert-betting-system/	1,726,697,516,000,000,000	text/html	crawl-data/CC-MAIN-2024-38/segments/1725700651941.8/warc/CC-MAIN-20240918201359-20240918231359-00738.warc.gz	647,579,469	23,847	# D’Alembert Betting System The D’Alembert is among the easiest betting systems that are applicable to the game of roulette. In fact, this system ranks second in terms of popularity after the Martingale. It bears many of the characteristics of the Martingale, which is not surprising considering the fact that both systems are based on negative betting progressions. Therefore, the D’Alembert dictates that players should increase the size of their stakes after a loss and decrease it after each winning bet. The origins of the D’Alembert system can be traced back to the 18th century, when the French mathematician Jean le Rond D’Alembert began his work on a statement of the fundamental laws of motion. The mathematician came to the conclusion that the external forces acting on a body and the inertial forces are a system which exists in equilibrium. This conclusion laid the foundations of Newton’s Second Law and became known as the D’Alembert Principle. ## How Does the D’Alembert Betting System Work When applied to games of chance like roulette, D’Alembert’s law of equilibrium leads to the idea that future outcomes of wheel spins can balance less likely results in the past. According to D’Alembert, the chances of a coin landing on tails increase if it has already landed on heads several consecutive times. In other words, if the ball lands in red pockets several times in a row, black is due to be spun and becomes a more likely outcome. This concept is also known as the Gambler’s fallacy. However, this serves as an indication that the D’Alembert system is based on the idea that players, who place even-money bets, will generate a profit if they win as often as they lose or more. When the flip of a coin is taken into consideration, 50% of the time, the result that will come up will be tails, and 50% of the time, it will be heads. Although this is what the figures dictate, probabilities are not the same in the gambling realm. What gambling enthusiasts should remember is that when they lay any of the roulette bets that pay even money, the chances of winning are not the same as with the flip of a coin. This is so because of the presence of the zero and the double-zero pockets in American-style roulette variants. The D’Alembert system is quite simple to master even by roulette novices, which comes to explain why it is so hugely popular among casino enthusiasts. If you are keen on the idea to give the betting system a trial run, the first thing you need to do is set a base betting unit depending on the overall size of your bankroll. It is important to mention that your betting unit should not exceed 5% of your bankroll for the session. A betting unit of 2% is considered the safest option, especially for gambling aficionados who are reluctant to take risks or do not have that bountiful bankroll. As it seems, some experienced roulette players advise going for a base bet unit that does not overstep the 1% threshold. Others go even further and recommend putting on the line an amount that is 0.50% of the funds you intend on using during your betting session. Since the base bet unit you will settle on will be the backbone of this roulette betting method, it is of crucial importance to pick it wisely due to the fact that it willdetermine the net profit you are expected to accumulate during your betting session. The D’Alembert system dictates that you should start by placing an even-money bet of precisely one betting unit. If luck is not on your side on the first spin and you lose, you need to increase your next wager with one betting unit. What you need to do is to keep on increasing your wagers with a single betting unit after each loss. After each winning bet, you are required to reduce your next stake with one betting unit. If your first bet is a winning one, you continue wagering with the same base betting unit until you lose. The idea here is that if a given player wins and loses roughly the same number of times, they will eventually turn a profit. It would be best to demonstrate how the D’Alembert betting method works in practice by providing an example. Let’s assume your bankroll is £250 and your betting unit is £5 or 2% of the overall sum you intend to join the roulette table with. You bet £5 on Red and lose. You increase your next wager with one betting unit, so you bet £10 on Red and lose again. At this point, you have lost £15. You increase your next wager on Red to £15 and win this time, thus collecting £15 in net profit. After this success, you are supposed to reduce the next wager with one betting unit to £10. You bet on Black and win again, generating a net profit of another £10. It becomes clear you have won £25 with your last two bets and have lost only £15 with your two losing wagers, so your net profit for this betting session is £10. As you can see, the D’Alembert system works efficiently as long as the number of winning bets coincides with or exceeds that of losing bets. Of course, there is the option to set a limit at which you will stop increasing the stakes after a loss and reduce the betting unit to its initial size h. This modification can help you minimise your losses if you happen to enter a longer losing streak. The D’Alembert System Spin Bet (units) Outcome Total Profit 1 5 LOSS -5 2 10 LOSS -15 3 15 WIN 0 4 10 WIN 10 5 5 WIN 15 6 5 LOSS 10 7 10 LOSS 0 8 15 WIN 15 9 10 WIN 25 10 5 LOSS 20 11 10 WIN 30 12 5 WIN 35 13 5 WIN 40 ## The Reverse D’Alembert As it can be expected from the name, it functions in the opposite way but is suitable for even-money bets in roulette. Players are again recommended to choose a base betting unit which should range between 2% and 5% of their overall bankrolls. Their first even-money bet should be one betting unit. According to the Reverse D’Alembert, players should increase their stakes by one betting unit after each winning bet and decrease it after they see a loss. So if your first bet of £5 on Red wins, your second bet should be no more than £10. If you succeed in predicting the winning number again, your next wager needs to be £15, and so on. When players lose a bet, they are required to decrease their next stake by one betting unit. So, if you lose your third bet of £15, you reduce the next wager to £10. One major advantage of the Reverse D’Alembert is that it enables players to minimise their losses. Those, who implement the Reverse D’Alembert are less likely to exhaust their bankrolls when they experience a lengthy losing streak. Besides, implementing this variation may prove to be rather profitable if the player wins several bets in a row. What gambling aficionados should take into account is that while employing the Reverse D’Alembert betting system, the results they will ultimately enjoy might not be that bountiful as the ones delivered by its original version. Yet, one of the biggest benefits roulette lovers will enjoy if they decide to give this betting method a shot is that they will be able to put it through its paces even if their bankroll is rather humble. Furthermore, if fortune does not smile on you and you end up on a sustained losing streak, you will not find yourself with no other solution but to risk a bigger chunk of your funds so as to make up for the losses you have seen in the previous rounds. As with the original version of the D’Alembert betting system, while making use of its reversed version gambling enthusiasts are advised to settle on a stopping point before they kick it off. As it seems, the better part of the gambling aficionados recommends starting the betting progression all over as soon as you augment the staked amount five times. Although taking the betting progression further might seem like an alluring idea because the potential payoffs will be more generous, players should bear in mind that the risk they will need to take will increase significantly. The Reverse D’Alembert System Spin Bet (units) Outcome Total Profit 1 5 WIN 5 2 10 WIN 15 3 15 LOSS 0 4 10 WIN 10 5 15 WIN 25 6 20 LOSS 5 7 15 LOSS -10 8 10 WIN 0 9 15 WIN 15 10 20 LOSS -5 11 15 WIN 10 12 20 WIN 30 13 25 WIN 55 ## How to Tinker the D’Alembert Betting System If you plan on putting the D’Alembert betting system to the proof, you should be aware that there are few more modifications that can be made to it. The results players might ultimately enjoy can turn out to be rather interesting, given that they do not increase the staked amount by one betting unit each time their wagers are settled as losing ones and instead bring the betting progression back to square one. The reason why many roulette mavens are unlikely to frown upon this variant of the betting system is that even if they place several losing wagers, this will not have such a major impact on their bankroll. If gambling enthusiasts decide to take advantage of this modified version of the roulette betting method, the winnings they are expected to generate should make up for the losses they have seen during their betting session. One factor gambling enthusiasts should not forget is the edge the casino gains over them whenever they play. In spite of the fact that the usage of this betting method will help players reduce the losses to a minimum, they should remember that ultimately, they will invariably lose to the casino. Still, the fact that the casino invariably holds the upper hand over players should not put you off from the betting system. In essence, if luck is on your side that day, you might enjoy a long winning streak that can bring you a massive payoff. ## Advantages of the D’Alembert Betting System Without question, the biggest advantage the D’Alembert system has to offer is its very simplicity. It does not require players to take notes like some other systems do. On the contrary, the system is quite easy to learn and incorporate at the roulette table. Another advantage results from the fact that while using this betting method, there is no steep increase in the bet size with this negative progression system. As wagers are increased with a single betting unit after each loss, a longer losing streak is less likely to drain your bankroll completely. On some occasions, the D’Alembert system also enables those who follow it to generate a consistent profit. The size of the wager after each loss is increased slowly which renders the system suitable for players who do not have that many funds to play with. Besides, the risk of reaching the table limit on a lengthy losing streak is much smaller if you implement the D’Alembert, which does not apply to some of the other betting methods. ## Disadvantages of the D’Alembert Betting System Similarly to the other betting systems we have discussed so far, the D’Alembert has a few faults. The system is most effective when it comes to generating short-term profits. Though consistent, the winnings will be far from life-changing which is to be expected from a system that involves a relatively low degree of risk. Then again, in order to turn a profit at all, players are expected to win and lose roughly the same number of times. Unfortunately, there is no guarantee that this will happen since the chances of winning and losing with even-money bets in roulette are not exactly 50%. The scales are tipped in favour of the casino because of the introduction of the zero pocket on the wheel. Even more so, if one plays American roulette, where there is an additional double-zero pocket, which further increases the house edge. Statistically speaking, players lose more often than they win which would make it more difficult for them to recover from their losses. Entering a longer losing streak is not that unlikely, either. If a given player suffers five or six losing bets in a row, there is no guarantee they will succeed in winning the same number of times to offset the losses. Essentially, players need to remember that neither the D’Alembert nor any other betting system can reduce the house edge or influence the spins. This is so because the outcome of each spin is not affected by the previous numbers which had been spun on the wheel. ## Dangers of Using the D’Alembert Betting System As we mentioned above, the D’Alembert betting system revolves around the idea that ultimately, the number of losing and winning wagers gambling enthusiasts have laid will balance. In spite of the fact that this claim makes perfect sense to roulette lovers, what they fail to take into consideration is that in order for the results to come out even, the period that should be taken into consideration will be dramatically prolonged. In essence, this is exactly where the main lacking point of the betting system stems from because the frequencies of the opposite results are unlikely to balance during the betting session of players unless they do not spend a huge amount of time at the roulette table. That is the reason why this idea is unworkable when the game of roulette is concerned. In the event that the number of winning and losing wagers does not balance during the time casino enthusiasts wager, they might end up with no winnings at all, which is certainly not a desirable scenario. That being said, what makes many roulette players give the betting system a chance is that unlike the Martingale betting method that is exceptionally aggressive, this is not the case with the D’Alembert betting system for the simple reason that gambling aficionados will not be prompted to double the staked amount whenever a loss incurs. As it was already mentioned, with the D’Alembert betting system, the amount you need to put on the line following a loss will be increased by a single bet unit, which undoubtedly adds up to a big difference. Although the D’Alembert betting system appeals to gambling aficionados because of its inherent simplicity, roulette lovers should not forget that its usage will not have any influence on their chances of win. The same applies to the house edge as no matter if you are bound to take advantage of the D’Alembert betting method or any of the other roulette betting systems available out there, losing to the house is unavoidable. ## Conclusion At the end of the day, many gambling aficionados are eager to try their hand at the D’Alembert betting system because there are not any complex calculations that should be made while adapting their stakes to the outcome of the previous round. Another thing that makes the betting method so immensely popular among casino enthusiasts is that it is a much safer alternative when compared to the rest of the betting systems that can be applied to the game of roulette because the amount they will need to risk will not soar. With that in mind, gambling aficionados will not stand such a good chance to walk away with a heftier payout. In a similar vein, making up for the losses they have seen might only be possible after placing several successful wagers, which can sometimes be hard. As you can see, using the D’Alembert betting system has its pros and cons that should be evaluated, but it is worth considering as an option if you are still not confident enough to make use of other riskier roulette betting methods.	3,274	15,187	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	3.65625	4	CC-MAIN-2024-38	latest	en	0.96043	[ 128000, 2, 423, 529, 84583, 3172, 531, 85085, 744, 271, 791, 423, 529, 84583, 3172, 531, 374, 4315, 279, 30689, 26243, 6067, 430, 527, 8581, 311, 279, 1847, 315, 49211, 13, 763, 2144, 11, 420, 1887, 21467, 2132, 304, 3878, 315, 23354, 1306, 279, 8290, 287, 1604, 13, 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https://www.naukri.com/code360/library/logistic-regression-introduction	1,725,723,076,000,000,000	text/html	crawl-data/CC-MAIN-2024-38/segments/1725700650883.10/warc/CC-MAIN-20240907131200-20240907161200-00849.warc.gz	896,447,996	56,988	1. Introduction 2. Methodology 3. Cost Function 4. 5. Evaluating the model 6. Implementation 7. 8. Key takeaways Last Updated: Mar 27, 2024 # Logistic Regression Introduction Arun Nawani 0 upvote ## Introduction Have you ever wondered how your email account accurately segregates regular emails, important emails, and spam emails? Itâ€™s not a very complex trick and weâ€™ll learn the secret behind it. This is done with a supervised learning model called Logistic regression (However, it can be done with other machine learning algorithms also, but for the sake of this blog, weâ€™ll stick to Logistic regression). Logistic regression is employed in supervised learning tasks. More specifically, it is used for classification tasks. We know that name throws some people off. But the regression in the logistic regression is slightly misleading. It is NOT a regression model. Logistic regression is a probabilistic function. That means it makes use of probabilities of events to make its prediction. ## Methodology Suppose we are given a task, say we are given a customerâ€™s banking history and are tasked to find if the customer can be sanctioned a loan. Basically we need to find if given a loan, will the customer default on payment or not. We can use logistic regression for this purpose. It will be a binary classification between â€˜Yesâ€™ or â€˜Noâ€™. Logistic regression makes use of a sigmoid function and it is of the form - We know the straight line equation - y = w0 + w1 We know the sigmoid function has a range between 0 and 1. So letâ€™s divide the above equation by 1-y. y / (1-y) : 0 for y = 0 and âˆž for y = 1 But we require our function to be between -âˆž to +âˆž. For that, weâ€™ll take logarithm so the new equation is: Log (y / (1-y)) = w0 + w1x Upon simplifying our final equation then becomes - Here y = predicted probability belonging to the default class( default class is 1(yes)) w0 + w1x = the linear model within logistic regression. Also, the function is of the form of a sigmoid function The Sigmoid function has a range between 0 and 1. And therefore forms an S-like curve. The logistic function predicts the probability of an outcome. Hence its value lies anywhere between 0 and 1. And thatâ€™s where it gets its name from. We choose a threshold value above which the final prediction would be 1 and 0 otherwise. Letâ€™s talk about the linear equation w0 + w1x within the logistic function. Why do we need the logistic regression function in the first place if it stems from linear regression? Itâ€™s because the linear regression equation isnâ€™t confined within a range, unlike logistic regression. And it would be a very difficult task to assign a threshold value for class membership for a linear regression function. Thus we feed the predicted value to a sigmoid function which makes it Logistic regression having a range between 0 and 1. Now since the range is between 0 and 1(no outliers) it would be convenient to do a probabilistic classification. It represents a linear relationship between the input features and the final output. Here x = input feature w0 = bias term w1 = weight associated with the input variable Now suppose we take 0.5 as our threshold value. That means A predicted value >0.5 from the logistic function would have the final prediction as 1 and, A predicted value â‰¤0.5 from the logistic function would have the final prediction as 0. This is also called the decision boundary. Plotting the graph clears what makes logistic regression different from linear regression. ## Cost Function We know the cost function tells us how erroneous our model is. In the case of linear regression, our cost function was: - Cost function = 1/N âˆ‘(y(i) - y(i)predicted )2 The linear regression cost function is used to find the average of errors in all predicted values corresponding to the actual value of y. But this cost function is suitable for logistic regression since logistic regression doesnâ€™t have a continuous output variable, unlike linear regression. Suppose we were to use this cost function for a logistic regression model that does a binary classification(0 and 1). This would make the predicted value and the actual value either 0 or 1. This way we may end up at the local minima instead of the global minima. Remember, the cost function is minimum at the global minima. So we would require a new cost function for logistic regression. And itâ€™s given below. Here y(i) = the actual output for ith training example hÎ¸x(i) = Predicted output for the ith training example by our model Now we know that looks a bit overwhelming but letâ€™s break it down term by term. Suppose we train a logistic regression model and it correctly predicts the value 1. So that means the predicted value of y would be closer to 1. In this case, the second term in the logistic regression cost function becomes 0 (1-y = 1-1 = 0). The first term would be- y * log(ypredicted)= 1log(~1) = ~0 ( low error) Since the value is close to 1, therefore the cost function would compute a low error. Refer the log and -log graph given below. Our predicted value was correct and therefore it makes sense for the error to be low. In case had our model predicted 0 instead of 1, it would have led to a higher error since the log function for an input value closer to 0 would make our error exponentially high. y log(ypredicted)= 1log(~0) = -âˆž (high error) Gradient descent is a technique of altering the weights of data points based on the cost function. LogLoss cost function is calculated at each input-output data point. • We take a partial derivative with respect to bias and weight to find the slope of the cost function at every point. • Gradient descent then updates the values bias and weight values based on slope. This is an iterative process. • The iterations are done until the gradient descent reaches the minimum cost. The rate at which the gradient descent reaches the minima is called the learning rate. Feature scaling is essential for gradient descent to work efficiently. We suggest you learn gradient descent in depth. You can refer to our blog on gradient descent if you wish to learn the math behind it. ## Evaluating the model Once we train the model it is necessary to evaluate how accurate our model really is. There are various evaluation metrics for this purpose. • Accuracy - It represents the number of correct predictions upon total number of predictions. • ROC AUC score it stands for Receiver Operating Characteristic Curve. The area under the ROC AUC curve describes the relationship between the true positive rate (ratio of samples that were correctly predicted belonging to the correct class) and the false positive rate (the ratio of samples for which we incorrectly predicted their class membership). More area under the curve signifies better predictions, hence a better model. ## Implementation Weâ€™ll start by importing the necessary libraries and the dataset. Here is the datasetThe dataset is about sales of a product via social media advertisement campaign. Our task at hand will be to predict if the item was purchased or not considering the given features. ``````import numpy as np import pandas as pd import matplotlib.pyplot as plt from sklearn.model_selection import train_test_split from math import exp plt.rcParams["figure.figsize"] = (10, 6) class LogisticRegression: def PltFun(self): plt.scatter(self.data['Age'], self.data['Purchased']) plt.show() # Split the training dataset and test dataset def Training(self): self.X_train, self.X_test, self.y_train, self.y_test = train_test_split(self.data['Age'], self.data['Purchased'], test_size=0.20,random_state=10) def normalize(self): self.X_train -= self.X_train.mean() # Method to make predictions def Sigmoid(self, b0, b1,x=None): if x is None: self.X_test-=self.X_test.mean() return np.array([1 / (1 + exp(-1b0 + -1b1i)) for i in self.X_test]) else: return np.array([1 / (1 + exp(-1b0 + -1b1i)) for i in x]) # Method to train the model def logistic_regression(self): # Initializing variables self.normalize() b0 = 0 b1 = 0 L = 0.001 epochs = 300 for epoch in range(epochs): y_pred = self.Sigmoid(b0, b1,self.X_train) D_b0 = -2 sum((self.y_train - y_pred) * y_pred * (1 - y_pred)) # Derivative of loss wrt b0 D_b1 = -2 * sum(self.X_train * (self.y_train - y_pred) * y_pred * (1 - y_pred)) # Derivative of loss wrt b1 # Update b0 and b1 b0 = b0 - L * D_b0 b1 = b1 - L * D_b1 return b0, b1 def Accuracy(self , y_pred): accuracy=0 for i in range(len(y_pred)): if y_pred[i] == self.y_test.iloc[i]: accuracy += 1 print(f"Accuracy = {accuracy / len(y_pred)}") def Plot_result(self,y_pred): plt.clf() plt.scatter(self.X_test, self.y_test) plt.scatter(self.X_test, y_pred, c="red") plt.show() def predictor(self,x): x = [1 if p >= 0.5 else 0 for p in x] return x My_model = LogisticRegression() My_model.PltFun() My_model.Training() b0,b1=My_model.logistic_regression() y_pred = My_model.Sigmoid(b0,b1) y_pred=My_model.predictor(y_pred) My_model.Plot_result(y_pred) My_model.Accuracy(y_pred)`````` The accuracy of the model is >80%. 1. What is a sigmoid function? Ans. The sigmoid function is given by 1 / ( 1 + e-z ) and its range is between 0 and 1. 2. Why is logistic regression called a probabilistic function? Ans. Logistic regression generates the probability of output given the input features. We then set a threshold value to classify the probability prediction. 3. Mention some real-life use cases of logistic regression. Ans. a. A bank can use a logistic regression model to predict if a customer would default on a loan b. A logistic regression model can be trained to tell if a tumour is benign or not. ## Key takeaways In this blog, we have thoroughly covered logistic regression. Its mathematical significance, its cost function, and its step-by-step python implementation. However, consider this a starting point. There is always room for improvement. If you want to ace your next data science interview, check out our machine learning courses curated by our faculty from Stanford University and industry experts. Happy Learning!! 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https://www.thestudentroom.co.uk/showthread.php?page=5&page=5&t=1986770	1,500,850,720,000,000,000	text/html	crawl-data/CC-MAIN-2017-30/segments/1500549424623.68/warc/CC-MAIN-20170723222438-20170724002438-00652.warc.gz	857,020,135	41,434	You are Here: Home >< Maths Stats 2 (S2) OCR (not MEI) 22nd June 2012 Watch 1. (Original post by studentofskhs) Yes . Btw how did you do 8iii)? I done 1-(p(x=0)+p(x=1)+p(x=2)) using s1 binominal rules. Not sure if i'm right though. I got a small value like 0.01 ish I think i did that it was a binomial of (20, 0.05) and then i did greater than or equal to 4 which is 1 - less than or equal to 3. Don't know if that's right though. 1) Random number generator 000 to 999 2 Var(V)=173 3) P(X>=7) = 0.0664 larger than 0.025 ACCEPT H0 4) i) independent ii) 0.114 iii) 0.422 iv) 0.268 5) 1.1153 < 1.645 ACCEPT H0 6) i) 0.0449 ii) 0.0494 7) ii) a=6 (k=1/72) iii) Var(X) = 1.35 8) i) CR >33.1 ii) Type I error iii) 0.0159 , mean not 30 because probability so small iv) mean is 33.58 Any thoughts guys? 3. (Original post by jamib0y) 1) Random number generator 000 to 999 2 Var(Vbar)=173 I think you mean Var(V), not V bar. 3) P(X>=7) = 0.0664 larger than 0.025 ACCEPT H0 4) i) independent ii) 0.114 iii) 0.422 iv) 0.268 5) 1.1153 < 1.645 ACCEPT H0 6) i) 0.0449 ii) 0.0494 7) ii) a=6 (k=1/72) iii) Var(X) = 1.35 8) i) CR >33.1 ii) Type I error iii) 0.0159 iv) mean is 33.58 Any thoughts guys? I've forgotten many answers - those that I remember, I agree with. 4. (Original post by CraigKirk) I've forgotten many answers - those that I remember, I agree with. Yeah, that's what I mean 5. (Original post by jamib0y) 1) Random number generator 000 to 999 2 Var(Vbar)=173 3) P(X>=7) = 0.0664 larger than 0.025 ACCEPT H0 4) i) independent ii) 0.114 iii) 0.422 iv) 0.268 5) 1.1153 < 1.645 ACCEPT H0 6) i) 0.0449 ii) 0.0494 7) ii) a=6 (k=1/72) iii) Var(X) = 1.35 8) i) CR >33.1 ii) Type I error iii) 0.0159 iv) mean is 33.58 Any thoughts guys? From vaguely remembering my answers, i think it all looks good. 6. for 8)iii), Does anyone know for sure whether you're supposed to put that the mean is 30 or not 30? 7. (Original post by Bwob) for 8)iii), Does anyone know for sure whether you're supposed to put that the mean is 30 or not 30? Yeah, I put down it's not 30 because the probability is so small 8. What did people say for 7 i)? I don't think I expressed myself very well. It's generally the questions with words involved I didn't like - again 8iii) I got 0.0155 or something, but I said since the probability is small, it probably is 30, which goes against what most people have been saying :/ 9. (Original post by Bb King) What did people say for 7 i)? I don't think I expressed myself very well. It's generally the questions with words involved I didn't like - again 8iii) I got 0.0155 or something, but I said since the probability is small, it probably is 30, which goes against what most people have been saying :/ Graph of , starting from 0 and ending at a. I commented that as X takes ranges closer to a, the probability of X increases. If ranges of X fall below 0 or above a, the probability of X is zero. 10. (Original post by JongKey) From vaguely remembering my answers, i think it all looks good. im so sillyy!! i tried to find the critcial region for question 3 and found x-1 =7 and being me said therfore x=7 and rejected h0 :/ . 3 markss gonee. sigh. think i got 68 or 67 from the looks of thingss.. but ive probs made some very silly mistakes :/ 11. Can you lose marks for giving a greater degree of accuracy than 3 significant figures? 12. (Original post by Matthew692692) Can you lose marks for giving a greater degree of accuracy than 3 significant figures? In general, no, as long as your answer rounds to the answer given in the mark scheme, or falls in a range given in the mark scheme. 13. (Original post by CraigKirk) In general, no, as long as your answer rounds to the answer given in the mark scheme, or falls in a range given in the mark scheme. Thank god for that, I usually give them to more :P 14. (Original post by Matthew692692) Thank god for that, I usually give them to more :P Yeah, I gave many of mine today to 4sfs, as in statistics mark schemes there often tend to be four decimal places in the answer. 15. (Original post by jamib0y) 1) Random number generator 000 to 999 2 Var(Vbar)=173 3) P(X>=7) = 0.0664 larger than 0.025 ACCEPT H0 4) i) independent ii) 0.114 iii) 0.422 iv) 0.268 5) 1.1153 < 1.645 ACCEPT H0 6) i) 0.0449 ii) 0.0494 7) ii) a=6 (k=1/72) iii) Var(X) = 1.35 8) i) CR >33.1 ii) Type I error iii) 0.0159 , mean not 30 because probability so small iv) mean is 33.58 Any thoughts guys? Everything correct, well done. Bits you've missed out: 1. Need to mention 15 unique numbers. 2. (i) should be Var(V), but you wouldn't lose a mark (ii) "We don't know the distribution of V"; "n>30 so CLT applies" 6. (i) Normal approximation is suitable as np and nq both exceed 5. 7. (i) positive quadratic graph between x=0 and x=a going through (0,0) and not extending below x=0 or above x=a. Then something along the lines of "X can take any value between 0 & a (but not including 0). X is more likely to take values closer to a than to 0." 16. (Original post by Mr M (jr)) Everything correct, well done. Bits you've missed out: 1. Need to mention 15 unique numbers. 2. (i) should be Var(V), but you wouldn't lose a mark (ii) "We don't know the distribution of V"; "n>30 so CLT applies" 6. (i) Normal approximation is suitable as np and nq both exceed 5. 7. (i) positive quadratic graph between x=0 and x=a going through (0,0) and not extending below x=0 or above x=a. Then something along the lines of "X can take any value between 0 & a (but not including 0). X is more likely to take values closer to a than to 0." Surely question 3 - it was a 1 tail test as it said 'Test at the 5% sig level that the study is correct'. I know it doesn't affect the overall result (I checked this) but you lose 2 marks for doing it as a 2 tail rather than 1 tail? 17. (Original post by Mr M (jr)) Everything correct, well done. Bits you've missed out: 1. Need to mention 15 unique numbers. 2. (i) should be Var(V), but you wouldn't lose a mark (ii) "We don't know the distribution of V"; "n>30 so CLT applies" 6. (i) Normal approximation is suitable as np and nq both exceed 5. 7. (i) positive quadratic graph between x=0 and x=a going through (0,0) and not extending below x=0 or above x=a. Then something along the lines of "X can take any value between 0 & a (but not including 0). X is more likely to take values closer to a than to 0." Yeah, thanks, I just didn't have the time to write all the above but got it all! Thanks! 18. (Original post by CraigKirk) Yeah, I gave many of mine today to 4sfs, as in statistics mark schemes there often tend to be four decimal places in the answer. Annoyingly for some reason I gave the final one to 3sfs though :/ I NEVER do that in stats!! 19. FFS, couldn't have gotten an easier paper and I still managed to muck it up; lost a good 10 marks at the least. Well, at least it'll contribute towards bringing the grade boundaries down (by whatever negligible value) for all of you concerned. 20. (Original post by Matthew692692) Annoyingly for some reason I gave the final one to 3sfs though :/ I NEVER do that in stats!! I nearly did that too, managed to see that right at the end when I was checking :P Updated: August 14, 2012 TSR Support Team We have a brilliant team of more than 60 Support Team members looking after discussions on The Student Room, helping to make it a fun, safe and useful place to hang out. This forum is supported by: Today on TSR Stuck for things to do this summer? 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http://www.chegg.com/homework-help/introduction-to-programming-with-c-plus-plus-2nd-edition-chapter-4-solutions-9780136097204	1,441,379,140,000,000,000	text/html	crawl-data/CC-MAIN-2015-35/segments/1440645353863.81/warc/CC-MAIN-20150827031553-00282-ip-10-171-96-226.ec2.internal.warc.gz	354,072,168	15,624	View more editions Introduction to Programming with C plus plus # TEXTBOOK SOLUTIONS FOR Introduction to Programming with C plus plus 2nd Edition • 660 step-by-step solutions • Solved by publishers, professors & experts • iOS, Android, & web Over 90% of students who use Chegg Study report better grades. 2013 Chegg Homework Help Survey SAMPLE SOLUTION Chapter: Problem: • Step 1 of 3 Count and compute the sum and average of numbers Program plan: • Check whether the integer numbers are positive or negative. • If the integer number is positive, increment the positive number count; otherwise, increment the negative number count. • Calculate the sum of numbers. • Calculate the average of numbers. • Display the positive number count, the negative number count, the sum of numbers, and the average of numbers. • Step 2 of 3 Program: /********************************************************** This program demonstrates how to determine the number of positive and negative numbers entered, and also to compute the sum and average of numbers. *********************************************************/ #include using namespace std; //Function main int main() { //Declare the variables int number; int sum=0; int count=0; int positive=0,negative=0; double ave=0; cout<<"Counting the positive and negative numbers and computing the average of numbers:"<<endl<<endl; Read the input value for ‘number’. cout<<"Enter the integer value, the program exits if the input is 0: "; cin>>number; This part of the coding is used to count the positive and negative numbers, and to calculate the sum and average of numbers. //Loop executes until the condition leads to false while(number!=0) { Check whether the number is positive or negative. /The if condition checks whether the number is greater than 0. If yes, then it increments the positive number count; otherwise, it increments the negative number count.*/ if(number>0) { positive++; } else { negative++; } Calculate the sum of numbers. sum+=number; Count the total number of inputs. //Increment the count variable count++; cin>>number; } Calculate the average of numbers. //Calculate the average ave=(double)sum/count; Display the results. //Print the result cout<<"Total numbers of inputs: "<<count<<endl; cout<<"The number of positives is: "<<positive<<endl; cout<<"The number of negatives is: "<<negative<<endl; cout<<"The sum is: "<<sum<<endl; cout<<"The average is: "<<ave<<endl; //Pause the system system("pause"); //Return the value return 0; } • Step 3 of 3 Output: Counting the positive and negative numbers, and computing the average of numbers: Enter the integer value, the program exits if the input is 0: 1 –5 8 –2 4 6 3 0 Total numbers of inputs: 7 The number of positives is: 5 The number of negatives is: 2 The sum is: 15 The average is: 2.14286 Corresponding Textbook Introduction to Programming with C plus plus \| 2nd Edition 9780136097204ISBN-13: 0136097200ISBN: Y Daniel LiangAuthors:	697	3,035	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	3.734375	4	CC-MAIN-2015-35	latest	en	0.686401	[ 128000, 860, 810, 47634, 198, 38255, 311, 39524, 449, 356, 5636, 5636, 271, 2, 16139, 37725, 98997, 50, 4716, 29438, 311, 39524, 449, 356, 5636, 5636, 220, 17, 303, 14398, 271, 6806, 220, 19274, 3094, 14656, 30308, 10105, 198, 6806, 328, 8905, 555, 36717, 11, 45724, 612, 11909, 198, 6806, 16433, 11, 8682, 11, 612, 3566, 198, 1959, 220, 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http://energocomplex.com/dln21/orthocenter-definition-geometry-5d75ac	1,627,930,802,000,000,000	text/html	crawl-data/CC-MAIN-2021-31/segments/1627046154356.39/warc/CC-MAIN-20210802172339-20210802202339-00709.warc.gz	15,748,715	8,002	orthocenter synonyms, orthocenter pronunciation, orthocenter translation, English dictionary definition of orthocenter. * The three heights (altitudes) of a triangle intersect at one point (are concurrent at a point), called the orthocentre of the triangle. Altitude of a Triangle, Definition & Example, Finding The Orthocenter, Acute Right & Obtuse Triangle - Duration: 11:15. It only takes a minute to sign up. translation and definition "orthocenter", English-Japanese Dictionary online. Here are three important theorems involving centroid, orthocenter, and circumcenter of a triangle. This is part of the series of posts on theorems in secondary school geometry.Proofs of the theorems and application problems will be provided in the next few posts. The line segment needs to intersect point C and form a right angle (90 degrees) with the "suporting line" of the side AB.Definition of "supporting line: The supporting line of a certain segment is the line The orthocenter is one of the triangle's points of concurrency formed by the intersection of the triangle's 3 altitudes.. Orthocenter Calculator is a free online tool that displays the intersection of the three altitudes of a triangle. This geometry video tutorial explains how to identify the location of the incenter, circumcenter, orthocenter and centroid of a triangle. This line containing the opposite side is called the extended base of the altitude. An altitude is the line perpendicular from a base that passes through the opposite vertex. The center of the incircle is called the incenter, and the radius of the circle is called the inradius.. The triangle is one of the most basic geometric shapes. Orthocenter of a Triangle In geometry, we learn about different shapes and figures. Definition of the Orthocenter of a Triangle. Orthocenter of a Triangle. Find the coordinates ofthe orthocenter of this triangle. Word of the day. This is a matter of real wonderment that the fact of the concurrency of altitudes is not mentioned in either Euclid's Elements or subsequent writings of the Greek scholars. Origin. Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. To download free study materials like NCERT Solutions, Revision Notes, Sample Papers and Board Papers to help you to score more marks in your exams. Step 1 : Incircle. In geometry, an altitude of a triangle is a line segment through a vertex and perpendicular to (i.e., forming a right angle with) a line containing the base (the side opposite the vertex). The orthocenter of a triangle is the point where the perpendicular drawn from the vertices to the opposite sides of the triangle intersect each other. Segment from a vertex that is perpendicular to the opposite side or line containing the opp. Orthocenter of a Triangle (Definition, How to Find, Video, & Examples) The orthocenter of a triangle, or the intersection of the triangle's altitudes, is not something that comes up in casual conversation. Mid 19th century. Define orthocenter. The timing of the first proof is still an open question; it is believed, though, that even the great Gauss saw it necessary to prove the fact. Here $$\text{OA = OB = OC}$$, these are the radii of the circle. Dealing with orthocenters, be on high alert, since we're dealing with coordinate graphing, algebra, and geometry, all … 2. Concurrent Math with the definitions A. I.e., the three heights of a triangle are cut in the orthocenter. It symbolizes from the capital letter H letter. Where all three lines intersect is the "orthocenter": Let's learn these one by one. forming a right angle with) a line containing the base (the opposite side of the triangle). The Organic Chemistry Tutor 4,974 Chemistry. Orthocenter doesn’t need to lie inside the triangle only, in case of an obtuse triangle, it lies outside of the triangle. Ruler. ... Deriving the barycentric coordinates of a triangle's orthocenter, using the areal definition of such coordinates. In right triangle, orthocenter is located on the triangle. Note: The orthocenter's existence is a trivial consequence of the trigonometric version Ceva's Theorem; however, the following proof, due to Leonhard Euler, is much more clever, illuminating and insightful.. Orthocenter Formula - Learn how to calculate the orthocenter of a triangle by using orthocenter formula prepared by expert teachers at Vedantu.com. See definitions & examples. The circumcenter of a triangle is the center of a circle which circumscribes the triangle.. (US orthocenter) Pronunciation ... sɛntə/ noun Geometry . are A (0, 0), N (6, 0), and D (–2, 8). Constructing Orthocenter of a Triangle - Steps. These three altitudes are always concurrent.In other, the three altitudes all must intersect at a single point , and we call this point the orthocenter of the triangle. It has several important properties and relations with other parts of the triangle, including its circumcenter, incenter, area, and more. In geometry, an altitude of a triangle is a straight line through a vertex and perpendicular to (i.e. Altitude in a triangle is a bisector lines for 3 angles in a triangle. An incircle is an inscribed circle of a polygon, i.e., a circle that is tangent to each of the polygon's sides. The Centroid is the point of concurrency of the medians of a triangle. Circumcenter. Three altitudes intersecting at the orthocenter. An altitude is the perpendicular segment from a vertex to its opposite side. Geometry Dictionary. Orthocenter definition is - the common intersection of the three altitudes of a triangle or their extensions or of the several altitudes of a polyhedron provided these latter exist and meet in a point. A geometrical figure is a predefined shape with certain properties specifically defined for that particular shape. A key part of learning is adding to your vocabulary. Orthocenter Draw a line segment (called the "altitude") at right angles to a side that goes to the opposite corner. razoo / rɑːˈzuː / noun. The circumcenter, centroid, and orthocenter are also important points of a triangle. Orthocenter Definition #In the diagram, O=Orthocenter(A,B,C) # The intersection of the three altitudes of the vertices # of a triangle whose vertices are Points A, B, C # hence the intersection of any two of them, its existence is proved by the OrthocenterExists Theorem. Compass. In obtuse triangle, orthocenter is … Concurrency is an excellent word to learn in geometry. The orthocenter of an acute triangle. Circumcenter definition, the center of a circumscribed circle; that point where any two perpendicular bisectors of the sides of a polygon inscribed in the circle intersect. BYJU’S online orthocenter calculator tool makes the calculation faster and it displays the orthocenter of a triangle in a fraction of seconds. Let's build the orthocenter of the ABC triangle in the next app. From ortho- + centre. To construct orthocenter of a triangle, we must need the following instruments. 1. The orthocenter is a term that is used exclusively within the scope of the geometry and refers to that point of intersection where converge the three altitudes of a triangle. orthocenter ( plural orthocenters ) ( geometry ) The intersection of the three lines that can be drawn flowing from the three corners of a triangle to a point along the opposite side where each line intersects that side at a 90 degree angle; in an acute triangle , it is inside the triangle; in an obtuse triangle , it is outside the triangle. Incenter: Where a triangle’s three angle bisectors intersect (an angle bisector is a ray that cuts an angle in half); the incenter is the center of a circle inscribed in (drawn inside) the triangle. Now, let us see how to construct the orthocenter of a triangle. In mathematics, it means a point shared by three or more lines. Circumcenter 3. The Orthocenter is the point of concurrency of the altitudes, or heights, as they are commonly called. The intersection of the extended base and the altitude is called the foot of the altitude. The orthocenter of a triangle is the intersection of the triangle's three altitudes. The common point of the perpendicular bisectors of a triangle B. Orthocenter It is the point where the three "altitudes" of a triangle meet and the orthocenter can be inside or outside of the triangle. Perpendicular Bisector of a Triangle 2. Proof of Existence. 1. This lesson focuses on points of concurrency in triangles. The point of intersection of the three perpendiculars drawn from the vertices of a triangle to the opposite sides. Orthocenter : Orthocenter is an intersection point of 3 altitudes of a triangle. Video Definition Centroid Incenter Circumcenter Orthocenter Facts. If four points form an orthocentric system, then each of the four points is the orthocenter of the other three. Orthocenter 4. See more. Ask … Geometry dictionary is the place where we can find meaning, definition and explanation etc for the geometric terms which are being often used by the students.Most of the students find it difficult to understand some geometric terms when they do theorems and problems on Geometry. Answers and explanations (–8, –6) The orthocenter of a triangle is the point where the three altitudes of the triangle intersect. An altitude of a triangle is perpendicular to the opposite side. In geometry, an orthocentric system is a set of four points on a plane, one of which is the orthocenter of the triangle formed by the other three.. n. The point of intersection of the three altitudes of a triangle. An "altitude" is nothing but a line that goes through a vertex (corner point) and is at right angles to the opposite side. In acute triangle, orthocenter is located inside the triangle. Construct triangle ABC whose sides are AB = 6 cm, BC = 4 cm and AC = 5.5 cm and locate its orthocenter. 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http://www.math.ucdenver.edu/~aengau/math3000/jeopardy.html	1,539,903,913,000,000,000	text/html	crawl-data/CC-MAIN-2018-43/segments/1539583512015.74/warc/CC-MAIN-20181018214747-20181019000247-00002.warc.gz	495,215,958	28,299	Sets & Cardinalities Mathematical Logic Relations & Functions Proof Techniques Results & Conjectures ### 100 What is the empty (null/void) set? The only set with zero cardinality. ### 100 What is a statement? A declarative sentence that is either true or false. ### 100 What is bijective? A relation and its inverse are functions if and only if they satisfy this property. ### 100 What is a counterexample? To disprove a statement, it suffices to give this. ### 100 What is a lemma? A mathematical result that is used as an auxiliary result, often to prove a theorem. ### 200 What is the set of rational numbers? Every element in this set can be written as the quotient of an integer and a natural number. ### 200 What it is a disjunction? The combination of two statements using a logical OR. ### 200 What is an equivalence relation? A relation that is reflexive, symmetric, and transitive. ### 200 What is QED (quod erat demonstrandum)? The English translation of this Latin abbreviation is "what had to be shown". ### 200 What is the Fundamental Theorem of Arithmetic? Every integer greater than 1 can be written as a unique product of prime numbers. ### 300 What are pairwise disjoint sets? A collection of nonempty subsets of a set A is called a partition of A if its elements contain all elements of A and satisfy this property. ### 300 What is the universal quantifier (∀)? This quantifier means FOR ALL. ### 300 What is surjective (onto)? If a function's range equals its codomain, then it has this property. ### 300 What is a proof by contradiction? To show that P ⇒ Q, suppose that P is true and Q is false. ### 300 What is Goldbach's Conjecture? Every even integer greater than 2 can be written as the sum of two prime numbers. ### 400 What are the integers modulo n? Given a positive integer n, this set contains all equivalence classes of integers that have the same remainder upon division by n. ### 400 What is a necessary condition? A requirement without which a conclusion is always false. ### 400 What is the identity? The composition of a bijective function with its own inverse. ### 400 What is without loss of generality (wlog)? A commmon justification to shorten a proof if different cases are identical or very similar. ### 400 What is Euclid's Lemma? Let m and n be integers, and p be a prime number: p \| mn ⇒ p \| m ∨ p \| n ### 500 What is aleph null? The cardinal number of each denumerable set. ### 500 What is the Modus Ponens? P ∧ (P ⇒ Q) ⇒ Q ### 500 What is a dictionary (alphabetical/lexicographic) order ? eight five four nine one seven six three two When sorting the numbers from 1 to 9, 1 < 2 but 2 > 3 > 4 > 5, 5 < 6 but 6 > 7 > 8, 8 < 9 but 9 < 1. ### 500 What is the Principle of Mathematical Induction? ( ∀ n ≥ n0: P(n) ) ≡ ( P(n0) ∧ ∀ k ≥ n0: P(k) ⇒ P(k+1) ) ### 500 What is the Schröder-Bernstein Theorem? If the cardinal number of a set A is both less-than-or-equal and greater-than-or-equal than the cardinal number of a set B, then A and B are numerically equivalent. # Math 3000 Abstract Math Jeopardy Press F11 for full-screen mode	814	3,128	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	3.75	4	CC-MAIN-2018-43	latest	en	0.912948	[ 128000, 31275, 612, 47014, 1385, 92102, 37201, 32467, 612, 24460, 38091, 66044, 18591, 612, 1221, 585, 1439, 271, 14711, 220, 1041, 271, 3923, 374, 279, 4384, 320, 2994, 14, 1019, 8, 743, 5380, 791, 1193, 743, 449, 7315, 56980, 488, 382, 14711, 220, 1041, 271, 3923, 374, 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https://brilliant.org/discussions/thread/prove-displaystyle-int_01-fraclogxlog1/	1,511,401,376,000,000,000	text/html	crawl-data/CC-MAIN-2017-47/segments/1510934806715.73/warc/CC-MAIN-20171123012207-20171123032207-00354.warc.gz	575,887,527	23,204	× # Prove this closed form of $$\displaystyle \int_0^1\frac{(\ln(1+x))^2\ln(x)\ln(1-x)}{1-x} \, dx$$ $\int_0^1\frac{(\ln(1+x))^2\ln(x)\ln(1-x)}{1-x} \, dx=\dfrac{7}{2}\zeta(3){(\ln 2)^2}-\dfrac{\pi^2}{6}{(\ln 2)^3}-\dfrac{\pi^2}{2}\zeta(3)+{6}\zeta(5)-\dfrac{\pi^4}{48}\ln2$ Prove that the equation above is true. Notation: $$\zeta(\cdot)$$ denotes the Riemann Zeta function. This was found in another mathematics form and it was unanswered there. This is a part of the set Formidable Series and Integrals. 1 year, 9 months ago MarkdownAppears as italics or _italics_ italics bold or __bold__ bold - bulleted- list • bulleted • list 1. numbered2. list 1. numbered 2. list Note: you must add a full line of space before and after lists for them to show up correctly paragraph 1paragraph 2 paragraph 1 paragraph 2 [example link](https://brilliant.org)example link > This is a quote This is a quote # I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world" # I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world" MathAppears as Remember to wrap math in $$...$$ or $...$ to ensure proper formatting. 2 \times 3 $$2 \times 3$$ 2^{34} $$2^{34}$$ a_{i-1} $$a_{i-1}$$ \frac{2}{3} $$\frac{2}{3}$$ \sqrt{2} $$\sqrt{2}$$ \sum_{i=1}^3 $$\sum_{i=1}^3$$ \sin \theta $$\sin \theta$$ \boxed{123} $$\boxed{123}$$ Sort by: Use this expansion, $\large \ln ^2 (1+x) = \sum_{r=0}^{\infty} \dfrac{H_{r} (-1)^r x^{r+1}}{r+1}$ Then it will be left to evaluate the integral $\large \int_{0}^{1} x^{r+1} \dfrac{\ln(x) \ln(1-x)}{1-x} dx$ This is derivative of beta function. - 1 year, 9 months ago How will you evaluate the resulting summation? - 1 year, 9 months ago wow nice one! Do you know how to prove it? - 1 year, 9 months ago Comment deleted Feb 29, 2016 I'll not give up on that problem. I have to solve it first. - 1 year, 9 months ago Integrate the generating function of harmonic number. - 1 year, 9 months ago I made it on my own and i dont know whether it is already there or not - 1 year, 9 months ago - 1 year, 9 months ago Now @Ishan Singh can solve this.. Integral can be written as: $\displaystyle \sum_{r,s,t\geq 0} \frac{H_r(-1)^r}{(r+1)(t+1)(r+s+t+3)^2}$ - 1 year, 8 months ago @Pi Han Goh add it to you set - 1 year, 9 months ago - 1 year, 9 months ago Thanks. Do try it. - 1 year, 9 months ago Hint : Convert into derivative of Beta function. - 1 year, 9 months ago Yup I did that but at some point, I have to take natural logarithm of (-1). Which is imaginary, but the closed form ain't contain any imaginary term. - 1 year, 9 months ago No. Take limit. For example this - 1 year, 9 months ago can u clearly explain how u applied the limit there? - 1 year, 9 months ago That will take $$2 -3$$ pages. I may post it when I'm free. - 1 year, 8 months ago You have to do something with the $$\ln^2 (1+x)$$ term and afterwards it gets converted into the limit. Another method is to use generating function of Harmonic numbers. - 1 year, 9 months ago I got imaginary term while using that Harmonic relation. You are referring this $$\sum_{n=1}^{\infty} H_{n} x^{n} = \dfrac{-ln(1-x)}{1-x}$$ no? - 1 year, 9 months ago Yes. I'm referring to that generating function. - 1 year, 9 months ago Kk I'll give another shot at Feynman's way. - 1 year, 9 months ago @Mark Hennings could you help us out? - 1 year, 9 months ago I got it. 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https://www.studypug.com/ca/en/math/grade10/cubic-and-cube-roots	1,485,278,096,000,000,000	text/html	crawl-data/CC-MAIN-2017-04/segments/1484560285001.96/warc/CC-MAIN-20170116095125-00377-ip-10-171-10-70.ec2.internal.warc.gz	974,293,923	12,167	# Cubic and cube roots It was said that he radical sign was first used by Rene Descartes, but long before he used this, it has been developed in Germany in the 1500s. The radical ( sign was invented to find the square root or the cube root, 4th root, 5th root etc., depending on the index of the radical sign. Now if you remember, in previous chapter, we learned all about the different number systems and we have mentioned about the rational and irrational numbers. Radicals can belong to either of which depending of course if the radicals would be able to fit into the definition of whichever group of number. In this lesson we will get to be able to identify whether or not a certain radical is rational. We will be specifically going to study about a square root or a cube root of a particular number. We will first look into what a square and a cube are since these would be the two fundamental concepts. In order to solve a square root, or a cube root, one must know how to square or cube a number. These will be thoroughly discussed in 4.1 and 4.2. For 4.3 to 4.8 we will be looking closer into how to simplify and evaluate these radicals through applying what we have learned from chapters on square and square roots, estimating square roots, and rational numbers from earlier grades. We would also be learning how to convert them into either entire radicals to mixed radicals or vice versa, by applying our knowledge of factoring out the perfect squares or the perfect cubes in the equation. Apart from converting the radicals from one form to another we are also going to learn more about how to combine different radicals and simplify them depending on the operation used and learn to rationalize a certain radical equation. ### Cubic and cube roots Whenever we see “roots”, let it be cubic roots or square roots, we know for sure that we will need to do prime factorization to find out the prime factors of the numbers. In this section, we use factors and multiples to find perfect cube whole numbers and cubic roots.	429	2,036	{"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 28, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	4.59375	5	CC-MAIN-2017-04	latest	en	0.953823	[ 128000, 2, 18521, 292, 323, 24671, 20282, 271, 2181, 574, 1071, 430, 568, 18336, 1879, 574, 1176, 1511, 555, 67527, 32285, 472, 288, 11, 719, 1317, 1603, 568, 1511, 420, 11, 433, 706, 1027, 8040, 304, 10057, 304, 279, 220, 3965, 15, 82, 13, 578, 18336, 320, 1879, 574, 36592, 311, 1505, 279, 9518, 3789, 477, 279, 24671, 3789, 11, 220, 19, 339, 3789, 11, 220, 20, 339, 3789, 5099, 2637, 11911, 389, 279, 1963, 315, 279, 18336, 1879, 382, 7184, 422, 499, 6227, 11, 304, 3766, 12735, 11, 584, 9687, 682, 922, 279, 2204, 1396, 6067, 323, 584, 617, 9932, 922, 279, 25442, 323, 61754, 5219, 13, 21254, 53703, 649, 9352, 311, 3060, 315, 902, 11911, 315, 3388, 422, 279, 74356, 1053, 387, 3025, 311, 5052, 1139, 279, 7419, 315, 54784, 1912, 315, 1396, 13, 763, 420, 18228, 584, 690, 636, 311, 387, 3025, 311, 10765, 3508, 477, 539, 264, 3738, 18336, 374, 25442, 13, 1226, 690, 387, 11951, 2133, 311, 4007, 922, 264, 9518, 3789, 477, 264, 24671, 3789, 315, 264, 4040, 1396, 382, 1687, 690, 1176, 1427, 1139, 1148, 264, 9518, 323, 264, 24671, 527, 2533, 1521, 1053, 387, 279, 1403, 16188, 19476, 13, 763, 2015, 311, 11886, 264, 9518, 3789, 11, 477, 264, 24671, 3789, 11, 832, 2011, 1440, 1268, 311, 9518, 477, 24671, 264, 1396, 13, 4314, 690, 387, 27461, 14407, 304, 220, 19, 13, 16, 323, 220, 19, 13, 17, 382, 2520, 220, 19, 13, 18, 311, 220, 19, 13, 23, 584, 690, 387, 3411, 12401, 1139, 1268, 311, 40821, 323, 15806, 1521, 74356, 1555, 19486, 1148, 584, 617, 9687, 505, 30732, 389, 9518, 323, 9518, 20282, 11, 77472, 9518, 20282, 11, 323, 25442, 5219, 505, 6931, 28711, 13, 1226, 1053, 1101, 387, 6975, 1268, 311, 5625, 1124, 1139, 3060, 4553, 74356, 311, 9709, 74356, 477, 17192, 46391, 11, 555, 19486, 1057, 6677, 315, 2144, 5620, 704, 279, 4832, 32440, 477, 279, 4832, 55204, 304, 279, 24524, 13, 35802, 505, 34537, 279, 74356, 505, 832, 1376, 311, 2500, 584, 527, 1101, 2133, 311, 4048, 810, 922, 1268, 311, 16343, 2204, 74356, 323, 40821, 1124, 11911, 389, 279, 5784, 1511, 323, 4048, 311, 25442, 553, 264, 3738, 18336, 24524, 382, 14711, 18521, 292, 323, 24671, 20282, 271, 65361, 584, 1518, 1054, 38265, 9520, 1095, 433, 387, 41999, 20282, 477, 9518, 20282, 11, 584, 1440, 369, 2771, 430, 584, 690, 1205, 311, 656, 10461, 8331, 2065, 311, 1505, 704, 279, 10461, 9547, 315, 279, 5219, 13, 763, 420, 3857, 11, 584, 1005, 9547, 323, 66160, 311, 1505, 4832, 24671, 4459, 5219, 323, 41999, 20282, 13, 128001 ]	[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ]
http://users.humboldt.edu/flashman/Presentations/MD.ICTCM.CV.3_12_16.I.html	1,544,457,549,000,000,000	text/html	crawl-data/CC-MAIN-2018-51/segments/1544376823348.23/warc/CC-MAIN-20181210144632-20181210170132-00571.warc.gz	328,920,435	128,972	28th International Conference on Technology in Collegiate Mathematics 2016 ¤ March 12, 2016 Mapping Diagrams for Complex Variable Functions Visualized Dynamically with GeoGebra 1:30 p.m. - 2:00 p.m. Part I Mapping Diagrams for Real Functions Complex Arithmetic Part II Complex Functions Part III Calculus for Complex Functions Martin Flashman Professor of Mathematics Humboldt State University http://users.humboldt.edu/flashman/Presentations/MD.ICTCM.CV.3_12_16.html Abstract: Mapping diagrams provide a valuable tool for visualizing complex variable functions. Using GeoGebra, demonstrated dynamic diagrams will allow users to see important properties of complex functions. Examples include complex arithmetic, key functions, and calculus concepts. Background and References to other work on Mapping Diagrams for real variables ¤ 1.1.Background: Mapping Diagrams for Real Functions 1.1.1 Real Functions. ¤ Understanding Real Functions: Tables, Mapping Diagrams, and Graphs 1.2. Real Linear Functions. ¤ Real Linear functions are the key to understanding calculus. Linear functions are traditionally expressed by an equation like : $f(x)= mx + b$. Mapping diagrams for real linear functions have one simple unifying feature- the focus point, determined by the numbers $m$ and $b$, denoted here by $[m,b]$ . Mapping Diagrams and Graphs of Real Linear Functions Visualizing real linear functions using mapping diagrams and graphs. This is a Java Applet created using GeoGebra from www.geogebra.org - it looks like you don't have Java installed, please go to www.java.com Notice how points on the graph pair with arrows and points on the mapping diagram. 2.1 A Review of Complex Numbers and Arithmetic: $\mathbb{C}$¤ Definition: The "number" $i$: To solve the equation $x^2 +1 =0$ we create a symbol $i$ and declare that $i^2 = -1$ so $i$ solves the equation and we write $$i = \sqrt{ -1 }.$$ Definitions: A complex number $z$ is a number that can be expressed in the form $z =a +bi$ where $a$ and $b$ are real numbers. If $z =a +bi$ then $a$ is called the real part of $z$ and $b$ (and sometime $bi$) is called the imaginary part of $z$. Examples: $3 + 2i$ and $-2 - i$. 2.2 Complex Geometry- The Complex Plane Complex numbers are identified with points in a cartesian plane by having $a +bi$ identified with the point with coordinates $(a,b)$ or with position vectors by identifying $a+bi$ with the vector $$. With this identification i is identified with the point (0,1) and -i is identified with (0,-1). 2.2.1 Complex Number Norm (Magnitude): The norm of z is defined by \|z\| = \|a+bi\| = \sqrt{ a^2 +b^2} 2.2.2 Polar Representation of z: ¤ Using trigonometry we have the identification:$$z = \|z\| \cos( \theta) + \|z\| \sin(\theta) i = \|z\| [\cos( \theta) + \sin(\theta) i] = \|z\| cis( \theta) $$where a = \cos( \theta ), b = \sin(\theta). The angle \theta determined by z can be measured in degrees or radians and restricted to be in a specific interval. For example \theta \in [0,2 \pi) or \theta \in (-\pi, \pi]. Thus the angle can be considered a function of z, called the argument of z: Arg(z) =\theta. 2.2.3 Exponential Representation of z: ¤ Consider the Taylor Series for the functions:$$e^x = 1 + x + \frac{x^2}2 +\frac{x^3}{3!} + \frac{x^4}{4!} + \frac{x^5}{5!} + \frac{x^6}{6!} + \frac{x^7}{7!} ...\cos(x) = 1 - \frac{x^2}2 + \frac{x^4}{4!} - \frac{x^6}{6!} + ...\sin(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} ...$$Then using \theta in radian measure:$$e^{i\theta}= \cos(\theta) + \sin(\theta)i = cis (\theta)$$and$$z =\|z\|e^{i \theta}$$. Note: When \theta =\pi this equation demonstrates that e^{\pi i} = \cos(\pi) + sin(\pi)i = -1. 2.3 Complex Arithmetic 2.3.1 Complex Addition: z_1+z_2 ¤ If z_1 = a_1 + b_1i and z_2 = a_2 + b_2i then$$z_1+z_2 =a_1 + a_2 + (b_1+ b_2) i Example: If $z_1 = 2+3i; z_2= 1 - i$ then $z_1+z_2= (2+1) + (3-1) i = 3 + 2i$. The addition can be thought of as vector addition - with separate addition of the real and imaginary parts. Addition can be visualized in the complex plane by using parallelograms. 2.3.2 Complex Multiplication: $z_1 \cdot z_2$ ¤ Algebraically:If $z_1 = 2+3i; z_2= 1 - i$ then $z_1 \cdot z_2= (2+3i) \cdot (1-i) = 2 \cdot 1 - 3\cdot i^2 + 3i \cdot 1 +2 \cdot (-i) = (2+3) +(3- 2)i = 5+i$. This is easier to understand complex multiplication geometrically using the polar or the exponential representation (in radian measure): $z_1=\|z_1\| [\cos( \theta_1) + \sin(\theta_1) i] = \|z_1\|e^{i\theta_1} ;\ z_2=\cdot \|z_1\| [\cos( \theta_2) + \sin(\theta_2) i] = \|z_2\|e^{i\theta_2}$ $z_1 \cdot z_2= \|z_1\| [\cos( \theta_1) + \sin(\theta_1) i] \cdot \|z_1\| [\cos( \theta_2) + \sin(\theta_2) i]$ $= \|z_1\|\cdot \|z_1\| [\cos( \theta_1) + \sin(\theta_1) i] \cdot [\cos( \theta_2) + \sin(\theta_2) i]$ $= \|z_1\|\cdot \|z_1\| [\cos( \theta_1)\cos( \theta_2) - \sin(\theta_1)\sin(\theta_2) + (\sin(\theta_1)\cos( \theta_2) + \sin(\theta_2) \cos(\theta_1) )i]$ $=\|z_1\|\cdot \|z_1\| [\cos( \theta_1 + \theta_2) + \sin(\theta_1 +\theta_2) i]$ $=\|z_1\|\cdot \|z_1\| cis (\theta_1 +\theta_2)$ or more simply using $\theta$ in radian measure: $z_1 \cdot z_2= \|z_1\|e^{i\theta_1}\cdot \|z_2\|e^{i\theta_2}$ $=\|z_1\|\cdot \|z_1\| e^{(\theta_1 +\theta_2)i}$ Example: If $z_1 = \sqrt 2\ cis(45°); z_2= 2\ cis(30°)$ then $z_1\cdot z_2= 2\sqrt 2\ cis (75 °)$.¤ 2.3.3 Complex Inverses: $\frac1z, z \ne 0$ ¤ Example: Express $\frac 1{2+3i}$ in the standard form of $a + bi$. Solution: $\frac 1{2+i} =\frac 1{2+i} \cdot \frac {2-i}{2-i} = \frac{2-i}{4+1} = \frac 2{5} - \frac 3{5}i$. Fact: If $a^2+b^2 \ne 0$ then $\frac1{a+bi} = \frac a{a^2 + b^2} - \frac b{a^2 + b^2}i$ 2.3.4 Geometry for $\frac1z, z \ne 0$ Geometric understanding of $\frac1z, z \ne 0$ comes best by using the polar or exponential representation. That is, if $z=\|z\|\ cis(\theta) = \|z\|e^{i\theta}$ then $\frac1z = \frac1{\|z\|} \ cis(-\theta) = \frac1{\|z\|} e^{-i\theta}$ On to Part II Complex Functions References:¤ Mapping Diagrams from A(lgebra) B(asics) to C(alculus) and D(ifferential) E(quation)s. A Reference and Resource Book on Function Visualizations Using Mapping Diagrams The Sensible Calculus Program	2,182	6,190	{"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	4.1875	4	CC-MAIN-2018-51	latest	en	0.794559	[ 128000, 1591, 339, 7327, 15217, 389, 12053, 304, 57665, 6629, 50895, 271, 679, 21, 198, 82638, 198, 28623, 220, 717, 11, 220, 679, 21, 198, 6950, 36361, 82, 369, 22872, 12689, 24460, 198, 9789, 1534, 33593, 2740, 449, 33010, 38, 51313, 198, 16, 25, 966, 281, 749, 13, 482, 220, 17, 25, 410, 281, 749, 382, 3744, 358, 39546, 36361, 82, 369, 8976, 24460, 22872, 94084, 3744, 8105, 22872, 24460, 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https://www.norecipejuststory.com/3369/157-8-lbs-to-kg-how-to-convert-pounds-to-kilograms.html	1,713,129,212,000,000,000	text/html	crawl-data/CC-MAIN-2024-18/segments/1712296816893.9/warc/CC-MAIN-20240414192536-20240414222536-00672.warc.gz	860,623,600	29,502	# 157.8 lbs to kg – How to Convert Pounds to Kilograms Source: bing.com Have you ever encountered the measurement of 157.8 pounds (lbs) and wanted to know its equivalent in kilograms (kg)? As different countries and industries use varying units of measurement, it is common to need to convert between units. In this article, we will discuss the conversion of 157.8 lbs to kg and provide insight into the usage and history of these units of measurement. ## Understanding Pounds and Kilograms Source: bing.com Pounds and kilograms are both units of mass that are used to measure the weight of an object. The pound is a unit of mass commonly used in the United States and a few other countries, whereas the kilogram is the standard measurement of mass in most other countries around the world. A pound is defined as 0.45359237 kilograms, whereas a kilogram is defined as the mass of a particular cylinder of platinum-iridium alloy kept at the International Bureau of Weights and Measures. ## Conversion of 157.8 lbs to kg Source: bing.com To convert 157.8 lbs to kg, we need to multiply the amount of pounds by the conversion factor of 0.45359237. Therefore, 157.8 lbs is equivalent to: 157.8 x 0.45359237 = 71.578 kg This means that 157.8 lbs is equal to 71.578 kg. To convert between pounds and kilograms, simply multiply the amount of pounds by 0.45359237 or divide the amount of kilograms by 2.20462. ## History of Pounds and Kilograms Source: bing.com The pound has a long history and was first used in ancient Rome. Originally, the pound was defined as 12 ounces, with each ounce being the weight of 437.5 grains of barley. This definition has since changed, with the pound now defined as 0.45359237 kg. The kilogram, on the other hand, was first defined in the late 18th century during the French Revolution. The French Academy of Sciences commissioned a group to create a new system of measurement that was based on natural standards. They defined the kilogram as the mass of 1 liter of water at its maximum density, which is 4°C. This definition has since been revised to be the mass of a particular cylinder of platinum-iridium alloy kept at the International Bureau of Weights and Measures. ## Usage of Pounds and Kilograms Source: bing.com As mentioned earlier, pounds are commonly used in the United States, while kilograms are the standard measurement of mass in most other countries. Industries such as agriculture, manufacturing, and automotive use these units of measurement for various purposes such as weighing products, determining shipping costs, and measuring the weight of vehicles. However, with the increasing globalization of trade and the internet, it is becoming more important to be able to convert between units of measurement to communicate effectively. Converting 157.8 lbs to kg is just one example of how the ability to convert between units of measurement is essential in modern life. ## Conclusion Converting 157.8 lbs to kg is a simple process that involves multiplying the amount of pounds by the conversion factor of 0.45359237. However, understanding the history and usage of pounds and kilograms provides insight into the reasons why these units of measurement are still used today. Whether for business, travel, or personal reasons, the ability to convert between units of measurement is a valuable skill that can be used in many different areas of life.	748	3,414	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	3.578125	4	CC-MAIN-2024-18	latest	en	0.949102	[ 128000, 2, 220, 10895, 13, 23, 29160, 311, 21647, 1389, 2650, 311, 7316, 393, 3171, 311, 38988, 56485, 271, 3692, 25, 293, 287, 916, 271, 12389, 499, 3596, 23926, 279, 19179, 315, 220, 10895, 13, 23, 16701, 320, 54044, 8, 323, 4934, 311, 1440, 1202, 13890, 304, 85402, 320, 7501, 12106, 1666, 2204, 5961, 323, 19647, 1005, 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https://magedkamel.com/18-effective-area-for-a-built-up-section	1,713,271,832,000,000,000	text/html	crawl-data/CC-MAIN-2024-18/segments/1712296817095.3/warc/CC-MAIN-20240416124708-20240416154708-00634.warc.gz	349,637,586	68,468	# 18-The Effective Area for a Built-Up Section. ## The Effective Area for a Built-Up Section. This post will check a practice problem for a built-up section, problem no.3-19 quoted from Prof. Mccormack’s book, fifth edition. It is required to compute the effective net area of the built-up section shown in Fig 3-19. The holes are punched for 3/4 inches. The U factor is given as equal to 0.90. ### The gross area for a Built-Up Section. The built-up section consists of two channels of C10x25 and two plates that have a dimension of 1/2×11 inches. The thickness of each plate is 1/2 inch and the length is 11 inches. We will estimate these four elements, since we have two channels we can estimate the gross area for one channel and multiply by two. Similarly, we can find the area of one plate and then multiply it by 2. From the table of areas for channels Table 1-5, we can find that for section C10x25 its area is equal to 7.35 inch^2. For a plate, it is easy to estimate, the area of a rectangle of (1/2″x11″), the area of one plate is equal to the product of (1/2×11) the result is 5.50 inch^2. The areas of the two channels are equal to (2×7.35)=14.70 inch^2. While the area of the two plates equals (25.50)=11.0 inch^2. The sum of areas will give us a total area of 14.70 inch^2, please refer to the next image for more details. The bolts used are of diameter 3/4 inches, to get the hole diameter we add 1/8 inch to the bolt diameter so the diameter of the hole will be equal to 7/8 inches. ### The net area for a Built-Up Section. For the Effective Area for a Built-Up Section, we need to find the net areas for the two channels and the net area for the two plates. 1-for the net area for plates use section (1-1). We have 4 bolts, and the area of these four bolts is equal to the hole diameter by the number of bolts multiplied by the thickness of the plate, this value will be deducted from the gross area of the plates. The same section 1-1, we can use to get the net area of the two channels, again we have four bolts connecting the flanges of the two channels to the upper and lower plates. We estimate the area of the holes by multiplying 4x the diameter of the holes by the thickness of the flange of the C channel. The final net area of the built-up section equals 22.42 square inches. The net area for channels is 13.74 square inches. The net area for the two plates is 9.25 square inches. Please refer to the next slide image for more details. ### The Effective Area for a Built-Up Section. The final value for the effective area for a built-up section will be equal to UAn, U is given as equal to 0.90, then the effective area of a built-up section will be equal to the product of (UxAn) or 0.9x(13.174+9.25)=20.18 inch^2. For staggered connection, please refer to posts 16 and 17 for the details of a solved problem Problem 3-4-3 for MC section-staggered bolts.	757	2,904	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	4.8125	5	CC-MAIN-2024-18	longest	en	0.924616	[ 128000, 2, 220, 972, 66106, 48023, 12299, 369, 264, 34154, 47197, 11360, 382, 567, 578, 48023, 12299, 369, 264, 34154, 47197, 11360, 382, 2028, 1772, 690, 1817, 264, 6725, 3575, 369, 264, 5918, 5352, 3857, 11, 3575, 912, 13, 18, 12, 777, 24116, 505, 8626, 13, 386, 641, 494, 474, 753, 2363, 11, 18172, 14002, 13, 1102, 374, 2631, 311, 12849, 279, 7524, 4272, 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https://educationexpert.net/mathematics/1651956.html	1,632,006,378,000,000,000	text/html	crawl-data/CC-MAIN-2021-39/segments/1631780056578.5/warc/CC-MAIN-20210918214805-20210919004805-00647.warc.gz	288,275,396	7,102	15 October, 08:44 # What is the answer to 7 (3+x+4) +1 1. 15 October, 09:07 0 Distribute the 7. =21+7x+28 Simplify by combining any terms that you can. = 49+7x To solve for X you need to get the variable alone. Start to do this by moving the 49. Subtract 49, what you do one side you have to also do to the other. So you have - 49=7x now to get the X alone divide both sides by seven. -7=x 2. 15 October, 10:02 0 7 (3+x+4) 21 + 7x + 28 7x + 49	167	449	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	4.0625	4	CC-MAIN-2021-39	latest	en	0.930812	[ 128000, 868, 6664, 11, 220, 2318, 25, 2096, 271, 2, 3639, 374, 279, 4320, 311, 220, 22, 320, 18, 38992, 10, 19, 696, 10, 16, 198, 16, 13, 220, 868, 6664, 11, 220, 2545, 25, 2589, 198, 15, 198, 35, 81233, 279, 220, 22, 382, 28, 1691, 10, 22, 87, 10, 1591, 62342, 1463, 555, 35271, 904, 3878, 430, 499, 649, 13, 284, 220, 2491, 10, 22, 87, 2057, 11886, 369, 1630, 499, 1205, 311, 636, 279, 3977, 7636, 13, 5256, 311, 656, 420, 555, 7366, 279, 220, 2491, 13, 94310, 220, 2491, 11, 1148, 499, 656, 832, 3185, 499, 617, 311, 1101, 656, 311, 279, 1023, 13, 2100, 499, 617, 482, 220, 2491, 28, 22, 87, 1457, 311, 636, 279, 1630, 7636, 22497, 2225, 11314, 555, 8254, 382, 12, 22, 26459, 198, 17, 13, 220, 868, 6664, 11, 220, 605, 25, 2437, 198, 15, 198, 22, 320, 18, 38992, 10, 19, 696, 1691, 489, 220, 22, 87, 489, 220, 1591, 271, 22, 87, 489, 220, 2491, 128001 ]	[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ]
http://www.wikihow.com/Checkmate-in-3-Moves-in-Chess	1,503,189,938,000,000,000	text/html	crawl-data/CC-MAIN-2017-34/segments/1502886105955.66/warc/CC-MAIN-20170819235943-20170820015943-00401.warc.gz	694,527,288	51,384	# wikiHow to Checkmate in 3 Moves in Chess You know the 2-move checkmate, or Fool's Mate, and you know the 4-move checkmate, or Scholar's Mate, but do you know the 3-move checkmate? Grab a friend, play white, and your next game of chess will take longer to set up than to play. You can achieve checkmate in three moves with capturing, or without capturing. For either of these methods to work requires some pretty bad play from your opponent, but maybe you can catch her cold at the start. ### Method 1 Getting Checkmate in Three Moves while Capturing 1. 1 Move your King Pawn forward to e4. In both of these methods the key piece for you is your Queen. The Queen is the piece that you are going to use to achieve the checkmate, so your first move should be to open up space for the Queen to move diagonally. Moving the King Pawn forward two spaces to square e4 achieves this (e4). • If you're unfamiliar with algebraic chess notation, check out the wikiHow article to brush up. • As well as freeing your queen, you need your opponent to expose their king. If black then moves their bishop pawn 2 spaces to f5 to tempt white, the checkmate in three moves is on! 2. 2 Capture your opponent's Pawn at f5. Now use your Pawn to capture your opponent's advanced Pawn by attacking on the diagonal. Notated, that's e4xf5. Here you are trying to encourage your opponent to move their Knight Pawn forward two spaces to g5, so it is alongside your Pawn. • This isn't a smart move from your opponent, but maybe you can lull her into it. • The idea of this move is to make sure nothing can block off your route to your opponent's King after you make your next move. 3. 3 Move your White Queen to h5 (Qh5). Checkmate! Now you can move your Queen on the diagonal to h5 and you have your opponents King pinned. That's game over! You'll notice that if your opponent hadn't moved their Pawn forward two in their last turn they could have blocked off your Queen by putting a pawn in her way by g6. • You really need your opponent to play into your hands to pull off this three-move checkmate. 4. 4 Call out checkmate! Now you can take the King with your Queen on the diagonal and celebrate a very swift victory. If your opponent has fallen into the trap they will likely be a bit annoyed, so don't gloat too much! ### Method 2 Getting Checkmate in Three Moves Without Capturing 1. 1 Move your King Pawn to d3. This is a very similar method to the previous one. You are basically aiming to get your opponent's Bishop and Knight Pawns forward one and two squares respectively, while freeing your Queen to enable it to move onto h5. The end result is the same as the previous method. • You are trying to tempt your opponent to move her Bishop and Knight Pawns. • You need you opponent to respond by bringing out her Bishop Pawn one square to f6. • It can also work if she moves her Knight Pawn forward two squares on this turn, as long as she moves the Bishop Pawn on her next move. 2. 2 Move your Queen Pawn forward to e4. The next move for you to make has to free up your Queen so it can get into a checkmate position on the next move. To do this, move the White King Pawn ahead two squares to e4. Now you have opened up an avenue for your Queen to reach h5. • In order to clear the way to your opponent's King you need her to move her Knight Pawn ahead two spaces to g5. 3. 3 Move the White Queen to h5 (Qh5). Checkmate! And that's it, you have trapped your opponent's King in the same position as the previous method, but this time you did it without even capturing a single piece. Game. Set. Match. Over.[1] • Again, this looks simple and it is. So don't expect it to work very often! • In theory, there are loads of variations on this. The key moves are getting your Queen to h5, and your opponent's Bishop and Knight Pawns out of the way of her King. ## Community Q&A Search • Can my opponent castle to get out of checkmate when my queen is on Qh5? wikiHow Contributor No, there are pieces in the way! It is also illegal to castle out of check. • What should I do if it doesn't work? wikiHow Contributor You can play another opening. It won't always work and it usually only works with players that are absolute beginners in the game. • When can my king swap with my rook? wikiHow Contributor Here are the conditions: when your bishop and knight are not in between your rook and king, your king and rook have not moved yet, there are no pieces attacking the space between your rook and king, and when doing so will not result in check. • What can I do if the opposite player doesn't move as I wish? wikiHow Contributor If the opponent doesn't make the moves that allow you to checkmate him/her in three moves, play another opening. It won't always work and it usually only works with players who are absolute beginners in the game. • How do I move the king? wikiHow Contributor You can move your king in any direction, just like the queen, but only one square at a time. Be careful where you move your king, however; the game is over if your opponent takes your king. • Can we revive pieces? wikiHow Contributor Some people adopt rules allowing players to recover captured pieces, but normally you can revive a captured piece only by promoting a pawn. • What is a checkmate? wikiHow Contributor You achieve checkmate (and win the game) by placing your opponent's king in check in such a way that the opponent cannot escape check in his/her next move. • How should I position the king? wikiHow Contributor For defensive purposes, the king is often "castled," which is explained in How to Castle in Chess. • Isn't there a mistake here? The first method has the king and queen on different squares to the second method. All the drawings above are correct. The king (with the cross on top) starts on a square of the opposite color. The queen starts on the matching-color square. • Is there such thing as a 1 move checkmate? wikiHow Contributor No, there isn't. The way these fast checkmates work is by moving through the kingside white square diagonal (comprising squares h5, g6, f7, and e8). In order to free up this diagonal, two pawn moves must occur. • What is check! N where should I move my piece to? How to check my opponent's piece? Not checkmate 200 characters left ## Warnings • For his to work you need an opponent who is either very cooperative, or perhaps not quite awake. • Be wary of trying this in a more serious game, as it not likely to come off. If they don't play right into your hand, the 3-move checkmate won't work. ## Things You'll Need • Chess board and pieces • Cooperative opponent	1,553	6,601	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	3.5	4	CC-MAIN-2017-34	latest	en	0.95798	[ 128000, 2, 29709, 4438, 311, 4343, 18543, 304, 220, 18, 57699, 304, 48487, 271, 2675, 1440, 279, 220, 17, 96794, 1817, 18543, 11, 477, 72635, 596, 44670, 11, 323, 499, 1440, 279, 220, 19, 96794, 1817, 18543, 11, 477, 25542, 596, 44670, 11, 719, 656, 499, 1440, 279, 220, 18, 96794, 1817, 18543, 30, 37294, 264, 4333, 11, 1514, 4251, 11, 323, 701, 1828, 1847, 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http://eng.yax.su/finlab/ir010/9/index.shtml	1,716,062,132,000,000,000	text/html	crawl-data/CC-MAIN-2024-22/segments/1715971057494.65/warc/CC-MAIN-20240518183301-20240518213301-00848.warc.gz	10,282,204	6,682	back start next [start] [1] [2] [3] [4] [5] [6] [7] [8] [ 9 ] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47] [48] [49] [50] [51] [52] [53] [54] [55] [56] [57] 9 the effect of inflation which reduces the purchasing power of money over time. Inflation-adjusted calculations will be discussed in Section 9.4. As a consequence of the above principle, it is obvious that two or more amounts of money payable at different points in time cannot be compared until all the amounts are accumulated or discounted to a common date. This common date is called the comparison date, and the equation which accumulates or discounts each payment to the comparison date is called the equation of value. One device which is often helpful in the solution of equations of value is the time diagram. A time diagram is a one-dimensional diagram in which units of time are measured along the one dimension and payments are placed on the diagram at the appropriate points. Note that payments in one direction are placed on the top of the diagram and payments in the other direction are placed on the bottom of the diagram. The comparison date is denoted by an arrow. Figure 2.1 is an example of a time diagram used in the solution of Example 2.4. The time diagram is not necessary in the solution of equations of value; it IS merely an aid in visualizing the problem. With some practice the reader can usually dispense with a time diagram on simpler problems. However, time diagrams are usually helpful in the solution of more complex problems. One of the properties of compound interest is that the choice of the comparison date makes no difference in the answer obtained. Thus, there is a different equation of value for each comparison date, but they all produce the same answer. This important property of compound interest will be illustrated in the solution of Example 2.4. The reader is cautioned that under other patterns of interest, e.g. simple interest or simple discount, the choice of a comparison date does affect the answer obtained. This illustrates once again the inherent inconsistency in using simple interest or simple discount. t;. The reader should be aware that the problems involving accumulated values and present values already considered in the first two chapters are examples of equations of value. Example 2.4 illustrates a more general type of problem. Example 2.4 In return for a promise to receive \$600 at the end of 8 fears, a person agrees to pay \$100 at once, \$200 at the end of 5 years, and to make a further payment at ffie end of 10 years. Find the payment at the end of 10 years if the nominal rate of interest is 8% convertible semiannually. We shall first work the problem with a comparison date of the present. The time diagram is shown in Figure 2.1. wording of a problem rrny be different depending upon the point of view. Examples and exercises phrased from both points of view appear, and the reader should not let the different phraseology be a source of confusion. As an example, recall the discussion in Section 1.7 involving the use of the words "paid" or "credited." To some readers the word "paid" may seem more normal from the vantage point of the borrower, while "credited" may seem more normal from the vantage point of the lender. Many other such examples could be cited - Complex financial transactions often involve more than two parties. For example, a business firm analyzing its rate of return on a major investment in a new plant is involved with a multiplicity of parties. However, the basic principles developed to analyze two-party transactions can readily be extended to analyze these more complex transactions. d) In practical applications involving interest the terminology can become confusing. Many terms have ambiguous meanings (e.g. see the discussion on the use of the word "discount" in Section 1.7). Furthermore, as we shall see in succeeding chapters some terms used unfortunately do not convey an intuitive description of the transactions involved (e.g. see the discussion on the terms "annuity-immediate" and "annuity-due" in Section 3.3). Finally, many parties involved in financial transactions simply do not always use terms with the precise meanings and definitions contained in this book. The reader is admonished in real-world applications to look beyond the stated terms and be certain to understand the exact nature of the financial transactions in question. There simply is not total consistency in terminology among the large and diverse number of parties involved in financial transactions involving interest. 2.5 EQUATIONS OF VALUE It is a fundamental principle in the theory of interest that the value of an amount of money at any given point in time depends upon the time elapsed since the money was paid in the past or upon the time which will elapse in the fiiture before it is paid. We have already seen this in many of the examples and exercises considered thus far in the first two chapters. This principle is often characterized as the recognition of the time value of money. This process would be in contrast to financial calculations not involving the effect of interest, in which case it would be said that such calculations do not recognize the time value of money. The reader is cautioned that "recognition of the time value of money" reflects the effect of interest, but not 8 9 10 600 Figure 2.1 Time Diagram for Example 2.4 Since interest is convertible semiannually, we will count time periods in half-years. The equation of value is 100 + 200v "> + Xvo = 600v 1 at 4% 600v- 100 - 200v< 600(.53391) - 100 - 200(.67556) .45639 = \$186.76. We could also have chosen a different comparison date and obtained a different equation of value. For example, if the comparison date were chosen to be the end of the 10th year, then the arrow in the time diagram would be under 10 and the equation of value would be 100(1.04)20 + 200(1.04)° + X = 600(1.04)" X = 600(1.04)" - 100(1.04)2° - 200(1.04)° = 600(1.16986)- 100(2.19112)-200(1.48024) = \$186.76. Thus, the same answer is obtained. The two equations of value are equivalent. If both sides of the first one are multiplied by (1.04)°, the second one is obtained. The reader can verify that if other comparison dates are chosen, the same answer is obtained. 2.6 UNKNOWN TIME As discussed in Section 2.4, if any three of the four basic quantifies entering into an interest problem are given, then the fourth can be determined. In this section we consider the situation in which the length of the investment period is the unknown. The best method of solving for unknown time involving a single payment is to use logarithms. This technique will be illustrated in Example 2.5. As indicated above in Section 2.4, we are assuming the availability of a pocket calculator with exponential and logarithmic functions. An alternative approach with less accuracy that can be used if such a calculator is unavailable is Unear inte olation in the interest tables. This technique is also illustrated in Example 2.5. A situation sometimes arises in which several payments made at various points in time are to be replaced by one payment numerically equal to the sum This approximation is denoted by t and is often called using the method of equated time. It is possible to prove that the value of t is always greater than the true value of t, or, alternatively, that the present value using the method of equated time is smaller than the true present value. Consider quantities each equal to v 2 quantities each equal to v, and so forth until there are s„ quantities each equal to v". The arithmetic mean of these quantities is Sjv + S2V + 5j + 2 + The geometric mean of these quantities is However, we know that the arithmetic mean of n positive numbers, not all of „ which are equal, is greater than the geometric mean, and thus we have ijv> + S2v + • • • + s/" 5j + 2 + + 5„ of the other payments. The problem is to find the point in time at which the one payment should be made such that it is equivalent in value to the payments made separately. Let amounts s,S2, . . ,s„ be paid at times fj, fj. • • • . respectively. The problem is to find time t, such tiiat + \$2 + • • • + paid at time t is equivalent to tiie payments of , • • . n made separately. The flindamental equation of value is (S, + :?2 + • • • + n> = + 2 + • • • + V" - which is one equation in one unknown t. As an exercise, the reader will be Risked to find an exact expression for t. As a first approximation, t is often calculated as a weighted average of the various times of payment, where the weights are the various amounts paid, i.e. llflfL = k=X (2.9) The left-hand side is the true present value which exceeds the present value given by the method of equated time on the right-hand side. Thus, the value of t from formula (2.9) is always greater than the true value of t from formula (2.8). The method of equated time is useful in analyzing the average length of financial transactions. This application will be discussed further in Section 9.8. Another interesting question often asked is how long it takes money to double at a given rate of interest. We can analyze this problem as follows: giving (1 + 0" = 2 n loggd + 0 = logg2 It is possible to derive an approximation to the exact result given in formula (2.10) as follows: log, 2 loged + 0 .6931 log,(l + 0 The second factor evaluated for i = 8% is 1.0395. Thus we have (2.11) This is frequently called the rule of 72, since it can be applied immediately by dividing 72 by the rate of interest expressed as a percentage (i.e. as 100/). The rule of 72 produces surprisingly accurate results over a wide range of interest rates. Illustrative values are provided in Table 2.1. Table 2.1 Length of Time It Takes Money to Double Rate of interest Rule of 72 Exact value 17.67 11.90 9.01 7.27 6.12 4.19 Example 2.5 Find the length of time necessary for \$1000 to accumulate to \$1500 if invested at 6% per annum compounded semiannually: (1) by use of logarithms, and (2) by interpolating in the interest tables. Let n be the number of half-years. The equation of value is 1000(1.03)" = 1500 (1.03)" =1.5. 1. Using logarithms n log J.03 = logl.5 log e 1-5 .405465 log, 1.03 Thus, the number of years is 6.859. .029559 = 13.717. 2. From the interest tables, (1.03)" = 1.46853 and (1.03)" = 1.51259, so that 14 13 < n < 14. Performing a linear interpolation 1.50000 - 1.46853 n = 13 -I- 1.51259 - 1.46853 = 13.714. Thus, the number of years is 6.857. This method is equivalent to the assumption of simple interest during the final fraction of an interest conversion period. This point was discussed in more detail in Section 2.2. Example 2.6 Payments of \$100, \$200, and \$500 are due at the ends of years 2, 3, and 8, respectively. Assuming an effective rate of interest of 5% per annum, find the point in time at which a payment of \$800 would be equivalent: (1) by the method of equated time, and (2) by an exact method. 1. By the method of equated time using formula (2.9) 100 • 2 200 • 3 -I- 500 100 -h 200 -I- 500 2. The exact equation of value is 800v = lOOv + 200v + 500v = 6 years. 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https://www.jiskha.com/display.cgi?id=1359662274	1,511,462,951,000,000,000	text/html	crawl-data/CC-MAIN-2017-47/segments/1510934806856.86/warc/CC-MAIN-20171123180631-20171123200631-00698.warc.gz	821,862,924	3,955	# Math PLZ HELP posted by . 1. Which of the following expressions is written in scientific notation? 1. 73.4 x 105 2. 0.09 × 107 3. 80 x 103 4. 4.22 x 10–3 2. Which of the following is 0.0000000708 written in scientific notation? 1. 7.08 x 10–8 2. 7.8 x 10–8 3. 708 x 10–10 4. 70.08 x 10–9 3. Which expression represents the largest number? 1. 40.1 x 10–6 2. 4.1 x 10–7 3. 0.411 x 10–7 4. 0.04001 x 10–5 4. Which expression is equal to 1/8000 written in scientific notation? 1. 8.0 x 103 2. 1.25 x 10–4 3. 125 x 10–5 4. 1.25 x 104 • Math PLZ HELP - if you know anything about scientific notation, #1,2 should be easy. For #3, convert all to same power of 10, then it is easy to compare. #4: 1/8000 = .000125 • Math PLZ HELP - 1.)A 2.)C 3.)B 4.)B ## Similar Questions 1. ### Math Use scientific notation to divide the following two numbers. Express the answer using scientific notation; retain at least three decimal places. 6.2 • 107 / 4.15 • 103 11. 17, 200 written in scientific notation is 1.7 x 10^5 21. 0.00105 written in scientific notation is 1.05 x 10^-3 are these correct? 3. ### Math Which of the following expressions is written in scientific notation? 4. ### Math Which of the following is 0.0000000708 written in scientific notation? 5. ### math 1. Which of the following expressions is written in scientific notation? 6. ### Math! Please Check My Awnser! Hello! I need some help with math! I have tried to do this but I am unsure about my answers. My I will identify my answers with a "<--". Also, if I have missed one, please tell me the right answer! Also, Thanks for your help in … 7. ### Homework help plz( Steve or Reiny) Determine if the number is written in scientific notation. If not, explain 32 * 10^4. (1 point) No; it is not written as a power of 10. No; the first factor is not a number between 1 and 10. Yes; the number is written in scientific … 8. ### Math Hi everyone! I am extremely struggling with this questions and was hoping someone would go over them with me. Thank you! 1. Which of the following expressions is written in scientific notation? 9. ### Algebra Which of the following expressions is written in scientific notation? 10. ### Algebra Which of the following is 0.0000000708 written in scientific notation? 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http://www.actuarialoutpost.com/actuarial_discussion_forum/showthread.php?s=1ed67fa957eae519c00653a0a26e312f&p=9130244	1,508,587,605,000,000,000	text/html	crawl-data/CC-MAIN-2017-43/segments/1508187824775.99/warc/CC-MAIN-20171021114851-20171021134851-00691.warc.gz	368,408,493	13,783	Actuarial Outpost Fall 2017 MLC Progress Thread Register Blogs Wiki FAQ Calendar Search Today's Posts Mark Forums Read FlashChat Actuarial Discussion Preliminary Exams CAS/SOA Exams Cyberchat Around the World Suggestions #451 Yesterday, 07:06 PM Jim Daniel Member SOA Join Date: Jan 2002 Location: Davis, CA College: Wabash College B.A. 1962, Stanford Ph.D. 1965 Posts: 2,558 Quote: Originally Posted by PerpetualMotion Can someone ELI5 the differences in these variances from Fall 2014 (ii) Drug A will be given to 1000 subjects comprising Cohort A. • With probability 80%, each subject in Cohort A will have q = 0.20 . • With probability 20%, each subject in Cohort A will have q = 0.05 . • Conditional on q, the lives have independent future lifetimes. I understand that everyone in the group will experience whichever q is chosen so we use (un?)conditional variance formula. I don't really know why this is though. (iii) Drug B will be given to 1000 subjects with independent future lifetimes, comprising Cohort B. The drug will affect different subjects differently and independently. • With probability 80%, any given subject in Cohort B will have q = 0.20 . • With probability 20%, any given subject in Cohort B will have q = 0.05 . This one you end up finding E(Q) and conditional variance of Q because any given subject can experience different Qs. I really just don't understand this topic, so if anyone can break it down I'd really appreciate it. Or if someone can point me to TIA videos or other online resources. This is a major flaw in my understanding of variance and I'm really trying to understand. ii) says there is an 80% probability that you have 1000 trials with q = 0.2 [so a Binomial RV with m=1000 and q = 0.2] and a 20% probability that you have 1000 trials with q = 0.05 [so a Binomial RV with m=1000 and q = 0.05]. Thus the distribution is an 80/20 mixture of these two Binomial RV's. Get the mean and second moment by mixing and then get the variance. [My website has a free study note on mixture distributions.] iii) says that each individual has an independent 80% probability of its q being 0.2 and a 20% probability of its q being 0.05. Thus, by mixing, each individual has a probability 0.8 (0.2) + (0.2) (0.05) =0.17 of death, and there are 1000 such individuals, so the number of deaths is Binomial with parameters 1000 and 0.17. So the mean and variance just come from this Binomial. __________________ Jim Daniel Jim Daniel's Actuarial Seminars www.actuarialseminars.com [email protected] #452 Yesterday, 07:15 PM PerpetualMotion Member SOA Join Date: Jan 2013 Location: Kentucky(Louisville) Studying for Exam MLC College: University of Louisville Alumni Favorite beer: Zombie Dust Posts: 325 Wow, that actually makes perfect sense. I'm going to check out that study note. I really appreciate it, I just know there's going to be a question like this on the exam __________________ P,FM MFE , C, MLC Modules 1-8 IA FA And NUH is the letter I use to spell Nutches Who live in small caves, known as Nitches, for hutches, These Nutches have troubles, the biggest of which is the fact there are many more Nutches than Nitches. #453 Yesterday, 07:37 PM Steveo1794 Member SOA Join Date: May 2016 Location: Ohio Studying for MLC College: Miami University (OH) Favorite beer: Anything Posts: 32 Quote: Originally Posted by PerpetualMotion Wow, that actually makes perfect sense. I'm going to check out that study note. I really appreciate it, I just know there's going to be a question like this on the exam Did this question last night... Confused the hell out of me but once I really thought about it, it made sense. I think about it like this: The scenario in b(ii) has five total groups of 1000 each. In four of those five groups, everyone has q=.20. In the other one, everyone has q=.05. The scenario in b(iii) is one group of 1000. 80% of the 1000 have q=.20 and the other 20% has q=.05. Hope this helps. --- Side note - Are we allowed to use pens on the written answer? Obviously not on the MC, but the WA? Last edited by Steveo1794; Yesterday at 07:40 PM.. #454 Yesterday, 07:42 PM Jim Daniel Member SOA Join Date: Jan 2002 Location: Davis, CA College: Wabash College B.A. 1962, Stanford Ph.D. 1965 Posts: 2,558 Both of Steveo1794's interpretations are incorrect. __________________ Jim Daniel Jim Daniel's Actuarial Seminars www.actuarialseminars.com [email protected] #455 Yesterday, 08:21 PM Steveo1794 Member SOA Join Date: May 2016 Location: Ohio Studying for MLC College: Miami University (OH) Favorite beer: Anything Posts: 32 I wasn't referring to the literal scenarios presented in the problem, I was referring to how to distinguish between the "each subject will have" and "any given subject". That phrasing confused me, and thinking about it like that helps me. Does the below explanation make sense? If not, please correct it. Scenario b(ii) is 80% chance that 100% of the cohort will have q=.20 and a 20% chance that 100% of the cohort will have q=.05. Scenario b(iii) is simply an 80% chance of a member of the cohort having q=.20 and a 20% chance of someone in the same cohort having q=.05. __________________ P FM MFE C MLC VEE FAP APC #456 Yesterday, 08:45 PM Jim Daniel Member SOA Join Date: Jan 2002 Location: Davis, CA College: Wabash College B.A. 1962, Stanford Ph.D. 1965 Posts: 2,558 Quote: Originally Posted by Steveo1794 I wasn't referring to the literal scenarios presented in the problem, I was referring to how to distinguish between the "each subject will have" and "any given subject". That phrasing confused me, and thinking about it like that helps me. Does the below explanation make sense? If not, please correct it. Scenario b(ii) is 80% chance that 100% of the cohort will have q=.20 and a 20% chance that 100% of the cohort will have q=.05. Scenario b(iii) is simply an 80% chance of a member of the cohort having q=.20 and a 20% chance of someone in the same cohort having q=.05. My problem with how you described it is the following: correct solutions based on the statements you gave yield incorrect answers for both cases. However, you may have interpreted your statements in an incorrect way that produced correct answers in boith cases. It's good to be lucky. Your explanation of b(ii) is correct. Your explanation of b(iii) may be correct, but the wording strikes me as a little confusing. It would be clearer to say: For each member of the cohort, there is an 80% chance that that member's q is 0.2 and a 20% chance that that member's q is 0.05, and the results are independent from member to member. By the way, I agree that the exam problem as stated is poorly worded, with the "any given subject" and "each subject" difficult to distinguish---and imagine what it's like for non-native English speakers! Of course, exam questions are often poorly worded, and it's not all that rare for them to use textbook terms to mean something different from their meaning in the textbook, or to be missing data, or for the wrong answer to be considered correct. Quality control truly sucks. Candidates need to be robust problem solvers in that the first part of the problem is to figure out what the problem really is. __________________ Jim Daniel Jim Daniel's Actuarial Seminars www.actuarialseminars.com [email protected] #457 Yesterday, 09:44 PM MySpaceTom SOA Join Date: Oct 2016 Posts: 7 The Written Answer instructions at the beginning of the exam state " Do not write answers to more than one question per sheet". Does this mean every part (a, b, and c) to question 1, must be on an individual sheet? Or, is it okay to answer parts a, b, and c all on the same sheet? #458 Yesterday, 10:38 PM PerpetualMotion Member SOA Join Date: Jan 2013 Location: Kentucky(Louisville) Studying for Exam MLC College: University of Louisville Alumni Favorite beer: Zombie Dust Posts: 325 Quote: Originally Posted by Jim Daniel By the way, I agree that the exam problem as stated is poorly worded, with the "any given subject" and "each subject" difficult to distinguish---and imagine what it's like for non-native English speakers! Of course, exam questions are often poorly worded, and it's not all that rare for them to use textbook terms to mean something different from their meaning in the textbook, or to be missing data, or for the wrong answer to be considered correct. Quality control truly sucks. Candidates need to be robust problem solvers in that the first part of the problem is to figure out what the problem really is. I'm the first to admit that life isn't fair but that really seems like bs from the soa standpoint. We put in hundreds of hours worth of work for each of these exams, they should be held to a higher standard. Some problems are unnecessarily complex and it just doesn't feel right. The further I progress, the more it all seems like a cash grab. I hope when I'm an FSA I get to write problems that actually tests relevant knowledge. Until then. __________________ P,FM MFE , C, MLC Modules 1-8 IA FA And NUH is the letter I use to spell Nutches Who live in small caves, known as Nitches, for hutches, These Nutches have troubles, the biggest of which is the fact there are many more Nutches than Nitches. #459 Today, 12:10 AM NoFunnyStuff Member SOA Join Date: Jul 2009 Location: Green Bay, WI Studying for MLC College: Georgia State University Alumni Posts: 164 Quote: Originally Posted by noonelikesmodules Doing a profit/gain question. Specifies surrender and mortality RATES. Doesn't specify about timing of surrenders. Do we always assume EoY surrenders? Most of the time yes, because if you have already paid for your insurance in BOY you would not surrender before the end of the year for something you have already paid for. Hope that this help. __________________ Prelim:P FM C MFE MLC VEE: FINANCE ECONOMICS STATISTICS Thread Tools Display Modes Linear Mode Posting Rules You may not post new threads You may not post replies You may not post attachments You may not edit your posts BB code is On Smilies are On [IMG] code is On HTML code is Off All times are GMT -4. 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beeadshobbycrafts.club	1,624,358,201,000,000,000	text/html	crawl-data/CC-MAIN-2021-25/segments/1623488517048.78/warc/CC-MAIN-20210622093910-20210622123910-00432.warc.gz	120,970,582	6,689	# How To Measure For A Chandelier ### Allow 2 to 3 inches of chandelier length per foot of wall height. How to measure for a chandelier. A general rule of thumb is to choose a chandelier that is 10 12 inches narrower than your table. So if you have a 10 ft ceiling multiply that by 3 and you get 30 or 30 diameter. So if your room measures 10 x 14 the diameter of the fixture should be about 24. For example if your room is 10 x 16 the sum of those equals 26. One standard method for determining the right size for a chandelier is to add the length and width dimensions of the room together measured in feet. Measure your room s length and width in feet and add those two numbers together. How to measure a room for a chandelier. Then use that number as the width in inches for your chandelier. To find this number you need to take a couple of measurements. So for each foot it translates to two and a half to three inches for your chandelier height. To determine height of chandelier. The sparkle and dazzle of a well placed chandelier defines and enhances the mood of any room. Measure the width and length of the room. The chandelier should be 26 wide. How to measure a proper sized chandelier should be related to the size of the room and the size of the table over which it is to hang. Measure from the installation point in the ceiling all the way to the floor. Multiply the height by approximately 3 1 foot wall height to 3 chandelier height per foot measurement. A chandelier that is too large can easily overpower a room and. Knowing the height of a chandelier will help you choose a chandelier that s proportional to the height of your space. This is the diameter of the chandelier that will best suit the room. I got eight feet. For example a foyer 12 feet x 12 feet in size calls for a 24 inch diameter chandelier fixture. A simple way to determine a chandelier size is to add the dimensions of the room together in feet and then convert the answer to inches. Measure the ceiling height from the point of installation point where you plan to hang the fixture to the floor. Here are a few more quick guidelines to help.	477	2,138	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	3.90625	4	CC-MAIN-2021-25	latest	en	0.891459	[ 128000, 2, 2650, 2057, 35204, 1789, 362, 921, 79221, 271, 14711, 27628, 220, 17, 311, 220, 18, 15271, 315, 523, 79221, 3160, 824, 4579, 315, 7147, 2673, 382, 4438, 311, 6767, 369, 264, 523, 79221, 13, 362, 4689, 6037, 315, 25015, 374, 311, 5268, 264, 523, 79221, 430, 374, 220, 605, 220, 717, 15271, 91529, 1109, 701, 2007, 13, 2100, 422, 499, 617, 264, 220, 605, 10702, 22959, 31370, 430, 555, 220, 18, 323, 499, 636, 220, 966, 477, 220, 966, 23899, 13, 2100, 422, 701, 3130, 11193, 220, 605, 865, 220, 975, 279, 23899, 315, 279, 12790, 1288, 387, 922, 220, 1187, 13, 1789, 3187, 422, 701, 3130, 374, 220, 605, 865, 220, 845, 279, 2694, 315, 1884, 17239, 220, 1627, 382, 4054, 5410, 1749, 369, 26679, 279, 1314, 1404, 369, 264, 523, 79221, 374, 311, 923, 279, 3160, 323, 2430, 15696, 315, 279, 3130, 3871, 17303, 304, 7693, 13, 35204, 701, 3130, 274, 3160, 323, 2430, 304, 7693, 323, 923, 1884, 1403, 5219, 3871, 13, 2650, 311, 6767, 264, 3130, 369, 264, 523, 79221, 13, 5112, 1005, 430, 1396, 439, 279, 2430, 304, 15271, 369, 701, 523, 79221, 382, 1271, 1505, 420, 1396, 499, 1205, 311, 1935, 264, 5743, 315, 22323, 13, 2100, 369, 1855, 4579, 433, 48018, 311, 1403, 323, 264, 4376, 311, 2380, 15271, 369, 701, 523, 79221, 2673, 13, 2057, 8417, 2673, 315, 523, 79221, 13, 578, 96170, 323, 61018, 273, 315, 264, 1664, 9277, 523, 79221, 19170, 323, 57924, 279, 20247, 315, 904, 3130, 382, 33336, 279, 2430, 323, 3160, 315, 279, 3130, 13, 578, 523, 79221, 1288, 387, 220, 1627, 7029, 13, 2650, 311, 6767, 264, 6300, 30387, 523, 79221, 1288, 387, 5552, 311, 279, 1404, 315, 279, 3130, 323, 279, 1404, 315, 279, 2007, 927, 902, 433, 374, 311, 15020, 13, 35204, 505, 279, 14028, 1486, 304, 279, 22959, 682, 279, 1648, 311, 279, 6558, 382, 96255, 279, 2673, 555, 13489, 220, 18, 220, 16, 4579, 7147, 2673, 311, 220, 18, 523, 79221, 2673, 824, 4579, 19179, 13, 362, 523, 79221, 430, 374, 2288, 3544, 649, 6847, 98887, 264, 3130, 323, 13, 58733, 279, 2673, 315, 264, 523, 79221, 690, 1520, 499, 5268, 264, 523, 79221, 430, 274, 55272, 311, 279, 2673, 315, 701, 3634, 13, 1115, 374, 279, 23899, 315, 279, 523, 79221, 430, 690, 1888, 7937, 279, 3130, 382, 40, 2751, 8223, 7693, 13, 1789, 3187, 264, 100016, 220, 717, 7693, 865, 220, 717, 7693, 304, 1404, 6880, 369, 264, 220, 1187, 17560, 23899, 523, 79221, 12790, 13, 362, 4382, 1648, 311, 8417, 264, 523, 79221, 1404, 374, 311, 923, 279, 15696, 315, 279, 3130, 3871, 304, 7693, 323, 1243, 5625, 279, 4320, 311, 15271, 13, 35204, 279, 22959, 2673, 505, 279, 1486, 315, 14028, 1486, 1405, 499, 3197, 311, 15020, 279, 12790, 311, 279, 6558, 382, 8586, 527, 264, 2478, 810, 4062, 17959, 311, 1520, 13, 128001 ]	[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ]
https://documen.tv/question/which-epression-is-equivalent-to-3-4m-2-2-m-5-a-10m-16-b-14m-4-c-16m-10-d-22350790-27/	1,670,520,608,000,000,000	text/html	crawl-data/CC-MAIN-2022-49/segments/1669446711344.13/warc/CC-MAIN-20221208150643-20221208180643-00331.warc.gz	250,594,595	18,508	## Which expression is equivalent to 3(4m – 2) – 2(m + 5) A. 10m – 16 B. 14m + 4 C. -16m + 10 D Question Which expression is equivalent to 3(4m – 2) – 2(m + 5) A. 10m – 16 B. 14m + 4 C. -16m + 10 D. 12m – 6 in progress 0 1 year 2021-07-27T16:04:03+00:00 2 Answers 9 views 0 A. 10m – 16 Step-by-step explanation: First, distribute 3 to all terms within the corresponding parenthesis, and -2 likewise: 3(4m – 2) = 3 * 4m = 12m 3 * -2 = -6 3(4m – 2) = 12m – 6 -2(m + 5) = -2 * m = -2m -2 * 5 = -10 -2(m + 5) = -2m -10 Combine like terms: 12m – 6 -2m – 10 12m – 2m – 6 – 10 (12m – 2m) + (-6 – 10) 10m – 16 ~ 10 m – 16 (Answer A) Step-by-step explanation: We use distributive property to eliminate parenthesis: 12 m – 6 – 2 m -10 and now combine like terms: [12 m – 2 m] – 6 – 10 10 m – 16 Therefore, answer option A. is the correct one	385	865	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	4.46875	4	CC-MAIN-2022-49	latest	en	0.689368	[ 128000, 567, 16299, 7645, 374, 13890, 311, 220, 18, 7, 19, 76, 1389, 220, 17, 8, 1389, 220, 17, 1278, 489, 220, 20, 8, 362, 13, 220, 605, 76, 1389, 220, 845, 426, 13, 220, 975, 76, 489, 220, 19, 356, 13, 482, 845, 76, 489, 220, 605, 423, 271, 14924, 271, 23956, 7645, 374, 13890, 311, 220, 18, 7, 19, 76, 1389, 220, 17, 8, 1389, 220, 17, 1278, 489, 220, 20, 696, 32, 13, 220, 605, 76, 1389, 220, 845, 271, 33, 13, 220, 975, 76, 489, 220, 19, 271, 34, 13, 482, 845, 76, 489, 220, 605, 271, 35, 13, 220, 717, 76, 1389, 220, 21, 271, 258, 5208, 220, 15, 198, 16, 1060, 220, 2366, 16, 12, 2589, 12, 1544, 51, 845, 25, 2371, 25, 2839, 10, 410, 25, 410, 220, 17, 38343, 220, 24, 6325, 220, 15, 271, 32, 13, 220, 605, 76, 1389, 220, 845, 271, 8468, 14656, 30308, 16540, 1473, 5451, 11, 16822, 220, 18, 311, 682, 3878, 2949, 279, 12435, 96456, 11, 323, 482, 17, 39022, 1473, 18, 7, 19, 76, 1389, 220, 17, 8, 80583, 18, 353, 220, 19, 76, 284, 220, 717, 76, 271, 18, 353, 482, 17, 284, 482, 21, 271, 18, 7, 19, 76, 1389, 220, 17, 8, 284, 220, 717, 76, 1389, 220, 21, 271, 12, 17, 1278, 489, 220, 20, 8, 80583, 12, 17, 353, 296, 284, 482, 17, 76, 271, 12, 17, 353, 220, 20, 284, 482, 605, 271, 12, 17, 1278, 489, 220, 20, 8, 284, 482, 17, 76, 482, 605, 271, 82214, 1093, 3878, 1473, 717, 76, 1389, 220, 21, 220, 4194, 12, 17, 76, 1389, 220, 605, 271, 717, 76, 1389, 220, 17, 76, 1389, 220, 21, 1389, 220, 605, 271, 7, 717, 76, 1389, 220, 17, 76, 8, 489, 10505, 21, 1389, 220, 605, 696, 605, 76, 1389, 220, 845, 271, 59729, 605, 296, 1389, 220, 845, 320, 16533, 362, 696, 8468, 14656, 30308, 16540, 1473, 1687, 1005, 2916, 6844, 3424, 311, 22472, 96456, 1473, 717, 296, 1389, 220, 21, 1389, 220, 17, 296, 482, 605, 271, 438, 1457, 16343, 1093, 3878, 1473, 58, 717, 296, 1389, 220, 17, 296, 60, 1389, 220, 21, 1389, 220, 605, 271, 605, 296, 1389, 220, 845, 271, 55915, 11, 4320, 3072, 362, 13, 220, 4194, 285, 279, 4495, 832, 128001 ]	[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ]
http://math.stackexchange.com/questions/154990/find-length-of-segment-in-triangle?answertab=oldest	1,462,267,492,000,000,000	text/html	crawl-data/CC-MAIN-2016-18/segments/1461860121090.75/warc/CC-MAIN-20160428161521-00202-ip-10-239-7-51.ec2.internal.warc.gz	187,001,935	17,142	# Find length of segment in triangle In triangle $\bigtriangleup ABC$, the known sides are: $AB=5$, $BC=6$ and $AC=7$. A circle passes through points $A$ and $C$, crosses straight lines $BA$ and $BC$ at points $K$ and $L$, which is non-vertex angle, respectively. Segment KL As to the incircle of the triangle ABC. Please find the length of the segment $KL$, and show me how to. - A picture would help. – Gigili Jun 7 '12 at 6:59 Proofreading would help even more. Is $S$ the same as $C$? Crosses what straight line? What does it mean to say that $L$ is a "non-vertex angle"? What does "respectively" mean is this context? Sorry, this question makes no sense at all. Please think about what you really mwean to ask, and edit the question accordingly. – Gerry Myerson Jun 7 '12 at 7:15 Much better. Almost there. What does "Segment KL As to the incircle..." mean? – Gerry Myerson Jun 7 '12 at 11:01 1. For any $\triangle ABC$ in which A circle passes through points $A$ and $C$, crosses straight lines $BA$ and $BC$ at points $K$ and $L$, which is non-vertex angle, respectively we have that $\triangle ABC \sim \triangle BKL$. Indeed, $AKLC$ is inscribed quadrilateral $\Rightarrow \angle AKL+\angle ACL=\pi$, but also we have $\angle AKL+\angle BKL=\pi \Rightarrow \angle ACL=\angle BKL$; $\angle KBL = \angle ABC$. As $\triangle ABC \sim \triangle BKL \Rightarrow \|BK\|=6x, \ \|BL\|=5x, \|KL\|=7x$. So we should find $x$ from additional condition of this problem. 2. I guess, that by Segment KL As to the incircle of the triangle ABC TS means following: For inscribe circle of $\triangle ABC$ - $KL$ is tangent line. In this case we can write, that $\|KL\|+\|AC\|=\|AK\|+\|LC\| \ \mathbf{^{)}} \Rightarrow$ $\Rightarrow 7x+7=(5-6x) + (6 - 5x)=11-11x \Rightarrow$ $\Rightarrow 18x=4 \Rightarrow x=\frac{2}{9}$. $\|KL\|=7x=\frac{14}{9}$ 3. $\mathbf{^{)}}$ As $AKLC$ is tangent quadrilateral thats why \|KL\|+\|AC\|=\|AK\|+\|LC\|. 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https://mathematiques-web.fr/en/areas-and-perimeters-math-exercises-in-5th-grade-corrected-in-pdf-15888	1,685,777,337,000,000,000	text/html	crawl-data/CC-MAIN-2023-23/segments/1685224649177.24/warc/CC-MAIN-20230603064842-20230603094842-00280.warc.gz	421,491,893	40,598	# Areas and perimeters: math exercises in 5th grade corrected in PDF. Figure areas with corrected exercises in fifth grade are always essential to the student’s understanding. Thus, the latter will have to be able to convert quantities but also to calculate the area usual figures such as the square, the rectangle, the rhombus, the parallelogram, the trapezoid or the area of a disk of radius R. You will have many exercises and problems to solve on the areas of geometric figures to develop solid skills in calculus and geometry. These work materials are similar to those in your textbook. In addition, the correction allows students to identify errors made in calculating the area of a figure in order to fill in gaps in mathematics and progress throughout the school year in fifth grade. ## Series of exercises on areas and perimeters ### Exercise #1: 1. the intruder: On the grid below, six figures have been drawn. Knowing that the unit of area is the square, calculate the area of each of the 6 figures and find the intruder. 2. Calculate the area of the following figures: ### Exercise 2 areas and perimeters : Using the square as a unit of area, give the area of rectangle ABCD and then the area of each of the parallelograms. ### Exercise #3: The figure below is a parallelogram. 1° Calculate its area. 2° Calculate its perimeter. ### Exercise #4: Consider the parallelogram below. ( a and d denote the heights ) . Circle the products that express thearea of this parallelogram? a $\times$ d c $\times$ d b $\times$ d a $\times$ b a $\times$ c b $\times$ c ### Exercise 5 areas and perimeters : ABCD is a square of side 5 cm. The two half-discs have diameters [AB] and [AD]. Calculate the area, in cm², of the blue surface to the nearest hundredth. ### Exercise #6: Calculate the area of the orange surface. ### Exercise 7 areas and perimeters : A disc has a diameter of 10 cm. Using the key of the calculator, give an approximate value of its area, in cm² : a. to the nearest unit; b. to the nearest hundredth. ### Exercise #8: Calculate the perimeter of the polygon ABCDE below. ### Exercise #9: It took 73.20 m of rope to install the three ropes of this boxing ring. How long is the side of this square ring? ### Exercise #10: Calculate the areas of these triangles, then arrange these areas in ascending order. ### Exercise #11: The orange and green surfaces are delimited by semicircles with centers D, B and E. Also, AD = 1.5 cm. Calculate the area, in cm², of the green surface to the nearest hundredth. ### Exercise #12: In handball, the goal area consists of two quarter discs and a rectangle. Calculate the area, in m², of this goal area to the nearest hundredth. 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In order to do well, you will have to start by reading the statement in its entirety. This will allow you to have a good overview of the subject… • 97 The corrected math exercises on geometry in space in 5th grade will help you understand the chapter. The student should be able to perform volume conversions but also know by heart the formula for the volume of a cube, a rectangle, a right block and a cylinder. In addition, many… Les dernières fiches mises à jour Voici la liste des derniers cours et exercices ajoutés au site ou mis à jour et similaire à areas and perimeters: math exercises in 5th grade corrected in PDF. . Mathématiques Web c'est 2 145 968 fiches de cours et d'exercices téléchargées. Copyright © 2008 - 2023 Mathématiques Web Tous droits réservés \| Mentions légales \| Signaler une Erreur \| Contact . 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https://brainly.com/question/358736	1,485,214,782,000,000,000	text/html	crawl-data/CC-MAIN-2017-04/segments/1484560283301.73/warc/CC-MAIN-20170116095123-00246-ip-10-171-10-70.ec2.internal.warc.gz	798,332,156	9,274	2015-03-17T18:27:56-04:00 Because 3 times 1/6 is 3/6 which can be simplified to 1/2 2015-03-17T18:41:46-04:00 ### This Is a Certified Answer Certified answers contain reliable, trustworthy information vouched for by a hand-picked team of experts. Brainly has millions of high quality answers, all of them carefully moderated by our most trusted community members, but certified answers are the finest of the finest. Because 1/2 ≠ 1/6. We know that 1/6 < 1/2, so we can set up an equation to see how many copies are needed for them to be equal. (1/6)x = 1/2 [(1/6)x] × 6 = [1/2] × 6 x = 6/2 = 3 This equation shows that 1/6 × 3 = 1/2, therefore we need 3 copies of 1/6 to equal 1 copy of 1/2.	242	696	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	4.0625	4	CC-MAIN-2017-04	latest	en	0.918444	[ 128000, 679, 20, 12, 2839, 12, 1114, 51, 972, 25, 1544, 25, 3487, 12, 2371, 25, 410, 198, 18433, 220, 18, 3115, 220, 16, 14, 21, 374, 220, 18, 14, 21, 902, 649, 387, 44899, 311, 220, 16, 14, 17, 198, 679, 20, 12, 2839, 12, 1114, 51, 972, 25, 3174, 25, 2790, 12, 2371, 25, 410, 271, 14711, 1115, 2209, 264, 36542, 22559, 271, 38034, 1908, 11503, 6782, 15062, 11, 57042, 2038, 348, 34170, 369, 555, 264, 1450, 2320, 19011, 2128, 315, 11909, 13, 31417, 398, 706, 11990, 315, 1579, 4367, 11503, 11, 682, 315, 1124, 15884, 87316, 555, 1057, 1455, 22542, 4029, 3697, 11, 719, 23759, 11503, 527, 279, 28807, 315, 279, 28807, 627, 18433, 220, 16, 14, 17, 4194, 126582, 220, 16, 14, 21, 382, 1687, 1440, 430, 220, 16, 14, 21, 366, 220, 16, 14, 17, 11, 779, 584, 649, 743, 709, 459, 24524, 311, 1518, 1268, 1690, 11236, 527, 4460, 369, 1124, 311, 387, 6273, 382, 7, 16, 14, 21, 51824, 284, 220, 16, 14, 17, 198, 9896, 16, 14, 21, 51824, 60, 25800, 220, 21, 284, 510, 16, 14, 17, 60, 4194, 18028, 220, 21, 198, 87, 284, 220, 21, 14, 17, 284, 220, 18, 271, 2028, 24524, 5039, 430, 220, 16, 14, 21, 4194, 18028, 220, 18, 284, 220, 16, 14, 17, 11, 9093, 584, 1205, 220, 18, 11236, 315, 220, 16, 14, 21, 311, 6273, 220, 16, 3048, 315, 220, 16, 14, 17, 13, 128001 ]	[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ]
https://math.answers.com/Q/How_do_you_work_the_prime_factor_problem	1,708,672,848,000,000,000	text/html	crawl-data/CC-MAIN-2024-10/segments/1707947474361.75/warc/CC-MAIN-20240223053503-20240223083503-00397.warc.gz	392,306,394	46,008	0 # How do you work the prime factor problem? Updated: 8/18/2019 Wiki User 14y ago 1. Find out all the factors for the given number 2. Select the prime numbers out of them 3. Try to divide the number by those prime numbers and see if any combination is possible. Ex: 8 as a prime factor multiple = 2 * 2 * 2 Wiki User 14y ago Earn +20 pts Q: How do you work the prime factor problem? Submit Still have questions? Related questions ### What is the LCM with prime factorization using a factor tree using numbers 3 and 4? That's a lot of extra work for this problem but here goes. 3 is already prime so it doesn't really have a factor tree or prime factorization. The prime factorization of 4 is 2 x 2 which looks like this in a factor tree.42,23 and 4 have no common prime factors, so the LCM is their product, 12 ### Is a factor a prime number? a factor is a factor , a prime number is a prime number, but a prime factor is a factor which is a prime number. it has to depend on what number number. exp 13 is a prime number ### How do you show work for finding prime numbers? Use a factor tree. ### Is three a prime factor? Yes. Any time three is a factor, it is a prime factor. ### Is 6 a prime factor of 132? It is a factor of 132, not a prime factor. ### What is a common prime factor? All numbers have factors. Some factors are prime numbers. A prime factor is a factor that is a prime number. A common prime factor is a prime factor that appears on the list of factors of two or more given numbers. ### Is 19 factor or prime? 19 is a prime factor. ### Is 35 a prime or a factor? 35 can be a factor, but it is not prime. ### Do factor trees always work for determining the greatest common factors? If you construct them correctly, factor trees always work to determine the prime factorization of a number. Once you compare the prime factorizations of two or more numbers, it is relatively easy to find the greatest common factor of them from there. ### What is the prime factor of 37? There is no factorization. 37 is a prime number.37 is already prime. Its only prime factor is itself. ### Is 83 a prime factor or composite factor? 81 is not prime. Its prime factorization is 3x3x3x3. ### Is 160 a prime factor or composite factor? 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https://www.reference.com/math/answer-multiplication-problem-called-1f10237e55b6f4b5	1,472,395,296,000,000,000	text/html	crawl-data/CC-MAIN-2016-36/segments/1471982939917.96/warc/CC-MAIN-20160823200859-00191-ip-10-153-172-175.ec2.internal.warc.gz	945,239,940	21,145	Q: # What is the answer to a multiplication problem called? A: The solution to a multiplication problem is called the "product." For example, the product of 2 and 3 is 6. When the word "product" appears in a mathematical word problem, it is a sign that multiplication is necessary. ## Keep Learning Credit: Mayr CC-BY 2.0 Each of the four basic arithmetical operations has a special term indicating the result. For addition, the word is "sum," for subtraction it is "difference," and for division it is "quotient." The word product comes from the Latin word "producere," meaning to "bring forth" or "draw out." The word quotient comes from the Latin word "quotiens," meaning "how many times." The word sum is also from Latin and is short for "summus," meaning "highest." Sources: ## Related Questions • A: You can find printable multiplication charts online on MathWorksheets4kids.com. On the left-hand side of the homepage, click on "Multiplication" under the heading "Basic Topics." Then click on "Multiplication Tables and Charts." Filed Under: • A: A partial product multiplication algorithm is the process in which each part of one number is multiplied by each part of another number, after which the products are added together. An understanding of place value is necessary to use this algorithm, and the largest numbers are multiplied first. Filed Under: • A: To perform partial product multiplication, you use the distributive property of numbers, multiplying each digit of a number by each digit of the other number and adding the results while taking the place value of each digit into account. One of the most common types of multiplication problems involves a pair of two-digit numbers. 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https://www.physicsforums.com/threads/center-of-mass-and-moment-of-inertia.224068/	1,624,288,434,000,000,000	text/html	crawl-data/CC-MAIN-2021-25/segments/1623488286726.71/warc/CC-MAIN-20210621151134-20210621181134-00179.warc.gz	863,310,887	14,501	# Center of Mass and Moment of Inertia ## Homework Statement Find the center of mass of the collection of mass points in Figure P.12 and then find the moment of inertia of the system about an axis through the center of mass and parallel to the y-axis. ## Homework Equations Center of Mass Moment of Inertia ## The Attempt at a Solution M1 = 1kg, r1 = (1i + 0j) m M2 = 2kg, r2 = (4i + 0j) m M3 = 3kg, r3 = (6i + 0j) m Mass total = 6kg X of CM = 1/6 (0 + 8 + 18) = 13/3 m Y of CM = 1/6 (0 + 0 + 0) = 0 m I = mR$$^{2}$$ I = 6 (4.333)$$^{2}$$ I = 112.6667 kgm$$^{2}$$ The answer was wrong. Thanks for the help #### Attachments • p10-12.gif 3 KB · Views: 369	258	664	{"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	3.640625	4	CC-MAIN-2021-25	latest	en	0.796431	[ 128000, 2, 5955, 315, 9346, 323, 40096, 315, 763, 41222, 271, 567, 83813, 22504, 271, 10086, 279, 4219, 315, 3148, 315, 279, 4526, 315, 3148, 3585, 304, 19575, 393, 13, 717, 323, 1243, 1505, 279, 4545, 315, 78552, 315, 279, 1887, 922, 459, 8183, 1555, 279, 4219, 315, 3148, 323, 15638, 311, 279, 379, 36421, 382, 567, 83813, 11964, 811, 271, 9577, 315, 9346, 198, 84917, 315, 763, 41222, 271, 567, 578, 44617, 520, 264, 12761, 271, 44, 16, 284, 220, 16, 7501, 11, 436, 16, 284, 320, 16, 72, 489, 220, 15, 73, 8, 296, 198, 44, 17, 284, 220, 17, 7501, 11, 436, 17, 284, 320, 19, 72, 489, 220, 15, 73, 8, 296, 198, 44, 18, 284, 220, 18, 7501, 11, 436, 18, 284, 320, 21, 72, 489, 220, 15, 73, 8, 296, 198, 26909, 2860, 284, 220, 21, 7501, 271, 55, 315, 18582, 284, 220, 16, 14, 21, 320, 15, 489, 220, 23, 489, 220, 972, 8, 284, 220, 1032, 14, 18, 296, 198, 56, 315, 18582, 284, 220, 16, 14, 21, 320, 15, 489, 220, 15, 489, 220, 15, 8, 284, 220, 15, 296, 271, 40, 284, 296, 49, 14415, 48922, 17, 92, 14415, 198, 40, 284, 220, 21, 320, 19, 13, 8765, 8, 14415, 48922, 17, 92, 14415, 198, 40, 284, 220, 7261, 13, 10943, 22, 21647, 76, 14415, 48922, 17, 92, 14415, 271, 791, 4320, 574, 5076, 13, 11361, 369, 279, 1520, 271, 827, 49484, 1392, 271, 6806, 281, 605, 12, 717, 16391, 198, 18, 26068, 9787, 25987, 25, 220, 19929, 128001 ]	[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ]
https://studysoup.com/tsg/16706/discrete-mathematics-and-its-applications-7-edition-chapter-2-3-problem-79e	1,660,822,923,000,000,000	text/html	crawl-data/CC-MAIN-2022-33/segments/1659882573193.35/warc/CC-MAIN-20220818094131-20220818124131-00173.warc.gz	485,210,755	15,127	× Get Full Access to Discrete Mathematics And Its Applications - 7 Edition - Chapter 2.3 - Problem 79e Get Full Access to Discrete Mathematics And Its Applications - 7 Edition - Chapter 2.3 - Problem 79e × # a) Show that if a set S has cardinality m, where m is a ISBN: 9780073383095 37 ## Solution for problem 79E Chapter 2.3 Discrete Mathematics and Its Applications \| 7th Edition • Textbook Solutions • 2901 Step-by-step solutions solved by professors and subject experts • Get 24/7 help from StudySoup virtual teaching assistants Discrete Mathematics and Its Applications \| 7th Edition 4 5 1 339 Reviews 24 3 Problem 79E Problem 79E a) Show that if a set S has cardinality m, where m is a positive integer, then there is a one-to-one correspondence between S and the set {1. 2,...,m). b) Show that if S and T are two sets each with m elements. where m is a positive integer, then there is a one-to-one correspondence between S and T. Step-by-Step Solution: Step 1 of 3 J N'= d ot70 d2, d'rt ojto.q'd h_ q'q...q ol'- d o fu yxv-tr^'' \-Jn larngE d bqSe ,' n Upo^mt /l- p,4,d., YnLt =d \- _4_h dn-Y!^ 4n4 : &.r.d Y : /' vn -/ -Yni --]il ct'01 h-h d' d:d ,7- - I -l* dvo Q^=o , A. h.'o Uo - nof rdef iro{ , th b) = ,r'orh... , o{^'^=6)* ) nYJ'i':l }J l^ G l-c )-e 11 Yx &,U # d rohn3I b1e* b {:;ua.{-wy \$e{s];,c\|e&rilfi',,kc * =nv i^1unq b =n,amkrJ borj#{rs {. b ^&\ t I^ n[r= 21 n{pi n&5= ,,nL'=' rf.) nr+j 'h[o) n lv10&ig d Ltnianv,wr+ d,,rh tlrtdx hasw 4-5 ou\ ,1d ,lo,.l 1 e"L,1lg%,,, T\* nwbw I Q: u'{dv\ ,\ (r,n^ 'lh+ b=+A) t "+-'&\ x ax I-X p -5 t'I "; +' w*4r.utr1,tlo oW ) Step 2 of 3 Step 3 of 3 ##### ISBN: 9780073383095 Discrete Mathematics and Its Applications was written by and is associated to the ISBN: 9780073383095. Since the solution to 79E from 2.3 chapter was answered, more than 390 students have viewed the full step-by-step answer. This textbook survival guide was created for the textbook: Discrete Mathematics and Its Applications, edition: 7. The answer to “a) Show that if a set S has cardinality m, where m is a positive integer, then there is a one-to-one correspondence between S and the set {1. 2,...,m).________________b) Show that if S and T are two sets each with m elements. where m is a positive integer, then there is a one-to-one correspondence between S and T.” is broken down into a number of easy to follow steps, and 58 words. The full step-by-step solution to problem: 79E from chapter: 2.3 was answered by , our top Math solution expert on 06/21/17, 07:45AM. This full solution covers the following key subjects: Where, correspondence, show, Integer, set. This expansive textbook survival guide covers 101 chapters, and 4221 solutions. ## Discover and learn what students are asking Calculus: Early Transcendental Functions : The Natural Logarithmic Function: Integration ?In Exercises 1-26, find the indefinite integral. $$\int \frac{4 x^{3}+3}{x^{4}+3 x} d x$$ Statistics: Informed Decisions Using Data : Applications of the Normal Distribution ?If X is a normal random variable with mean 40 and standard deviation 10 and P(X < 38) = 0.4207, then P(X ? 38) = _________ . Statistics: Informed Decisions Using Data : The Normal Approximation to the Binomial Probability Distribution ?In a binomial experiment with n trials and probability of success p, if ________, the binomial random variable X is approximately normal with ?x= and Statistics: Informed Decisions Using Data : Inference about Measures of Central Tendency ?Write a paragraph that describes the logic of the test statistic in a right-tailed sign test. Statistics: Informed Decisions Using Data : Inference about the Difference between Two Medians: Dependent Samples ?Effects of Exercise To determine the effectiveness of an exercise regimen, a physical therapist randomly selects 10 women to participate in a study. 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https://jeopardylabs.com/play/2nd-grade-math-review-495	1,680,139,362,000,000,000	text/html	crawl-data/CC-MAIN-2023-14/segments/1679296949093.14/warc/CC-MAIN-20230330004340-20230330034340-00590.warc.gz	383,273,336	9,756	Addition Subtraction Word Problems What is the missing number? 100 42 + 67 = What is 109? 100 98 - 32 = What is 66? 100 Jake has 95 baseball cards. He gives 24 cards to his little sister. How many cards does Jake have left? What is 71 baseball cards? 100 8 - 2 = 9 - ___ What is 6 ? 200 58 + 32 = What is 90? 200 81 - 65 = What is 16? 200 Heidi is having a party. She needs to make 75 cookies. So far she has only made 56 cookies. How many more cookies does Heidi need to make? What is 19 cookies? 200 12 - ___ = 9 - 4 What is 5? 300 39 + 52 = What is 91? 300 75 - 49 = What is 26? 300 Mrs. Dallman has 37 books in her library. Miss Stern has 26 more books in her library. How many books does Miss Stern have in her library? What is 63 books? 300 19 - 8 = ___ - 10 What is 21? 400 45 + 38 = What is 83? 400 60 - 29 = What is 31? 400 1st grade has 46 students, 2nd grade has 30 more students. How many students does 2nd grade have? What is 76 students? 400 15 - ____ = 20 - 10 What is 5? 500 65 + 25 = What is 90? 500 82 - 47 = What is 35? 500 Jenna read some pages in her book on Sunday. She read 27 pages on Monday. Now she is on page 50 in her book. How many pages did she read on Sunday? What is 23 pages? 500 20 - 5 = ___ - 5 What is 15? 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https://convertoctopus.com/2936-years-to-seconds	1,718,700,055,000,000,000	text/html	crawl-data/CC-MAIN-2024-26/segments/1718198861747.70/warc/CC-MAIN-20240618073942-20240618103942-00891.warc.gz	156,565,159	8,072	## Conversion formula The conversion factor from years to seconds is 31556952, which means that 1 year is equal to 31556952 seconds: 1 yr = 31556952 s To convert 2936 years into seconds we have to multiply 2936 by the conversion factor in order to get the time amount from years to seconds. We can also form a simple proportion to calculate the result: 1 yr → 31556952 s 2936 yr → T(s) Solve the above proportion to obtain the time T in seconds: T(s) = 2936 yr × 31556952 s T(s) = 92651211072 s The final result is: 2936 yr → 92651211072 s We conclude that 2936 years is equivalent to 92651211072 seconds: 2936 years = 92651211072 seconds ## Alternative conversion We can also convert by utilizing the inverse value of the conversion factor. In this case 1 second is equal to 1.0793167066353E-11 × 2936 years. Another way is saying that 2936 years is equal to 1 ÷ 1.0793167066353E-11 seconds. ## Approximate result For practical purposes we can round our final result to an approximate numerical value. We can say that two thousand nine hundred thirty-six years is approximately ninety-two billion six hundred fifty-one million two hundred eleven thousand seventy-two seconds: 2936 yr ≅ 92651211072 s An alternative is also that one second is approximately zero times two thousand nine hundred thirty-six years. ## Conversion table ### years to seconds chart For quick reference purposes, below is the conversion table you can use to convert from years to seconds years (yr) seconds (s) 2937 years 92682768024 seconds 2938 years 92714324976 seconds 2939 years 92745881928 seconds 2940 years 92777438880 seconds 2941 years 92808995832 seconds 2942 years 92840552784 seconds 2943 years 92872109736 seconds 2944 years 92903666688 seconds 2945 years 92935223640 seconds 2946 years 92966780592 seconds	474	1,819	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, 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http://sci.renewable.media/shapes	1,601,241,180,000,000,000	text/html	crawl-data/CC-MAIN-2020-40/segments/1600401578485.67/warc/CC-MAIN-20200927183616-20200927213616-00074.warc.gz	121,668,812	6,504	# Shapes ## Areas of 2D shapes These are some common two-dimensional shapes and the formulae for calculating their areas: ### Triangle The area of a triangle is half its height times its base: \$A = ½h⋅b\$ This works for all triangles. If you made a copy of a triangle, and cut up the copy so that you made a rectangle with the original triangle, you would see that the rectangle has the dimensions of the height and base of the triangle. The area is therefore half of this rectangle! ### Heron's formula The area of a triangle can also be calculated using Heron's formula: \$\$A = √{s(s-a)(s-b)(s-c)}\$\$ Where \$a\$, \$b\$ and \$c\$ are le lengths of the sides of the triangle, and \$s = {a+b+c}/2\$ Square: \$A = L^2\$, where L is the length of one side. Rectangle: \$A = L ⋅ W\$, where L is the length, and W is the width. Rhomboid (elongated diamond): \${pq}/2\$, where p and q are the lengths of the two diagonals. Rhombus ('pushed over rectangle'): \$L ⋅ H\$, where L is the length and H is the height (perpendicular to L). ## Triangles In two dimensions (such as on a flat piece of paper), the angles of a triangle all add up to 180°. The equilateral triangle has all three side and all three angles equal. The angles are 60° each. The isoceles triangle has two sides the same length. This requires two angles to be same as well. Trapezium: one set of two parallel sides Isoceles trapezium: two sides equal length, the other two sides parallel Parallelogram: two sets of parallel and equal length sides Rhombus: two sets of two parallel sides, 4 equal lengths, two sets of identical angles ## Lines of Symmetry A useful way to describe a shape is to state its number of reflection lines of symmetry. These are the number of straight lines that may be drawn through a shape, across which a reflection of the shape would result in an identical shape. ### Triangle lines of symmetry Triangles may have zero, one or three lines of symmetry, depending on their type. An equilateral triangle has 3 lines of symmetry. Equilateral triangles have 3 lines of symmetry Line of symmetry for an isoceles triangle An isoceles has only one. Other types of triangles have no lines of symmetry, so their orientation is unique. A square reflects across a horizontal line through its centre, and a vertical line through its centre. It also has lines of symmetry across its diagonals. It therefore has four lines of reflection symmetry. A rhombus, like a rectangle, has 2 lines of symmetry A rhombus and a rectangle have only 2 lines of symmetry. A parallelogram has no lines of symmetry. ## Rotational Symmetry A shape which has rotational or radial symmetry is one which, when rotated 360° reassumes an identical form to the starting position one or more times. Order of rotational symmetry: the number of times an object takes an identical shape while being rotated through 360°. e.g. a square has rotational symmetry of 4, a rhombus 2, a triangle 3. ## Site Index ### Latest Item on Science Library: The most recent article is: Trigonometry View this item in the topic: Vectors and Trigonometry and many more articles in the subject: ### Mathematics Mathematics is the most important tool of science. The quest to understand the world and the universe using mathematics is as old as civilisation, and has led to the science and technology of today. Learn about the techniques and history of mathematics on ScienceLibrary.info. ### Great Scientists #### Dmitri Mendeleev 1834 - 1907 Dmitri Mendeleev, 1834 - 1907, was a Russian chemist who developed the modern Periodic Table of Elements. ### Quote of the day... We've long since won the argument about climate change; we've just lost the fight. 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http://clay6.com/qa/2855/show-that-the-matrix-a-begin-1-1-5-1-2-1-5-1-3-end-is-a-symmetric-matrix-	1,529,790,716,000,000,000	text/html	crawl-data/CC-MAIN-2018-26/segments/1529267865250.0/warc/CC-MAIN-20180623210406-20180623230406-00178.warc.gz	70,878,243	26,205	# Show that the matrix $A = \begin{bmatrix} 1 & -1 & 5 \\ -1 & 2 & 1 \\ 5 & 1 & 3 \end{bmatrix}$ is a symmetric matrix. ## 1 Answer Toolbox: • If A_{i,j} be a matrix m*n matrix , then the matrix obtained by interchanging the rows and column of A is called as transpose of A. • A square matrix A=[a$_{ij}$] is said to be symmetric if A'=A that is $[a_{ij}]=[a_{ji}]$ for all possible value of i and j. Given $A = \begin{bmatrix} 1 & -1 & 5 \\ -1 & 2 & 1 \\ 5 & 1 & 3 \end{bmatrix}$ Transpose of a matrix can be obtained by interchanging the rows and columns. $A'=\begin{bmatrix}1 & -1 & 5\\-1 & 2 & 1\\5 & 1 & 3\end{bmatrix}$ A square matrix is said to be symmetric if A'=A. $\Rightarrow [a_{ji}]=[a_{ij}]$ for all values of i & j. $a_{21}=-1=a_{12}$ $a_{31}=5=a_{13}$ $a_{32}=1=a_{23}$ $a_{11}=1=a_{11}$ $a_{22}=2=a_{22}$ $a_{33}=3=a_{33}$ Hence $a_{ji}=a_{ij}$ Therefore A is symmetric matrix. answered Apr 12, 2013 1 answer 1 answer 1 answer 1 answer 1 answer 1 answer 1 answer	379	988	{"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	4.375	4	CC-MAIN-2018-26	latest	en	0.640252	[ 128000, 2, 7073, 430, 279, 6303, 400, 32, 284, 1144, 7413, 90, 65, 18602, 92, 220, 16, 612, 482, 16, 612, 220, 20, 26033, 482, 16, 612, 220, 17, 612, 220, 16, 26033, 220, 20, 612, 220, 16, 612, 220, 18, 1144, 408, 90, 65, 18602, 32816, 374, 264, 55443, 6303, 382, 567, 220, 16, 22559, 271, 7896, 2054, 512, 6806, 1442, 362, 15511, 72, 10540, 92, 387, 264, 6303, 296, 24942, 6303, 1174, 1243, 279, 6303, 12457, 555, 958, 52813, 279, 7123, 323, 3330, 315, 362, 374, 2663, 439, 52023, 315, 362, 627, 6806, 362, 9518, 6303, 362, 5941, 64, 6535, 90, 3251, 32816, 60, 374, 1071, 311, 387, 55443, 422, 362, 61385, 32, 430, 374, 400, 58, 64, 15511, 3251, 92, 67228, 64, 15511, 7910, 26516, 3, 369, 682, 3284, 907, 315, 602, 323, 503, 627, 22818, 198, 3, 32, 284, 1144, 7413, 90, 65, 18602, 92, 220, 16, 612, 482, 16, 612, 220, 20, 26033, 482, 16, 612, 220, 17, 612, 220, 16, 26033, 220, 20, 612, 220, 16, 612, 220, 18, 1144, 408, 90, 65, 18602, 92, 26101, 89298, 315, 264, 6303, 649, 387, 12457, 555, 958, 52813, 279, 7123, 323, 8310, 627, 3, 32, 6, 35533, 7413, 90, 65, 18602, 92, 16, 612, 482, 16, 612, 220, 20, 3505, 12, 16, 612, 220, 17, 612, 220, 16, 3505, 20, 612, 220, 16, 612, 220, 18, 59, 408, 90, 65, 18602, 92, 26101, 32, 9518, 6303, 374, 1071, 311, 387, 55443, 422, 362, 61385, 32, 627, 59836, 27338, 510, 64, 15511, 7910, 92, 67228, 64, 15511, 3251, 26516, 3, 369, 682, 2819, 315, 602, 612, 503, 627, 40662, 15511, 1691, 92, 11065, 16, 25222, 15511, 717, 92, 26101, 40662, 15511, 2148, 52285, 20, 25222, 15511, 1032, 92, 26101, 40662, 15511, 843, 52285, 16, 25222, 15511, 1419, 92, 26101, 40662, 15511, 806, 52285, 16, 25222, 15511, 806, 92, 26101, 40662, 15511, 1313, 52285, 17, 25222, 15511, 1313, 92, 26101, 40662, 15511, 1644, 52285, 18, 25222, 15511, 1644, 92, 26101, 39, 768, 400, 64, 15511, 7910, 52285, 64, 15511, 3251, 92, 26101, 55915, 362, 374, 55443, 6303, 627, 57824, 5186, 220, 717, 11, 220, 679, 18, 271, 16, 4320, 271, 16, 4320, 271, 16, 4320, 271, 16, 4320, 271, 16, 4320, 271, 16, 4320, 271, 16, 4320, 128001 ]	[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ]
http://www.mathacademy.com/pr/prime/browse.asp?LT=L&ANCHOR=space0000000000000000000000000&LEV=&TBM=&TAL=&TAN=&TBI=&TCA=&TCS=&TDI=&TEC=&TFO=&TGE=&TGR=&THI=&TNT=&TPH=&TST=&TTO=&TTR=&TAD=	1,369,198,846,000,000,000	text/html	crawl-data/CC-MAIN-2013-20/segments/1368701314683/warc/CC-MAIN-20130516104834-00018-ip-10-60-113-184.ec2.internal.warc.gz	598,316,439	6,029	BROWSE ALPHABETICALLY LEVEL: Elementary Advanced Both INCLUDE TOPICS: Basic Math Algebra Analysis Biography Calculus Comp Sci Discrete Economics Foundations Geometry Graph Thry History Number Thry Phys Sci Statistics Topology Trigonometry space – Tarski Truth Theorem space Any abstract set with a structure defined on it, such as an order relation, metric, etc.Cf. Euclidean space, Hilbert space, metric space, topological space. sphere A closed surface, all points of which are equidistant from a given point, called the center. In 3-dimensional Euclidean space, the equation of a sphere of radius r and center (h, j, k) isThe term sphere may also refer to the solid bounded by this surface, and the interior is then called the open sphere of radius r.More generally, a sphere may be defined as the set of points in n-dimensional space (or any metric space) equidistant from a given point. The unit sphere in n-dimensional space is typically denoted S n - 1. Thus, the unit sphere in ordinary 3-space is denoted S2, and the unit circle in the plane is denoted S1. square A regular polygon having four equal sides and four right angles. stationary set If a is an ordinal, a set S in a is called stationary if S has non-empty intersection with every closed unbounded subset of a. story problem ARTICLE A mathematical problem presented as a real-world situation. See the article for problem solving techniques. subset A set A is a subset of a set B if every element of A is also an element of B. If in addition B is a subset of A, then A = B, but if not then A may be said to be a proper subset of B.Cf. superset. subtract To subtract a number m from a number n is to calculate the difference of m and n. If m is less than n we take the positive difference, otherwise we take the negative of the difference. This is tantamount to adding the negative of m to n. successor In a structure with an order relation defined upon it, the successor of an element a is the least element greater than a, if such exists.Cf. predecessor. sumset Given a set A, the sumset of A, denoted byis the set containing all of the elements of the elements of A, that is, it is the union of the elements of A. sumset axiom An axiom of set theory which states that if A is any set, then the sumset of A is also a set. sup Abbreviation of supremum. superset A set A is a superset of a set B if every element of B is an element of A.Cf. subset. supplemental angles Two angles are supplemental if they add up to 180 degrees (p radians).Cf. complementary angles. supremum The supremum of any subset of a linearly ordered set is the least upper bound of the subset. In particular, the supremum of any set of numbers is the smallest number in the set which is greater than or equal to every number in the set. In a complete linear order the supremum of any bounded set always exists.Cf. infimum, least upper bound axiom. surd (rare) An irrational root of a number, e.g., the square root of two. Related article: Irrationality of the Square Root of 2 surjection A surjective function, i.e., a function that maps at least one element of its domain to each element of its range.Cf. injection, bijection. Suslin tree Set Theory: For a an infinite cardinal, an a-Suslin tree is a tree T such that \|T\| = a, and every chain and every antichain of T has cardinality less than a.Cf. Aronszajn tree. symbolic logic Logic reduced to syntax, i.e., which works only with uninterpreted symbols. The two most often used kinds of symbolic logic are the propositional calculus and the predicate calculus. symmetric difference The symmetric difference of two sets A and B is the set of those elements that are in either A or B but not both.Cf. intersection, union. symmetric relation A relation “ ~ ” on a set X is symmetric if for any two elements x and y in X we have x ~ y if and only if y ~ x. The relation “ ~ ” is called asymmetric if for any two elements x and y we have that x ~ y implies it is not true that y ~ x, and it is called antisymmetric if whenever x ~ y and y ~ x then x = y. Note that a relation may be neither symmetric, asymmetric, nor antisymmetric.Cf. reflexive relation, transitive relation. Tarski Truth Theorem Let T be a mathematical theory. Denote formulas of T by j, y, etc., and let Tj be the statement that “the sentence j is true in T ” Choose a canonical numbering of the formulas of T (e.g., Gödel numbering) that assigns to each sentence of T a unique positive integer, and denote the positive integer associated with a sentence j by ‘j’. Then there is no formula y in T such that y(‘j’)Tj. In other words, “truth” in a theory T is not definable in T. space – Tarski Truth Theorem HOME \| ABOUT \| CONTACT \| AD INFO \| PRIVACYCopyright © 1997-2013, Math Academy Online™ / Platonic Realms™. Except where otherwise prohibited, material on this site may be printed for personal classroom use without permission by students and instructors for non-profit, educational purposes only. All other reproduction in whole or in part, including electronic reproduction or redistribution, for any purpose, except by express written agreement is strictly prohibited. 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https://artofproblemsolving.com/wiki/index.php?title=2012_AIME_I_Problems/Problem_6&diff=next&oldid=48045	1,679,539,377,000,000,000	text/html	crawl-data/CC-MAIN-2023-14/segments/1679296944606.5/warc/CC-MAIN-20230323003026-20230323033026-00435.warc.gz	141,358,574	11,945	# Difference between revisions of "2012 AIME I Problems/Problem 6" ## Problem 6 The complex numbers $z$ and $w$ satisfy $z^{13} = w,$ $w^{11} = z,$ and the imaginary part of $z$ is $\sin{\frac{m\pi}{n}}$, for relatively prime positive integers $m$ and $n$ with $m Find $n.$ ## Solutions ### Solution 1 Substituting the first equation into the second, we find that $(z^{13})^{11} = z$ and thus $z^{142} = 1.$ So $z$ must be a $142$nd root of unity, and thus the imaginary part of $z$ will be $\sin{\frac{2m\pi}{142}} = \sin{\frac{m\pi}{71}}$ for some $m$ with $0 \le m < 142.$ But note that $71$ is prime and $m<71$ by the conditions of the problem, so the denominator in the argument of this value will always be $71$ and thus $n = \boxed{071.}$ ### Solution 2 Note that $w^{143}=w$ and similar for $z$, and they are not equal to $0$ because the question implies the imaginary part is positive. Thus $w^{142}=z^{142}=1$, so each is of the form $sin(2 \pi k/142)$ where $k$ is a positive integer between $0$ and $141$ inclusive. This simplifies to $sin(pi*k/71)$, and $071$ is prime, so it is the only possible denominator, and thus is the answer.	366	1,153	{"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 32, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	4.4375	4	CC-MAIN-2023-14	latest	en	0.857419	[ 128000, 2, 56180, 1990, 54862, 315, 330, 679, 17, 362, 5778, 358, 45635, 14, 32298, 220, 21, 1875, 567, 22854, 220, 21, 271, 791, 6485, 5219, 400, 89, 3, 323, 400, 86, 3, 27651, 400, 89, 48922, 1032, 92, 284, 289, 4884, 400, 86, 48922, 806, 92, 284, 1167, 4884, 323, 279, 51052, 961, 315, 400, 89, 3, 374, 59060, 16319, 36802, 38118, 90, 76, 59, 2554, 15523, 77, 3500, 55976, 369, 12309, 10461, 6928, 26864, 400, 76, 3, 323, 400, 77, 3, 449, 400, 76, 7531, 400, 77, 2475, 271, 567, 23508, 271, 14711, 12761, 220, 16, 271, 3214, 3781, 10831, 279, 1176, 24524, 1139, 279, 2132, 11, 584, 1505, 430, 5035, 89, 48922, 1032, 5525, 48922, 806, 92, 284, 1167, 3, 323, 8617, 400, 89, 48922, 10239, 92, 284, 220, 16, 2475, 2100, 400, 89, 3, 2011, 387, 264, 400, 10239, 3, 303, 3789, 315, 31426, 11, 323, 8617, 279, 51052, 961, 315, 400, 89, 3, 690, 387, 59060, 16319, 36802, 38118, 90, 17, 76, 59, 2554, 15523, 10239, 3500, 284, 1144, 16319, 36802, 38118, 90, 76, 59, 2554, 15523, 6028, 3500, 3, 369, 1063, 400, 76, 3, 449, 400, 15, 1144, 273, 296, 366, 220, 10239, 2475, 2030, 5296, 430, 400, 6028, 3, 374, 10461, 323, 400, 76, 27, 6028, 3, 555, 279, 4787, 315, 279, 3575, 11, 779, 279, 48012, 304, 279, 5811, 315, 420, 907, 690, 2744, 387, 400, 6028, 3, 323, 8617, 400, 77, 284, 1144, 80175, 90, 24508, 13, 32816, 271, 14711, 12761, 220, 17, 271, 9290, 430, 400, 86, 48922, 10290, 52285, 86, 3, 323, 4528, 369, 400, 89, 55976, 323, 814, 527, 539, 6273, 311, 400, 15, 3, 1606, 279, 3488, 24897, 279, 51052, 961, 374, 6928, 13, 14636, 400, 86, 48922, 10239, 52285, 89, 48922, 10239, 52285, 16, 55976, 779, 1855, 374, 315, 279, 1376, 400, 16319, 7, 17, 1144, 2554, 597, 14, 10239, 15437, 1405, 400, 74, 3, 374, 264, 6928, 7698, 1990, 400, 15, 3, 323, 400, 9335, 3, 29408, 13, 1115, 15858, 9803, 311, 400, 16319, 52364, 43566, 14, 6028, 15437, 11, 323, 400, 24508, 3, 374, 10461, 11, 779, 433, 374, 279, 1193, 3284, 48012, 11, 323, 8617, 374, 279, 4320, 13, 128001 ]	[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ]
https://www.varsitytutors.com/hspt_math-help/concepts/arithmetic?page=6	1,582,165,246,000,000,000	text/html	crawl-data/CC-MAIN-2020-10/segments/1581875144498.68/warc/CC-MAIN-20200220005045-20200220035045-00366.warc.gz	948,452,143	46,512	# HSPT Math : Arithmetic ## Example Questions ### Example Question #51 : Concepts Fill in the blank with the proper sign. __________ Explanation: In order to solve this problem, we will use a number line. Numbers to the left of zero on the line are less than numbers to the right. Therefore: ### Example Question #11 : Understand An Inequality On A Number Line: Ccss.Math.Content.6.Ns.C.7a Fill in the blank with the proper sign. __________ Explanation: In order to solve this problem, we will use a number line. Numbers to the left of zero on the line are less than numbers to the right. Therefore: ### Example Question #51 : Concepts Fill in the blank with the proper sign. __________ Explanation: In order to solve this problem, we will use a number line. Numbers to the left of zero on the line are less than numbers to the right. Therefore: ### Example Question #51 : Negative Numbers Fill in the blank with the proper sign. __________ Explanation: In order to solve this problem, we will use a number line. Numbers to the left of zero on the line are less than numbers to the right. Therefore: ### Example Question #52 : Concepts Fill in the blank with the proper sign. __________ Explanation: In order to solve this problem, we will use a number line. Numbers to the left of zero on the line are less than numbers to the right. Therefore: ### Example Question #53 : Concepts Fill in the blank with the proper sign. __________ Explanation: In order to solve this problem, we will use a number line. Numbers to the left of zero on the line are less than numbers to the right. Therefore: ### Example Question #11 : Understand An Inequality On A Number Line: Ccss.Math.Content.6.Ns.C.7a Fill in the blank with the proper sign. __________ Explanation: In order to solve this problem, we will use a number line. Numbers to the left of zero on the line are less than numbers to the right. Therefore: ### Example Question #51 : Concepts Fill in the blank with the proper sign. __________ Explanation: In order to solve this problem, we will use a number line. Numbers to the left of zero on the line are less than numbers to the right. Therefore: ### Example Question #471 : The Number System Fill in the blank with the proper sign. __________ Explanation: In order to solve this problem, we will use a number line. Numbers to the left of zero on the line are less than numbers to the right. Therefore: ### Example Question #11 : Understand An Inequality On A Number Line: Ccss.Math.Content.6.Ns.C.7a Fill in the blank with the proper sign. __________	580	2,608	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	4.3125	4	CC-MAIN-2020-10	latest	en	0.834709	[ 128000, 2, 34514, 2898, 4242, 551, 94084, 271, 567, 13688, 24271, 271, 14711, 13688, 16225, 674, 3971, 551, 76872, 271, 14788, 304, 279, 10321, 449, 279, 6300, 1879, 382, 4067, 25174, 70869, 1473, 644, 2015, 311, 11886, 420, 3575, 11, 584, 690, 1005, 264, 1396, 1584, 13, 35813, 311, 279, 2163, 315, 7315, 389, 279, 1584, 527, 2753, 1109, 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https://www.cuemath.com/maths/pre-number-math/	1,585,938,321,000,000,000	text/html	crawl-data/CC-MAIN-2020-16/segments/1585370515113.54/warc/CC-MAIN-20200403154746-20200403184746-00057.warc.gz	843,240,922	19,491	In the verge of coronavirus pandemic, we are providing FREE access to our entire Online Curriculum to ensure Learning Doesn't STOP! # Pre-number Math Go back to 'Maths' ## Introduction to Pre-number Math Building pre-number math skills is a prerequisite to understanding numbers. So even before young children (aged 3 to 5 years) start with numbers, a fair amount of time should be invested in building these pre-number skills. Pre-number skills like matching, sorting, classifying, ordering and comparing will set the stage to build a strong number sense. Numbers should be introduced only after this. It’s after the numbers that the number names can be introduced. Because once the child is familiar with the numbers, associating it with the spelling is easier. ## The Big Idea At Cuemath pre-number math skills are built-in preschool years. It begins by identifying and working with objects like colour counters, pattern blocks and number blocks. Then moving on to working with dots. Counting the dots and associating it with the right numbers. After the child has gained decent proficiency over this they move to counting and identifying only numbers. Thus, laying a strong foundation for number sense. ## How do I understand? A preferable way to introduce pre-number skills is by using physical objects like pattern blocks. Children can be asked to match, sort, classify pattern blocks. From the concrete (pattern blocks) one can move to matching and sorting images. Children can now be introduced to different quantities. An association between the number and the quantity needs to be formed. This association will help the children to understand the largeness or smallness of a number. Avoid introducing numbers in its abstract form. Rote memorization of numbers without visualizing its magnitude won’t build a good number sense. ## How to Visualise numbers using counters? This video explains how numbers can be logically introduced. Instead of rote memorizing them here is an interesting way to understand numbers. Using counters will strengthen the number sense. ## Sub Topics Here are a few links that will take you through the journey that every Cuemath student undertakes in the pursuit of understanding Pre-Number Math along with practice worksheets:	445	2,278	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	4.09375	4	CC-MAIN-2020-16	latest	en	0.909353	[ 128000, 644, 279, 59818, 315, 33333, 28522, 11, 584, 527, 8405, 16655, 2680, 311, 1057, 4553, 8267, 75306, 311, 6106, 21579, 49932, 956, 46637, 2268, 2, 5075, 26939, 4242, 271, 11087, 1203, 311, 4194, 364, 8991, 82, 3961, 567, 29438, 311, 5075, 26939, 4242, 271, 31233, 864, 26939, 7033, 7512, 374, 264, 80884, 311, 8830, 5219, 13, 2100, 1524, 1603, 3995, 2911, 320, 3359, 220, 18, 311, 220, 20, 1667, 8, 1212, 449, 5219, 11, 264, 6762, 3392, 315, 892, 1288, 387, 29091, 304, 4857, 1521, 864, 26939, 7512, 13, 5075, 26939, 7512, 1093, 12864, 11, 29373, 11, 538, 7922, 11, 22106, 323, 27393, 690, 743, 279, 6566, 311, 1977, 264, 3831, 1396, 5647, 13, 35813, 1288, 387, 11784, 1193, 1306, 420, 13, 1102, 753, 1306, 279, 5219, 430, 279, 1396, 5144, 649, 387, 11784, 13, 9393, 3131, 279, 1716, 374, 11537, 449, 279, 5219, 11, 4189, 1113, 433, 449, 279, 43529, 374, 8831, 382, 567, 578, 6295, 52101, 271, 1688, 27560, 336, 589, 864, 26939, 7033, 7512, 527, 5918, 3502, 61905, 1667, 13, 1102, 12302, 555, 25607, 323, 3318, 449, 6302, 1093, 12745, 32632, 11, 5497, 10215, 323, 1396, 10215, 13, 5112, 7366, 389, 311, 3318, 449, 32094, 13, 4605, 287, 279, 32094, 323, 4189, 1113, 433, 449, 279, 1314, 5219, 13, 220, 4194, 6153, 279, 1716, 706, 18661, 15326, 63239, 927, 420, 814, 3351, 311, 26060, 323, 25607, 1193, 5219, 13, 14636, 11, 35744, 264, 3831, 16665, 369, 1396, 5647, 382, 567, 2650, 656, 358, 3619, 1980, 32, 70668, 1648, 311, 19678, 864, 26939, 7512, 374, 555, 1701, 7106, 6302, 1093, 5497, 10215, 13, 15394, 649, 387, 4691, 311, 2489, 11, 3460, 11, 49229, 5497, 10215, 13, 5659, 279, 14509, 320, 14676, 10215, 8, 832, 649, 3351, 311, 12864, 323, 29373, 5448, 13, 15394, 649, 1457, 387, 11784, 311, 2204, 33776, 13, 1556, 15360, 1990, 279, 1396, 323, 279, 12472, 3966, 311, 387, 14454, 13, 1115, 15360, 690, 1520, 279, 2911, 311, 3619, 279, 4143, 24639, 477, 2678, 2136, 315, 264, 1396, 13, 35106, 33018, 5219, 304, 1202, 8278, 1376, 13, 432, 1295, 16420, 2065, 315, 5219, 2085, 9302, 4954, 1202, 26703, 2834, 1431, 1977, 264, 1695, 1396, 5647, 382, 567, 2650, 311, 20796, 1082, 5219, 1701, 32632, 1980, 2028, 2835, 15100, 1268, 5219, 649, 387, 74145, 11784, 13, 12361, 315, 938, 668, 16420, 4954, 1124, 1618, 374, 459, 7185, 1648, 311, 3619, 5219, 13, 12362, 32632, 690, 20259, 279, 1396, 5647, 382, 567, 3804, 41994, 271, 8586, 527, 264, 2478, 7902, 430, 690, 1935, 499, 1555, 279, 11879, 430, 1475, 27560, 336, 589, 5575, 4194, 1263, 531, 2094, 304, 279, 33436, 315, 8830, 5075, 12, 2903, 4242, 3235, 449, 6725, 68625, 25, 128001 ]	[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ]
https://capablemachine.wordpress.com/tag/bernoulli/	1,656,848,997,000,000,000	text/html	crawl-data/CC-MAIN-2022-27/segments/1656104240553.67/warc/CC-MAIN-20220703104037-20220703134037-00134.warc.gz	215,941,004	19,899	# Tag: Bernoulli • ## A brief Introduction to Probability Distribution for Machine Learning Probability Distributions are prevalent in many fields, namely, computer science, stock market, astronomy and economics. In this blog we are going to see different probability distribution for Machine Learning and their properties. Also, we will discuss key statistical points for the simple models. Note, all the discussion in this blog is based on the assumption that all data points are independent and identically distributed. “A probability distribution for Machine Learning is a statistical method that describes all the possible values and likelihoods that a random variable can take within a given interval.” Always remember the issue of choosing an appropriate distribution relates to the problem of model selection. ###### Discrete Probability Distributions It is distribution of all possible values of a discrete random variable together with an indication of their probabilities. Some examples of well known discrete probability distribution for Machine Learning are: • Bernoulli distributions. • Poisson distribution. • Uniform distribution. • Multinomial distributions. ###### Bernoulli Distribution Bernoulli Distribution is widely use for distribution of categorical outcomes. Concept behind logistic regression is best example of Bernoulli distribution. Let’s consider you have toss a coin ‘c’ , c ∈ {0, 1}. such that c = 1 representing heads and c = 0 representing tails. Let the probability of head is denoted by ‘μ’ ,such that: P(x=1 \| μ) = μ ; 0 =< μ =< 1 So, P(x=0 \| μ) = 1 – μ The probability distribution for flipping a coin is therefore can be written as: Bern (c \| μ) = μc (1 -μ)1-c This is known as Bernoulli Distribution. From this equation we can easily said that this distribution is normalize. Now, suppose we have a data set X = {c1, c2, c3,….cn} of observed value of c. So, for this we can construct maximum likelihood function, which is function of μ. Such that P( X \| μ) is given as: P(X \| μ) = ∏ P(c \| μ) = ∏ μc (1 -μ)1-c where; c = {c1, c2,…….cn} Now, we can estimate a value for μ by maximizing the logarithm of the likelihood and it is given as: lnP(X \| μ) = ΣP(cn \| μ) = Σ{cn ln μ + (1 + cn) ln(1 – μ)} Now if you remember, this is a same equation we have used in logistic regression for the error calculation. ###### Poisson Distribution A Poisson distribution is a measure of how many times an event is likely to occur within given period of time. It’s super handy because it’s pretty simple to use and is applicable for tons of things, there are a lot of interesting processes that boil down to “events that happen in time or space.” So, the probability for total ‘E’ events in ‘X’ period of time is given as: P(E events in X period of time) = e(-( E / T ) * X) * { (E / T) * X)k} / K! Let λ = (E / T) * X) Then, P(E events in X period of time) = eλ * {λ}k/ K! Where T is time for each event, X is total time interval and λ is a rate parameter which is the expected number of events in the interval, i.e. (E / T) * X). ###### Continuous Probability Distributions It is distribution of all possible values of a continuous random variable together with an indication of their probabilities. Some examples of well-known continuous probability distribution for Machine Learning are: • Normal or Gaussian distribution. • Power-law distribution. • Pareto distribution. ###### Gaussian Distribution The Gaussian, also known as normal distribution, is a widely used model for the distribution of continuous variables. When you work on large data it is observed that most of the data is closer to mean and the very less frequent data is observed towards the extremes, which is nothing but a gaussian distribution, and this is what central limit theorem tells us. In the case of a single variable ‘x’, gaussian distribution is given as: Where μ is the mean and σ² is the variance. For the single real variable, the distribution that maximizes the entropy is the Gaussian. Also, the Gaussian Distribution arises is when we consider the sum of multiple random variables. They may used to find non-linear regression as well as to reduce dimensionality by identifying which dimensions of a dataset have larger variance. Lots of phenomenon in nature follows the Gaussian Distribution. Like, Our height, weight, blood pressure etc. Hence it is a widely used distribution and favorite of many data scientists. 😉 I hope this article helped you in your data science journey. Was it explanatory? If you have any doubts and want to see more articles on distributions, please do write in the comment section. ###### Written by – Sarang Deshmukh Co – Founder and Developer @capablemachine.com \| Performance Engineer @AMDOCS	1,074	4,776	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	4.09375	4	CC-MAIN-2022-27	latest	en	0.895739	[ 128000, 2, 12633, 25, 14502, 283, 41076, 271, 6806, 7860, 362, 10015, 29438, 311, 87739, 35009, 369, 13257, 4194, 48567, 271, 89564, 423, 18478, 527, 46941, 304, 1690, 5151, 11, 32125, 11, 6500, 8198, 11, 5708, 3157, 11, 82213, 323, 28989, 13, 763, 420, 5117, 584, 527, 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https://nethercraft.net/kmap-calculator/	1,721,323,830,000,000,000	text/html	crawl-data/CC-MAIN-2024-30/segments/1720763514848.78/warc/CC-MAIN-20240718161144-20240718191144-00865.warc.gz	352,677,759	12,855	# Kmap Calculator ## Kmap Calculator: A Helpful Tool for Simplifying Boolean Functions When working with Boolean functions in digital electronics, simplifying expressions can be a tedious and time-consuming task. This is where a Kmap calculator comes in handy. A Karnaugh map, or Kmap for short, is a graphical method used to simplify Boolean algebra expressions. By using a Kmap calculator, you can easily reduce complex Boolean functions to their simplest form, making circuit designing and analysis more efficient and error-free. ## How Does a Kmap Calculator Work? A Kmap calculator takes as input the truth table of a Boolean function and creates a Karnaugh map representation of the function. The Kmap is then used to identify patterns and groupings of ones in the truth table, which can be used to simplify the function. The calculator applies the rules of Boolean algebra to combine adjacent ones in the map, resulting in a simplified expression that is equivalent to the original function. Also Check This Residual Calculator ## Benefits of Using a Kmap Calculator There are several advantages to using a Kmap calculator when working with Boolean functions: • Efficiency: Simplifying Boolean expressions manually can be time-consuming and prone to errors. A Kmap calculator automates the simplification process, saving time and reducing the risk of mistakes. • Accuracy: By applying the rules of Boolean algebra systematically, a Kmap calculator ensures that the simplified expression is equivalent to the original function. • Visual representation: Kmaps provide a visual representation of the Boolean function, making it easier to identify patterns and groupings for simplification. • Ease of use: Kmap calculators are user-friendly and require minimal input from the user, making them accessible to beginners and experts alike. Also Check This Reflection Calculator ## How to Use a Kmap Calculator Using a Kmap calculator is a straightforward process that involves the following steps: 1. Enter the truth table of the Boolean function into the calculator. 2. Generate the Karnaugh map representation of the function. 3. Identify patterns and groupings of ones in the map. 4. Combine adjacent ones to simplify the expression. 5. Verify the simplified expression with the original function to ensure correctness. ## Applications of Kmap Calculators Kmap calculators are commonly used in digital electronics and computer science for simplifying Boolean functions in circuit design, logic gates, and programming. They are also utilized in telecommunications, control systems, and signal processing for optimizing algorithms and reducing computational complexity. Also Check This Double Angle Calculator ## Conclusion In conclusion, a Kmap calculator is a valuable tool for simplifying Boolean functions in digital electronics. By automating the simplification process and providing a visual representation of the function, Kmaps make it easier to analyze and design circuits accurately. Whether you are a student learning Boolean algebra or a professional working in the field of digital electronics, a Kmap calculator can help streamline your work and improve efficiency. 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https://math.answers.com/other-math/What_is_2000000_divided_by_1000000	1,709,424,859,000,000,000	text/html	crawl-data/CC-MAIN-2024-10/segments/1707947476137.72/warc/CC-MAIN-20240302215752-20240303005752-00341.warc.gz	377,791,757	45,318	0 # What is 2000000 divided by 1000000? Updated: 4/28/2022 Wiki User 12y ago 2 Mara Denesik Lvl 10 3y ago Wiki User 12y ago 2000000/1000000 = 2 Earn +20 pts Q: What is 2000000 divided by 1000000? Submit Still have questions? Related questions ### What is 1000000 plus 2000000? 1000000 + 2000000 = 3,000,000 2000000 ### What is 2 percent in 1000000? 2 percent of 2000000 = 40000 2% of 2000000 = 2% * 2000000 = 2%/100% * 2000000 = 0.02 * 2000000 = 40000 ### How many kilograms are there in two million milligrams? There are 1000000 milligrams in one kilogram. Therefore, 2000000 milligrams is equal to 2000000/1000000 = 2 kilograms. 1/2 80000 20000 ### What is x divided by 2000000 if x equals 00002? 00002/2000000 = 2/2000000 = 0.000001 14072.6147 ### 2 kilometers are to how many milimeters? 1 kilometre = 1000000 millimetres so 2 km = 2*1000000 = 2000000 millimetres. Simple! ### How many micrograms are in 2 grams? There are 1000000 micrograms in one gram. Therefore, 2 grams is equal to 2 x 1000000 = 2000000 micrograms.	389	1,051	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	3.9375	4	CC-MAIN-2024-10	latest	en	0.818782	[ 128000, 15, 271, 2, 3639, 374, 220, 1049, 931, 15, 18255, 555, 220, 1041, 931, 15, 1980, 16593, 25, 220, 19, 14, 1591, 14, 2366, 17, 271, 54996, 2724, 271, 717, 88, 4227, 271, 17, 271, 44, 5169, 9973, 288, 1609, 271, 43, 14878, 220, 605, 198, 18, 88, 4227, 271, 54996, 2724, 271, 717, 88, 4227, 271, 1049, 931, 15, 14, 1041, 931, 15, 284, 220, 17, 271, 96359, 489, 508, 31093, 198, 48, 25, 3639, 374, 220, 1049, 931, 15, 18255, 555, 220, 1041, 931, 15, 5380, 9066, 198, 24205, 617, 4860, 5380, 11948, 4860, 271, 14711, 3639, 374, 220, 1041, 931, 15, 5636, 220, 1049, 931, 15, 1980, 1041, 931, 15, 489, 220, 1049, 931, 15, 284, 220, 18, 11, 931, 11, 931, 271, 1049, 931, 15, 271, 14711, 3639, 374, 220, 17, 3346, 304, 220, 1041, 931, 15, 1980, 17, 3346, 315, 220, 1049, 931, 15, 284, 220, 3443, 410, 220, 17, 4, 315, 220, 1049, 931, 15, 284, 220, 17, 4, 353, 220, 1049, 931, 15, 284, 220, 17, 54481, 1041, 4, 353, 220, 1049, 931, 15, 284, 220, 15, 13, 2437, 353, 220, 1049, 931, 15, 284, 220, 3443, 410, 271, 14711, 2650, 1690, 85402, 527, 1070, 304, 1403, 3610, 2606, 84209, 1980, 3947, 527, 220, 1041, 931, 15, 2606, 84209, 304, 832, 15395, 13255, 13, 15636, 11, 220, 1049, 931, 15, 2606, 84209, 374, 6273, 311, 220, 1049, 931, 15, 14, 1041, 931, 15, 284, 220, 17, 85402, 382, 16, 14, 17, 271, 4728, 410, 271, 1049, 410, 271, 14711, 3639, 374, 865, 18255, 555, 220, 1049, 931, 15, 422, 865, 17239, 220, 931, 2437, 1980, 931, 2437, 14, 1049, 931, 15, 284, 220, 17, 14, 1049, 931, 15, 284, 220, 15, 13, 931, 4119, 271, 6860, 5332, 13, 22638, 22, 271, 14711, 220, 17, 41668, 527, 311, 1268, 1690, 7625, 55336, 1980, 16, 44987, 265, 284, 220, 1041, 931, 15, 2606, 86366, 417, 779, 220, 17, 13437, 284, 220, 17, 9, 1041, 931, 15, 284, 220, 1049, 931, 15, 2606, 86366, 417, 13, 9170, 2268, 14711, 2650, 1690, 8162, 51870, 527, 304, 220, 17, 34419, 1980, 3947, 527, 220, 1041, 931, 15, 8162, 51870, 304, 832, 23882, 13, 15636, 11, 220, 17, 34419, 374, 6273, 311, 220, 17, 865, 220, 1041, 931, 15, 284, 220, 1049, 931, 15, 8162, 51870, 13, 128001 ]	[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ]
https://alchetron.com/Density-(polytope)	1,713,664,640,000,000,000	text/html	crawl-data/CC-MAIN-2024-18/segments/1712296817699.6/warc/CC-MAIN-20240421005612-20240421035612-00236.warc.gz	75,736,196	24,254	# Density (polytope) Updated on Edit Like Comment In geometry, the density of a polytope represents the number of windings of a polytope, particularly a uniform or regular polytope, around its center. It can be visually determined by counting the minimum number of facet or face crossings of a ray from the center to infinity. The density is constant across any continuous interior region of a polytope that crosses no facets. For a non-self-intersecting (acoptic) polytope, the density is 1. ## Contents Tessellations with overlapping faces can similarly define density as the number of coverings of faces over any given point. ## Polygons The density of a star polygon is the number of times that the polygonal boundary winds around its center; it is the winding number of the boundary around the central point. For a regular star polygon {p/q}, the density is q. It can be visually determined by counting the minimum number of edge crossings of a ray from the center to infinity. ## Polyhedra Arthur Cayley used density as a way to modify Euler's polyhedron formula (VE + F = 2) to work for the regular star polyhedra, where dv is the density of a vertex figure, df of a face and D of the polyhedron as a whole: dv VE + df F = 2D For example, the great icosahedron, {3, 5/2}, has 20 triangular faces (df = 1), 30 edges and 12 pentagrammic vertex figures (dv = 2), giving 2·12 − 30 + 1·20 = 14 = 2D. This implies a density of 7. The unmodified Euler's polyhedron formula fails for the small stellated dodecahedron {5/2, 5} and its dual great dodecahedron {5, 5/2}, for which VE + F = −6. The regular star polyhedra exist in two dual pairs, with each figure having the same density as its dual: one pair (small stellated dodecahedron—great dodecahedron) has a density of 3, while the other (great stellated dodecahedron–great icosahedron) has a density of 7. Hess further generalised the formula for star polyhedra with different kinds of face, some of which may fold backwards over others. The resulting value for density corresponds to the number of times the associated spherical polyhedron covers the sphere. This allowed Coxeter et al. to determine the densities of the majority of the uniform polyhedra. For hemipolyhedra, some of whose faces pass through the center, the density cannot be defined. Non-orientable polyhedra also do not have well-defined densities. ## Polychora There are 10 regular star polychora or 4-polytopes (called the Schläfli–Hess polychora), which have densities between 4, 6, 20, 66, 76, and 191. They come in dual pairs, with the exception of the self-dual density-6 and density-66 figures. ## References Density (polytope) Wikipedia	700	2,689	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	3.578125	4	CC-MAIN-2024-18	latest	en	0.911385	[ 128000, 2, 73710, 320, 34535, 998, 375, 696, 16593, 389, 198, 4126, 198, 13246, 198, 10906, 271, 644, 17484, 11, 279, 17915, 315, 264, 10062, 998, 375, 11105, 279, 1396, 315, 10160, 826, 315, 264, 10062, 998, 375, 11, 8104, 264, 14113, 477, 5912, 10062, 998, 375, 11, 2212, 1202, 4219, 13, 1102, 649, 387, 43395, 11075, 555, 26060, 279, 8187, 1396, 315, 45607, 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http://www.oceannavigator.com/Web-Exclusives-2013/Voltage-systems-Ohms-Law-and-DC-DC-converters/	1,568,838,359,000,000,000	text/html	crawl-data/CC-MAIN-2019-39/segments/1568514573331.86/warc/CC-MAIN-20190918193432-20190918215432-00480.warc.gz	317,767,788	10,982	Email Print # Voltage systems, Ohms Law, and DC-DC converters Oct 1, 2013 Voltage systems: Boats up to about 60 feet in length generally have a 12 Volt DC electrical system. This is largely due to the fact that many pieces of boating electrical gear are based on automotive/industrial electrical devices and so it is cheaper to use these mass produced items. Larger boats tend to have 24-Volt DC electrical systems for the same reason of keeping costs down and increasing savings. This may sound a bit confusing, but after we bone up on Ohms Law this seeming paradox will become clear. Ohms Law deals with the relationship of Voltage (E), Current (I), Resistance (R) and Power (P) of an electrical circuit and if we know any two quantities we can plug them into the formula and come up with the third. The various forms of Ohms Law follow: E=IR, I=E/R, R=E/I and P=IE, I=P/E, E=P/I. For our purposes the most important formula is the one dealing with current (I) because it will determine what size wire we need to operate safely. Notice that current in the Power formula I=P/E is a function of the power in watts divided by the system voltage, so if we double the voltage our current will be cut in half. Since electrical wire gauge is determined by its current carrying capacity, increasing voltage allows us to decrease the size of electrical cabling and save money. Example #1: P=240 Watts, E=12 VDC so I=240/12 which equals 20 amps current, but if we go to a 24-volt system then it works out as in Example #2: P=240 Watts, E=24 VDC so I=240/24 which equals 10 Amps current. Since a longer boat will need longer cable lengths it is doubly important to keep the cable gauge to a safe minimum or the costs will skyrocket and that is the big advantage of using a 24-VDC system versus 12 VDC. But what if your boat has a 24-VDC electrical system and you need to install a new electronic device that is built for 12 VDC? Again the current is the problem because I=E/R and given the same resistance if you tap into 24 VDC with a device designed for 12 VDC then you will double the current and most likely smoke your electronics. This situation can be resolved in one of three ways: 1) using a device that is rated for 10 to 35 VDC, 2) tapping off 12 volts from a 24 volt system, or 3) by using a 24 Volt to 12 Volt DC-DC converter. The simplest solution is to check the data plate on your new electronic device and hope that it is rated for 10-35 VDC in which case you could just connect it into your 24 Volt system. If this is not the case then you can tap off 12 volts from the middle of the 24-volt stack, but I suggest that this only be done on an emergency basis for the following reasons. Tapping will cause disproportionate charging of the batteries, overcharging one and undercharging the other which leads to shortened battery life. Also, you need to use the proper size wiring and fuse or circuit breaker it to protect the wiring from overheating during a short, etc. If you still want to tap off the 24-volt stack, then hook the positive conductor to where the two batteries are connected together (+ to -) and connect the ground lead to the output black lead. Again this method is not recommended. The optimum solution to our problem is to procure and install a device known as a 24 Volt to 12 Volt DC converter. These devices are rated for peak output and continuous output current. Just make sure that the continuous output rating matches your constant load requirements. In order to cut down on any potential RF interference from a DC-DC converter make sure to buy one that is listed as FCC “Class B” which should keep interference to a minimum. A comparison price for these taken from the latest West Marine catalog is \$79.99 for the 6 Amp model, \$119.99 for the 12 Amp model, and \$ 969.99 for the 50 Amp model. As you can see the price varies considerably depending on the continuous output current rating. If your boat has a NMEA 2000 network it may be as easy as tapping into the 12 Volt trunk line using a tee and running a drop cable to your 12-volt device. Most newer marine electronics have provision for hooking into an NMEA network so it should be an easy installation. You can buy NMEA 2000 cables and accessories from West Marine and they even have a NMEA 2000 starter kit for only \$87.99. Well, fellow sailors, enough for his time, so until we talk again: “Clear Skies and Following Seas!” Edit Module Old to new \| New to old Oct 2, 2013 02:58 pm Posted by Arnold Fine article, thank you for writing it! I've always remember Ohm's Law as W (watts) = V (volts) x A (amps). My silly memory hook for this is War=Vetrans x Adminstration. Oct 2, 2013 03:45 pm Posted by Tim Thanks, Arnold. 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http://www.statisticshowto.com/partial-correlation/	1,534,345,465,000,000,000	text/html	crawl-data/CC-MAIN-2018-34/segments/1534221210133.37/warc/CC-MAIN-20180815141842-20180815161842-00229.warc.gz	553,035,846	10,687	# Partial Correlation & Semi-Partial: Definition & Example Contents: ## What is Partial Correlation? Partial correlation measures the strength of a relationship between two variables, while controlling for the effect of one or more other variables. For example, you might want to see if there is a correlation between amount of food eaten and blood pressure, while controlling for weight or amount of exercise. It’s possible to control for multiple variables (called control variables or covariates). However, more than one or two is usually not recommended because the more control variables, the less reliable your test. Partial correlation has one continuous independent variable (the x-value) and one continuous dependent variable (the y-value); This is the same as in regular correlation analysis. In the blood pressure example above, the independent variable is “amount of food eaten” and the dependent variable is “blood pressure”. The control variables — weight and amount of exercise — should also be continuous. ## Notation A period in the subscript separates the correlated variables and the controlled for variables. For example, correlating caloric intake (X1) against blood pressure (X2), while controlling for weight (X3), is written as: r12.3 Alternatively, a bar is used instead of a period and subscript: r(1,2\|3). ## Running the Test The correlation coefficient, r, is also used to show the results from partial correlation. Like the regular correlation coefficient, rpartial returns a value from -1 to 1. Graphs showing a correlation of -1, 0 and +1 Partial correlation is usually carried out by running multiple regression analysis. Some software programs include partial correlation. For example, in SPSS choose Analyze > Correlations > Partial. ## How to Interpret the Result If the partial correlation, r12.3, is smaller than the simple (two-variable) correlation r12, but greater than 0, then variable 3 partly explains the correlation between X and Y. ## Semi-Partial Correlation Semi-partial correlation is almost the same as partial. In fact, many authors use the two terms to mean the same thing. However, others do make the following subtle distinction: With semi-partial correlation, the third variable holds constant for either X or Y but not both; with partial, the third variable holds constant for both X and Y. For example, the semi partial correlation statistic can tell us the particular part of variance, that a particular independent variable explains. It explains how one specific independent variable affects the dependent variable, while other variables are controlled for to prevent them getting in the way. To find it, calculate the correlation between the dependent variable and the residual of the prediction of one independent variable by the others. ## Example Suppose we use a set of data (from a 2002 paper from Abdi et al.) which lists three variables over six children. Each child was tested for memory span (Y) and speech rate (X2), and their age was also noted. A correlation statistic was desired which predicts Y (memory span) from X1 and X2 (age and speech rate). Normally, in a situation where X1 and X2 were independent random variables, we’d find out how important each variable was by computing a squared coefficient of correlation between X1 and X2 and the dependent variable Y. We would know that these squared coefficients of correlation were equal to the square multiple coefficient of correlation. But in a case like ours, X1 and X2 are anything but independent. Speech rate is highly dependent on age, and so using the squared coefficient will count the contributions of each variable several times over. ## References 1. Abdi, Herve. Part (Semi Partial) and Partial Regression Coefficients. Retrieved from https://www.utdallas.edu/~herve/Abdi-PartialRegressionCoefficient2007-pretty.pdf on May 8, 2018 2. Abdi, H., Dowling, W.J., Valentin, D., Edelman, B., & Posamentier M. (2002). Experimental Design and research methods. Unpublished manuscript. Richardson: The University of Texas at Dallas, Program in Cognition. 3. Brannick, M. Partial and Semipartial Correlation. Retrieved from http://faculty.cas.usf.edu/mbrannick/regression/Partial.html on May 8, 2018 4. Sharpa, J. (2007). Business Stats. Pearson Education India. 5. STATISTICA Help. Semi-Partial (or Part) Correlation. Retrieved from http://documentation.statsoft.com/STATISTICAHelp.aspx?path=glossary/GlossaryTwo/S/SemiPartialorPartCorrelation on May 8, 2018 6. Weatherburn, C. (1949). A First Course Mathematical Stats. CUP Archive. ------------------------------------------------------------------------------ Need help with a homework or test question? With Chegg Study, you can get step-by-step solutions to your questions from an expert in the field. If you'd rather get 1:1 study help, Chegg Tutors offers 30 minutes of free tutoring to new users, so you can try them out before committing to a subscription. If you prefer an online interactive environment to learn R and statistics, this free R Tutorial by Datacamp is a great way to get started. If you're are somewhat comfortable with R and are interested in going deeper into Statistics, try this Statistics with R track.	1,121	5,237	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	3.90625	4	CC-MAIN-2018-34	latest	en	0.892221	[ 128000, 2, 25570, 4563, 23013, 612, 55738, 12, 38414, 25, 20288, 612, 13688, 271, 15147, 1473, 567, 3639, 374, 25570, 4563, 23013, 1980, 38414, 26670, 11193, 279, 8333, 315, 264, 5133, 1990, 1403, 7482, 11, 1418, 26991, 369, 279, 2515, 315, 832, 477, 810, 1023, 7482, 13, 1789, 3187, 11, 499, 2643, 1390, 311, 1518, 422, 1070, 374, 264, 26670, 1990, 3392, 315, 3691, 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http://www.slideshare.net/irvianarozi/pembahasan-soal-ipa-26834734	1,464,234,395,000,000,000	text/html	crawl-data/CC-MAIN-2016-22/segments/1464049275437.19/warc/CC-MAIN-20160524002115-00075-ip-10-185-217-139.ec2.internal.warc.gz	823,697,973	36,102	Upcoming SlideShare × # Pembahasan soal ipa 241 views 191 views Published on 0 Likes Statistics Notes • Full Name Comment goes here. Are you sure you want to Yes No • Be the first to comment • Be the first to like this Views Total views 241 On SlideShare 0 From Embeds 0 Number of Embeds 0 Actions Shares 0 6 0 Likes 0 Embeds 0 No embeds No notes for slide ### Pembahasan soal ipa 1. 1. PEMBAHASAN SOAL IPA 1. Besaran fisika/ Besaran Pokok (BP)/Besaran Turunan (BT) Satuan / (SI) / (Non SI) Keterangan Panjang (BP) meter (SI) Benar Suhu (BP) celsius (non SI) Salah Kecepatan (BT) km/jam (non SI) Salah Berat (BP) gram (non SI) Salah Kunci Jawaban : A 2. Data dari gambar gelas ukur : Vair + benda = 85 ml V air = 60 ml V benda = 25 ml Kunci Jawaban : C 3. Data dari gambar neraca tiga lengan : mbenda = 125 gram ρ = 7,9 g/cm³ → V = g/cm³7,9 gram125 = 15,95 cm³ Kunci Jawaban = A 4. Dari gambar : Hal ini dilakukan agar kawat tidak mudah putus di malam hari ketika suhu rendah . Berarti : A. → salah B. → salah C. → salah D. → benar Kunci Jawaban : D 5. - Proses A – B = Proses Kenaikan suhu - ∆t = 100°C - 60°C = 40°C - Q = m . c. ∆t = 2 . 4200 . 40 = 336.000 joule Kunci Jawaban : A V m =ρ ρ m V = 2. 2. 6. Contoh gerak Jenis gerak 1 GLBB diperlambat 2 GLBB dipercepat 3 GLBB diperlambat 4 GLBB diperlambat 5 GLBB dipercepat Kunci Jawaban : D (1), (3) dan (5) 7. ρ = 1 g/cm³ = 1.000 kg/m³ g = 10 m/s² h = 60 cm – 35 cm = 25 cm = 0,25 m Ph = 1.000 kg/m³ . 10 m/s² . 0,25 m = 2.500 N/m² Kunci Jawaban : D 8. Dari Gambar ; Baterai : energi kimia → energi listrik Voltmeter : energi listrik → energi gerak Jadi : Energi kimia → energi listrik → energi gerak Kunci Jawaban : B 9. 3 detik menghasilkan 18 getaran Hz satuan dari frekuensi Frekuensi : jumlah getaran setiap detik Jadi : detik3 getaran18 =f = 6 get/det = 6 Hz Kunci Jawaban : B 10. Data dari gambar : s = 20 meter w = 600 joule W = F . S = F = S W = meter20 joule600 = 30 m Kunci Jawaban : B 11. Resultan gaya = 25 N – 5 N = 20 N hgPh ××= ρ 3. 3. Perpindahan (s) = 10 m W = F x S = 20 N x 10 m = 200 N.m = 200 joule Kunci Jawaban : A 12. A. Prinsip kerjanya : bidang miring B. Prinsip kerjanya : gerak dipermudah oleh roda C. Prinsip kerjanya : tuas D. Prinsip kerjanya : bidang miring Kunci Jawaban : C 13. T = 0,4 sekon f = T 1 = 4,0 1 = 4 10 Hz = 2,5 Hz Kunci Jawaban : D 14. Bunyi dapat merambat pada : zat padat, zat cair, dan gas Bunyi merambat pada zat padat lebih cepat dari zat cair dan lebih cepat dari gas Kunci Jawaban : D 15. Data dari gambar : So = 12 cm f = 10 cm SofSifSiSo 111111 −=→=+ = 12 1 10 1 − = 60 56 − Si 1 = 60 1 Si = 60 Kunci Jawaban : D 16. f = 12 cm So = 15 cm 4. 4. SofSifSiSo 111111 −=→=+ = 15 1 12 1 − = 60 45 − = 60 1 Si = 60 cm Kunci Jawaban : D 17. Dari data pada gambar : Gambar I : Titik bayangan jatuh di depan retina → cacat matanya : MIOPI Gambar II : setelah pakai kacamata titik bayangan jatuh tepat di retina Berarti lensa kacamata yang dipakai berlensa : cekung (-) Kunci Jawaban : D 18. Agar sisir plastik bermuatan listrik maka harus digosok dengan kain wol Dari peristiwa penggosokan terjadi : Elektron kain wol berpindah ke sisir plastik sehingga sisir plastik kelebihan elektron. Jadi sisir plastik bermuatan listrik negatif (-) Kunci Jawaban : B 19. Rtotal = 2 Ω + 3 Ω = 5 Ω I yang mengalir : Ω5 V02 = 4 A Volt menunjuk angka : V = I x R2 = 4 x 3 = 12 V Kunci Jawaban : C 20. Rtotal : Ω= × = ++ ×+ 15 54 15 69 663 6)63( I = A R V total 7,1 9 15 54 15 6 15 54 6 ==×== Kunci Jawaban : C 21. Biaya = daya x waktu x tarif = 000.1 800.3051004 Rpjamwatt ×××× = Rp. 48.000,00 Kunci Jawaban : C 22. Berdasarkan arah gosokan dan jenis kutub yang digosokkan : Besi PQ : Ujung P → kutub utara Ujung Q → kutub selatan Besi RT : Ujung R → kutub utara Ujung T → kutub selatan Kunci Jawaban : C 23. Berdasarkan arah arus pada lilitan dan kaidah tangan kanan, maka : 5. 5. Baja KL : Ujung K → kutub utara Ujung L → kutub selatan Sifat magnet : Baja → permanen Kunci Jawaban : A 24. Nama nama planet berdasarkan gambar : 1. Merkurius 5. Yupiter 2. Venus 6. Saturnus 3. Bumi 7. Uranus 4. Mars 8. Neptunus Saturnus berada diantara Yupiter (5) dan Uranus (7) Kunci Jawaban : C 25. Posisi kedua gambar : Gambar I : pasang maksimum Gambar II : pasang maksimum atau pasang purnama Kunci Jawaban : D 26. Kunci Jawaban : D Ciri mahluk hidup pada gambar adalah seekor ayam sedang mengerami telurnya adalah berkembangbiak 27. Kunci Jawaban : B Ciri tumbuhan dikotil dan monokotil No Organ Monokotil Dikotil 1 Akar Serabut Tumggang 2 Batang Tidak berkambium, tidak bercabang Letak jaringan pembuluh tersebar Berkambium, dan bercabang Letak jaringan pembuluh teratur 3 Daun Tulang daun lurus berbentuk pita Tulang daun menyirip atau menjari 4 Bunga Jumlah mahkota/kelopak berkelipatan 3 Jumlah mahkota/kelopak berkelipatan 2, 4, atau 5 5 Biji Lembaga berkeping satu Lembaga berkeping 2 28. 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https://mrburkemath.blogspot.com/2024/02/january-2024-algebra-2-part-ii.html	1,713,091,328,000,000,000	text/html	crawl-data/CC-MAIN-2024-18/segments/1712296816879.25/warc/CC-MAIN-20240414095752-20240414125752-00121.warc.gz	379,019,124	29,479	## Monday, February 19, 2024 ### January 2024 Algebra 2, Part II This exam was adminstered in January 2024. More Regents problems. ### Algebra 2 January 2024 Part II: Each correct answer will receive 2 credits. Partial credit can be earned. One mistake (computational or conceptual) will lose 1 point. A second mistake will lose the other point. It is sometimes possible to get 1 point for a correct answer with no correct work shown. 25. Factor the expression x3 + 4x2 - 9x - 36 completely. Factor by grouping, and then factor the quadratic you get after the first step. There are two ways to group, and either should work in any question of this kind. x3 + 4x2 - 9x - 36 (x3 + 4x2) - (9x + 36) x2(x + 4) - 9(x + 4) (x2 - 9)(x + 4) (x + 3)(x - 3)(x + 4) You can also switch the two middle terms around. This is just the way I learned it, so I usually do it, especially if it helps me avoid factoring out a minus sign. x3 - 9x + 4x2 - 36 (x3 - 9x) + (4x2 - 36) x(x2 - 9) + 4(x2 - 9) (x + 4)(x2 - 9) (x + 4)(x + 3)(x - 3) Note: This was Very Similar to Question 25 on the Auguest 2023 Regents. Right down to the (x + 3)(x - 3). 26. Determine if x + 4 is a factor of 2x3 + 10x2 + 4x - 16. Explain your answer. If (x + 4) is a factor of the polynomial, then the value of the polynomial must be 0 when x = -4. 2(-4)3 + 10(-4)2 + 4(-4) - 16 = 0 Since the expression is equal to zero when x = -4, then (x + 4) must be a factor. You could also solve this using polynomial division. (x + 4) divides evenly, with no remainder, so it is a factor. 27. An initial investment of \$1000 reaches a value, V(t), according to the model V(t) = 1000(1.01)4t, where t is the time in years. Determine the average rate of change, to the nearest dollar per year, of this investment from year 2 to year 7. Calculate V(7) and V(2). Subtract them and divide by 7 - 2, which is 5. You are looking for the rate of change (or slope, if you prefer). V(7) = 1000(1.01)4(7) = 1321.29 V(2) = 1000(1.01)4(2) = 1082.86 Rate of change = (1321.29 - 1082.86) / 5 = 47.686, which is \$48 to the nearest dollar. 28. When ( 1 / ∛(y2) ) y4 is written in the form yn, what is the value of n? Justify your answer. Use the laws of exponents to change the radical into a fraction. The combine the terms. ( 1 / ∛(y2) ) y4 ( 1 / (y2/3) y4 (y-2/3) y4 y10/3 n = 10/3. 29. The heights of the members of a ski club are normally distributed. The average height is 64.7 inches with a standard deviation of 4.3 inches. Determine the percentage of club members, to the nearest percent, who are between 67 inches and 72 inches tall. They don't use the chart with the normal distribution and all the standard deviations marked off any more. They just assume that you have and will use a calculator for this. You need to use the normalcdf function. Enter the command normalcdf(67,72,64.7,4.3) and you will get .2515... or 25%. All of the numbers that go into the command are in the question. Lower bound, upper bound, median, standard deviation. 30. The explicit formula an = 6 + 6n represents the number of seats in each row in a movie theater, where n represents the row number. Rewrite this formula in recursive form. A recursive function needs an initial value (a1) and an equation for an is terms of an-1. The inition value a1 = 12. Then an = an-1 + 6, because the common difference (rate of change) is 6. 31.Write (2xi3 - 3y)2) in simplest form. Square the binomial, substitute the powers of i, and Combine Like Terms. (2xi3 - 3y)2) (2xi3 - 3y)(2xi3 - 3y) 4x2i6 - 6xyi3 - 6xyi3 + 9y2 -4x2 - 12xyi3 + 9y2 -4x2 + 12xyi + 9y2 32. A survey was given to 1250 randomly selected high school students at the end of their junior year. The survey offered four post-graduation options: two-year college, four-year college, military, or work. Of the 1250 responses, 475 chose a four-year college. State one possible conclusion that can be made about the population of high school juniors, based on this survey This seems almost too simple a problem. If you divide 475/1250, you get .38 or 38%. One conclusion you can draw is that the population of high school juniors that would chose a four-year college would probably be about 38% and 62% would choose a different option. End of Part II How did you do? More to come. Comments and questions welcome. More Regents problems. ### I also write Fiction! You can now order my newest book Burke's Lore, Briefs: A Heavenly Date / My Damned Best Friend, written by Christopher J. Burke, which contains the aforementioned story and a bonus story. Order the softcover or ebook at Amazon. Also, check out In A Flash 2020, by Christopher J. Burke for 20 great flash fiction stories, perfectly sized for your train rides. Available in softcover or ebook at Amazon. 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https://gmatclub.com/forum/if-m-4-2-what-is-the-value-of-m-a-m-0-a-m2-8m-2781.html	1,511,336,938,000,000,000	text/html	crawl-data/CC-MAIN-2017-47/segments/1510934806509.31/warc/CC-MAIN-20171122065449-20171122085449-00683.warc.gz	635,453,747	42,253	It is currently 22 Nov 2017, 00:48 GMAT Club Daily Prep Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email. Customized for You we will pick new questions that match your level based on your Timer History Track every week, we’ll send you an estimated GMAT score based on your performance Practice Pays we will pick new questions that match your level based on your Timer History Events & Promotions Events & Promotions in June Open Detailed Calendar if \|m+4\|=2, what is the value of m? (a) m <0 (a) m2 +8m + Author Message Manager Joined: 27 Jul 2003 Posts: 122 Kudos [?]: 58 [0], given: 0 Location: Singapore if \|m+4\|=2, what is the value of m? (a) m <0 (a) m2 +8m + [#permalink] Show Tags 05 Oct 2003, 23:34 1 This post was BOOKMARKED 00:00 Difficulty: (N/A) Question Stats: 100% (00:03) correct 0% (00:00) wrong based on 11 sessions HideShow timer Statistics This topic is locked. If you want to discuss this question please re-post it in the respective forum. if \|m+4\|=2, what is the value of m? (a) m <0 (a) m2 +8m + 12=0 Kudos [?]: 58 [0], given: 0 SVP Joined: 03 Feb 2003 Posts: 1603 Kudos [?]: 308 [0], given: 0 Show Tags 05 Oct 2003, 23:44 m=-2 or -6 (1) not suff (2) m2 +8m + 12=0 or m=-2 or -6 not suff E Kudos [?]: 308 [0], given: 0 Non-Human User Joined: 09 Sep 2013 Posts: 15562 Kudos [?]: 283 [0], given: 0 Re: if \|m+4\|=2, what is the value of m? (a) m <0 (a) m2 +8m + [#permalink] Show Tags 21 Oct 2016, 22:14 Hello from the GMAT Club BumpBot! 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(a) m <0 (a) m2 +8m + [#permalink] 21 Oct 2016, 22:14 Display posts from previous: Sort by	723	2,203	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	3.796875	4	CC-MAIN-2017-47	latest	en	0.879639	[ 128000, 2181, 374, 5131, 220, 1313, 4723, 220, 679, 22, 11, 220, 410, 25, 2166, 271, 21287, 835, 10349, 13690, 49169, 271, 13359, 499, 369, 1701, 279, 9198, 482, 420, 11084, 5507, 649, 16430, 701, 5178, 323, 4284, 810, 6725, 4860, 13, 1226, 617, 53200, 499, 311, 13690, 49169, 24271, 4669, 2613, 382, 10480, 1534, 198, 2000, 1472, 271, 906, 690, 3820, 502, 4860, 430, 2489, 701, 2237, 3196, 389, 701, 18232, 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https://wiki-helper.com/question/a-dealer-deals-in-two-products-amp-y-he-has-kitu1-00-000-to-invest-amp-space-to-store-80-pieces-38094948-89/	1,631,847,753,000,000,000	text/html	crawl-data/CC-MAIN-2021-39/segments/1631780054023.35/warc/CC-MAIN-20210917024943-20210917054943-00180.warc.gz	649,763,879	16,330	## A dealer deals in two products X & Y. He has ₹1,00,000/- to invest & space to store 80 pieces. Product X costs ₹ 2500/- &a Question A dealer deals in two products X & Y. He has ₹1,00,000/- to invest & space to store 80 pieces. Product X costs ₹ 2500/- & product Y costs ₹ 1000/- per unit. Construct the LPP and find the number of units of each product to be purchased. in progress 0 1 month 2021-07-31T04:49:37+00:00 2 Answers 0 views 0 8jwhsbdjr8jd sides susme sidn 2. Given : A dealer deals in two products X & Y. He has ₹1,00,000/- to invest & space to store 80 pieces. Product X costs ₹ 2500/- & product Y costs ₹ 1000/- per unit. To Find : Construct the LPP and the number of units of each product to be purchased. Solution: Let say unit to be bought for product X are x and product Y are y space to store 80 pieces x + y ≤ 80 He has ₹1,00,000/- to invest => 2500x + 1000y ≤ 100000 => 25x + 10y ≤ 1000 => 5x + 2y ≤ 200 x + y ≤ 80 5x + 2y ≤ 200 x = 40 , y = 0 x = 40/3 , y = 200/3 x = 0 , y = 80 Now further conditions are not provided that what is best to buy	396	1,099	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	3.859375	4	CC-MAIN-2021-39	latest	en	0.885832	[ 128000, 567, 362, 24204, 12789, 304, 1403, 3956, 1630, 612, 816, 13, 1283, 706, 90891, 16, 11, 410, 11, 931, 24572, 311, 2793, 612, 3634, 311, 3637, 220, 1490, 9863, 13, 5761, 1630, 7194, 90891, 220, 5154, 15, 24572, 612, 64, 271, 14924, 271, 32, 24204, 12789, 304, 1403, 3956, 1630, 612, 816, 13, 1283, 706, 90891, 16, 11, 410, 11, 931, 24572, 311, 2793, 612, 3634, 311, 3637, 220, 1490, 9863, 13, 5761, 1630, 7194, 90891, 220, 5154, 15, 24572, 612, 2027, 816, 7194, 90891, 220, 1041, 15, 24572, 824, 5089, 13, 19196, 279, 445, 4505, 323, 1505, 279, 1396, 315, 8316, 315, 1855, 2027, 311, 387, 15075, 382, 258, 5208, 220, 15, 198, 16, 2305, 220, 2366, 16, 12, 2589, 12, 2148, 51, 2371, 25, 2491, 25, 1806, 10, 410, 25, 410, 220, 17, 38343, 220, 15, 6325, 220, 15, 271, 23, 73, 1336, 82, 9117, 60013, 23, 44490, 11314, 4582, 2727, 14414, 77, 271, 17, 13, 16644, 551, 362, 24204, 12789, 304, 1403, 3956, 1630, 612, 816, 13, 1283, 706, 90891, 16, 11, 410, 11, 931, 24572, 311, 2793, 612, 3634, 311, 3637, 220, 1490, 9863, 13, 5761, 1630, 7194, 90891, 220, 5154, 15, 24572, 612, 2027, 816, 7194, 90891, 220, 1041, 15, 24572, 824, 5089, 382, 1271, 7531, 551, 19196, 279, 445, 4505, 323, 271, 1820, 1396, 315, 8316, 315, 1855, 2027, 311, 387, 15075, 382, 37942, 1473, 10267, 2019, 5089, 311, 387, 11021, 369, 2027, 1630, 527, 865, 220, 4194, 438, 2027, 816, 527, 379, 271, 8920, 311, 3637, 220, 1490, 9863, 271, 87, 489, 379, 220, 4194, 126863, 220, 1490, 271, 1548, 706, 90891, 16, 11, 410, 11, 931, 24572, 311, 2793, 271, 2228, 220, 5154, 15, 87, 220, 4194, 10, 220, 1041, 15, 88, 38394, 220, 1041, 931, 271, 2228, 220, 914, 87, 489, 220, 605, 88, 38394, 220, 1041, 15, 271, 2228, 220, 20, 87, 220, 4194, 10, 220, 17, 88, 38394, 220, 1049, 271, 87, 489, 379, 220, 4194, 126863, 220, 1490, 271, 20, 87, 220, 4194, 10, 220, 17, 88, 38394, 220, 1049, 271, 87, 284, 220, 1272, 1174, 379, 284, 220, 15, 271, 87, 284, 220, 1272, 14, 18, 220, 4194, 11, 379, 284, 220, 4194, 1049, 14, 18, 271, 87, 284, 220, 15, 17529, 1174, 220, 4194, 88, 284, 220, 1490, 271, 7184, 4726, 4787, 527, 539, 3984, 430, 1148, 374, 1888, 311, 3780, 128001 ]	[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ]
https://www.slideserve.com/tessa/physics-2102	1,701,682,213,000,000,000	text/html	crawl-data/CC-MAIN-2023-50/segments/1700679100527.35/warc/CC-MAIN-20231204083733-20231204113733-00217.warc.gz	1,123,466,458	19,762	1 / 15 # Physics 2102 Physics 2102 Gabriela Gonz á lez. b. a. Physics 2102 . Circuits. DC circuits: resistances in series. Two resistors are “in series” if they are connected such that the same current flows in both. ## Physics 2102 E N D ### Presentation Transcript 1. Physics 2102 Gabriela González b a Physics 2102 Circuits 2. DC circuits: resistances in series Two resistors are “in series” if they are connected such that the same current flows in both. The “equivalent resistance” is a single imaginary resistor that can replace the resistances in series. “Walking the loop” results in :E –iR1-iR2-iR3=0  i=E/(R1+R2+R3) In the circuit with the equivalent resistance, E –iReq=0  i=E/Req Thus, 3. Multiloop circuits: resistors in parallel Two resistors are “in parallel” if they are connected such that there is the same potential drop through both. The “equivalent resistance” is a single imaginary resistor that can replace the resistances in parallel. “Walking the loops” results in :E –i1R1=0, E –i2R2=0, E –i3R3=0 The total current delivered by the battery is i = i1+i2+i3 = E/R1+ E/R2+ E/R3. In the circuit with the equivalent resistor, i=E/Req. Thus, 4. Resistors and Capacitors ResistorsCapacitors Key formula: V=iR Q=CV In series: same current same charge Req=∑Rj 1/Ceq=∑1/Cj In parallel: same voltage same voltage 1/Req=∑1/Rj Ceq=∑Cj 5. DC circuits • Loop rule: when walking along a loop, add potential differences across each element, and make the total equal to zero when you come back to the original point. • Junction rule: at every junction, total current is conserved. • Problem strategy: • Replace resistors in series and in parallel with their equivalent resistors • Draw currents in every wire, and label them • Write the loop rule for each loop (or for the loop that involves your question) • Write the junction rules for the currents. • Solve the equations for the currents. • Answer the question that was asked. 6. Bottom loop: (all else is irrelevant) 12V 8W Example Which resistor gets hotter? 7. Example • Which circuit has the largest equivalent resistance? • Assuming that all resistors are the same, which one dissipates more power? • Which resistor has the smallest potential difference across it? 8. Assume the batteries are ideal, and have emf E1=12V, E2=9V, and R=3W. Which way will the current flow? Which battery us doing positive work? If the potential at A is 0V, what is the potential at B? How much power is dissipated by the resistor? How much power is dissipated (or absorbed) by the batteries? Example 9. Example Find the equivalent resistance between points (a) F and H and (b) F and G. (Hint: For each pair of points, imagine that a battery is connected across the pair.) 10. Non-ideal batteries • You have two ideal identical batteries, and a resistor. Do you connect the batteries in series or in parallel to get maximum current through R? • Does the answer change if you have non-ideal (but still identical) batteries? 11. Light bulbs • If all batteries are ideal, and all batteries and lightbulbs are identical, in which arrangements will the lightbulbs as bright as the one in circuit X? • Does the answer change if batteries are not ideal? 12. i(t) E/R RC circuits: charging a capacitor In these circuits, current will change for a while, and then stay constant. We want to solve for current as a function of time i(t). The charge on the capacitor will also be a function of time: q(t). The voltage across the resistor and the capacitor also change with time. To charge the capacitor, close the switch on a. E + VR(t)+VC(t) =0 E -i(t)R -q(t)/C = 0 E - (dq(t)/dt) R - q(t)/C =0 A differential equation for q(t)! The solution is: q(t) = CE(1-e-t/RC) And then i(t) = dq/dt= (E/R) e-t/RC Time constant=RC 13. i(t) + C E/R - RC circuits: discharging a capacitor Assume the switch has been closed on a for a long time: the capacitor will be charged with Q=CE. +++ --- Then, close the switch on b: charges find their way across the circuit, establishing a current. VR+VC=0 -i(t)R+q(t)/C=0 => (dq/dt)R+q(t)/C=0 Solution: q(t)=q0e-t/RC=CEe-t/RC i(t) = dq/dt = (q0/RC) e-t/RC = (E/R) e-t/RC 14. Example The three circuits below are connected to the same ideal battery with emf E. All resistors have resistance R, and all capacitors have capacitance C. • Which capacitor takes the longest in getting charged? • Which capacitor ends up with the largest charge? • What’s the final current delivered by each battery? • What happens when we disconnect the battery? 15. Example In the figure, E= 1 kV, C = 10 µF, R1 = R2 = R3 = 1 MW. With C completely uncharged, switch S is suddenly closed (at t = 0). • What’s the current through each resistor at t=0? • What’s the current through each resistor after a long time? • How long is a long time? 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https://classbasic.com/second-term-examination-mathematics-for-primary-1-basic-1-maths-exam-questions/	1,723,699,019,000,000,000	text/html	crawl-data/CC-MAIN-2024-33/segments/1722641151918.94/warc/CC-MAIN-20240815044119-20240815074119-00247.warc.gz	128,268,715	14,341	# Second Term Examination Mathematics for Primary 1 (Basic 1) Maths Exam Questions MATHEMATICS SECOND TERM EXAMINATION PRIMARY 1 (BASIC 1) ### MATHS EXAM QUESTIONS SECTION A – Choose the correct answer from the options. 1. ( 24 + 10 ) + 10 [a] 54 [b] 64 [c] 44 2. Fatimah has 465 bags and dashes raphael 142 bag. How many bags are left. [a] 323 [b] 332 [c] 233 3. 1,5,7,9,11, 13, 15, 17 can be referred to as an __________ number. [a] high [b] odd [c] even 4. Two hundred and fourteen can be written as __________. [a] 214 [b] 241 [c] 421 5. How many 3’s are in 12? [a] 10 [b] 2 [c] 4 6. Which of this is an even number? [a] 2, 4, 6, 8 [b] 4, 3, 2, 1 [c] 8, 1, 5, 3, 2 7. 6 8 9 stands for [a] hundreds [b] tens [c] units 8. Write 234 in expanded form. [a] 200 + 300 + 4 [b] 200 + 30 + 4 [c] 200 + 3 + 40 9. Find the product of 33 and 2. [a] 77 [b] 88 [c] 66 10. A dozen is __________. [a] 12 [b] 14 [c] 24 11. 3 2 1 + 1 4 2 [a] 346 [b] 463 [c] 643 12. 9 4 6 – 1 2 4 [a] 212 [b] 248 [c] 822 13. 23 + __________ = 53 [a] 40 [b] 50 [c] 30 14. 67 – 48 = [a] 49 [b] 115 [c] 19 15. 32 x 2 [a] 54 [b] 74 [c] 64 SECTION B – Attempt all questions 1. Add up these with remaining a. 20 + 4 b. 10 + 8 c. 440 + 3 d. 220 + 6 2. Subtraction of three digit number. a. H T U b. H T U 6 9 8 8 4 7 – 5 3 3 – 2 4 6 c. H T U d. H T U 4 4 6 5 4 9 – 2 1 6 – 2 2 2 3a. Write out odd number from 1 to 39 b. Write out even number from 2 to 40 4a. Multiply these a. 1 0 x 2 b. 3 0 x 3 c. 4 3 x 2 d. 1 6 x 2 5. Write the value of each underlined digit. a. 4 9 6 b. 2 4 2 c. 9 6 9 6. How many 4s in 16? c. How many 2s in 20? d. 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http://mathforum.org/library/drmath/view/55690.html	1,519,603,228,000,000,000	text/html	crawl-data/CC-MAIN-2018-09/segments/1518891817523.0/warc/CC-MAIN-20180225225657-20180226005657-00386.warc.gz	228,977,145	3,298	Associated Topics \|\| Dr. Math Home \|\| Search Dr. Math ### Truth Tables: And, Or, Implies, Not ``` Date: 06/10/2001 at 22:08:15 From: Lisa Subject: Truth Tables I have looked in the textbook about how to do truth tables, and it really doesn't explain or show how to do it. Can you please email me an explanation and some examples? Sincerely, Lisa ``` ``` Date: 06/11/2001 at 17:39:13 From: Doctor Tony Subject: Re: Truth Tables Hi Lisa, Thanks for writing to Ask Dr. Math. Essentially, truth tables allow you to evaluate every possibility for a given logic statement. Each of the basic operations (and, or, implies, not) has a truth table associated with it. The easiest way to explain is through examples. Let's form the truth tables for the set of operations I listed above, and then build a truth table for an example logic statement. Example 1: ~p (not p) We only have one logic variable, p, which can either be true or false. The first column of the truth table will contain all possible values of p. The second column will give the appropriate value for ~p. p \| ~p ----------- T \| F F \| T That is the truth table. This was a pretty simple one, since it only involved one logic variable. Example 2: p^q (p and q) Now there are two logic variables, p and q; therefore, our truth table will have 4 rows, since there are four possible arrangements of p and q. In general, if there are n logic statements involved, there will be 2^n rows in the truth table. p q \| p^q -------------- T T \| T T F \| F F T \| F F F \| F So we see that p^q is only T if both p and q are T - it makes sense. Example 3: p\/q (p or q) p q \| p\/q --------------- T T \| T T F \| T F T \| T F F \| F We see that p\/q is only F if both p and q are F. Example 4: p->q (p implies q) p q \| p->q --------------- T T \| T T F \| F F T \| T F F \| T Now let's do a more complicated example. Let's make a truth table for the logic statement ~p^(q\/r): p q r \| ~p \| q\/r \| ~p^(q\/r) --------------------------------- T T T \| F \| T \| F T T F \| F \| T \| F T F T \| F \| T \| F T F F \| F \| F \| F F T T \| T \| T \| T F T F \| T \| T \| T F F T \| T \| T \| T F F F \| T \| F \| F As the last column shows, this statement is true for three out of the eight possible truth states of the logic variables. I hope this helps. If you're still stuck, please feel free to write back. - Doctor Tony, The Math Forum http://mathforum.org/dr.math/ ``` Associated Topics: High School Logic Search the Dr. Math Library: Find items containing (put spaces between keywords): Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words Submit your own question to Dr. Math Math Forum Home \|\| Math Library \|\| Quick Reference \|\| Math Forum Search	913	2,970	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 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http://www.tpub.com/math2/86.htm	1,586,193,993,000,000,000	text/html	crawl-data/CC-MAIN-2020-16/segments/1585371656216.67/warc/CC-MAIN-20200406164846-20200406195346-00017.warc.gz	301,748,790	5,939	Expectation Custom Search EXPECTATION Expectation is the average of the values you would get in conducting an experiment or trial exactly the same way many times. In this discussion of expectation, we will consider two types. One is a numerical expectation and the other is a mathematical expectation. Numerical Expectation If you tossed a coin 50 times, you would expect the coin to fall heads (on the average) about 25 times. Your assumption is explained by the following definition of numerical expectation: If the probability of success in one trial is p, and k is the total number of trials, then kp is the expected number of successes in the k trials. In the above example of tossing the coin 50 times, the probability of heads (successes) is where Substituting values in the equation, we find that = 25 EXAMPLE: A die is rolled by a player. What is the expectation of rolling a 6 in 30 trials? SOLUTION: The probability of rolling a 6 in 1 trial is and the number of rolls is k=30 therefore, In words, the player would expect (on the average) to roll a 6 five times in 30 rolls. Mathematical Expectation We will define mathematical expectation as follows: If, in the event of a successful result, amount a, is to be received and the probability of success of that event is p, then ap is the mathematical expectation. If you were to buy 1 of 500 raffle tickets for a video recorder worth \$325.00, what would be your mathematical expectation? In this case, the product of the amount you stand to win and the probability of winning is where a = amount you stand to win p = probability of success and Then, by substitution Thus, you would not want to pay more than 65 cents for the ticket, unless, of course the raffle were for a worthy cause. EXAMPLE: To entice the public to invest in their development, Sunshine Condominiums has offered a prize of \$2,000 to 1 randomly selected family out of the first 1,000 families that participate in the condominium's tour. 1. What would be each family's mathematical expectation? 2. Would it be worthwhile for the Jones family to spend \$3.00 in gasoline to drive to Sunshine Condominiums to take the tour? SOLUTION: 1. 2. No; since \$3.00 is \$1.00 over their expectation of \$2.00, it would not be worthwhile for the Jones family to take the tour. PRACTICE PROBLEMS: 1. A box contains 7 slips of paper, each numbered differently. A girl makes a total of 50 draws, returning the drawn slip after each draw. a. What is the probability of drawing a selected numbered slip in 1 drawing? b. How many times would the girl expect to draw the single selected numbered slip in the 50 draws? 2. In a winner-take-all tournament among four professional tennis players, the prize money is \$500,000. Joe Conners, one of the tennis players, figures his probability of winning is 0.20. a. What is his mathematical expectation? b. Would he be better off if he made a secret agreement with the other tennis players to divide the prize money evenly regardless of who wins? ANSWERS: 2. a. \$100,000 b. 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https://ncatlab.org/nlab/show/Stirling%27s+approximation	1,713,684,390,000,000,000	application/xhtml+xml	crawl-data/CC-MAIN-2024-18/segments/1712296817729.87/warc/CC-MAIN-20240421071342-20240421101342-00659.warc.gz	393,324,675	8,736	Contents Idea Stirling’s approximation, or Stirling’s formula, gives an asymptotic approximation to the factorial function in terms of elementary functions. Statement Define $n!$ for $n \in \mathbb{N}$ as usual by recursion: $0! = 1$ and $(n+1)! = (n+1) \cdot n!$. Stirling’s approximation says $n! \sim \frac{n^n \sqrt{2\pi n}}{e^n}$ or in other words that $\underset{n \to \infty}{\lim}\; \frac{n^n \sqrt{2\pi n}}{e^n \cdot n!} = 1.$ Proof Many proofs are known. The following is meant to bring out the essential kinship between this formula and the famous Gaussian integral identity $\int_{-\infty}^\infty e^{-x^2}\; d x = \sqrt{\pi},$ or equivalently $\int_{-\infty}^\infty e^{-x^2/2}\; d x = \sqrt{2\pi}.$ First we establish an elementary fact, accessible to a student of elementary calculus. Proposition For some constant $C$, $n! \sim C\frac{n^n \sqrt{n}}{e^n}.$ Proof Consider a trapezoidal sum estimate? for the integral $I_n = \int_1^n \log x\; dx = \left. x\log x - x\right\|_1^n = n \log n - n + 1 = 1 + \log \frac{n^n}{e^n}.$ Partitioning the interval $[1, n]$ into subintervals of unit length, the corresponding trapezoidal sum is $T_n = \frac{\log 1 + \log 2}{2} + \frac{\log 2 + \log 3}{2} + \cdots + \frac{\log (n-1) + \log n}{2} = \log (n!) - \frac1{2} \log n = \log \frac{n!}{\sqrt{n}}.$ $T_n$ underestimates $I_n$ because the graph of $\log$ is “concave down” (meaning $-\log x$ is a convex function). In other words, the error terms $\int_k^{k+1} \log x\; dx - \frac{\log k + \log (k+1)}{2} \qquad (1)$ are positive, so that the sum of these error terms from $k=1$ to $n-1$, which is $I_n - T_n$, increases with $n$. However, using the error term estimate for the trapezoidal rule, the term (1) is on the order of $1/k^2$ (the maximum value of $\|(\log)''(x)\| = 1/x^2$ over $[k, k+1]$). Since $\sum 1/k^2$ converges, we see that the increasing sequence $I_n - T_n$ has a uniform upper bound and therefore a limit. Hence the limit $\underset{n \to \infty}{\lim} (I_n - 1) - T_n = \underset{n \to \infty}{\lim} \log \frac{n^n}{e^n} - \log \frac{n!}{\sqrt{n}} = \underset{n \to \infty}{\lim} \log \frac{n^n \sqrt{n}}{e^n \cdot n!}$ exists, and therefore $n! \sim C\frac{n^n \sqrt{n}}{e^n}$ for some constant $C$. The following lemma makes reference to the Gamma function. Lemma $\frac{\sqrt{2\pi}}{C} = \underset{n \to \infty}{\lim} \frac{\Gamma(n + \frac1{2})}{\Gamma(n) \cdot \sqrt{n}}$; in particular, this limit exists. Proof From $n! \sim C\frac{n^n \sqrt{n}}{e^n}$, we easily derive $\frac{(2n)!}{n! \cdot n!} \sim \frac{2^{2n} \sqrt{2n}}{C n}$ so that $\frac1{C} \sim \frac{(2n)!}{2^{2n} n! \cdot n!} \cdot \frac{n}{\sqrt{2n}} = \frac{(2n-1)(2n-3) \ldots 3 \cdot 1}{2 \cdot (2n-2)(2n-4) \ldots 2} \cdot \frac1{\sqrt{2n}}.$ Then, using $\Gamma\left(\frac1{2}\right) = 2\int_0^\infty e^{-x^2} dx = \sqrt{\pi}$, we have $\frac{\sqrt{2\pi}}{C} \sim \frac{(2n-1)(2n-3) \ldots 3 \cdot 1}{2 \cdot (2n-2)(2n-4) \ldots 2} \cdot \frac{\sqrt{\pi}}{\sqrt{n}} = \frac{\left(n-\frac1{2}\right)\left(n-\frac{3}{2}\right) \ldots \left(\frac1{2}\right) \Gamma\left(\frac1{2}\right)}{(n-1)! \cdot \sqrt{n}} = \frac{\Gamma\left(n + \frac1{2}\right)}{\Gamma(n) \cdot \sqrt{n}}.$ Proposition $\underset{n \to \infty}{\lim} \frac{\Gamma(n + \frac1{2})}{\Gamma(n) \cdot \sqrt{n}} = 1$. Proof From log-convexity of $\Gamma$ (see here), we may derive $\frac{\Gamma(x-\frac1{2})}{\Gamma(x - 1)} \leq \frac{\Gamma(x)}{\Gamma(x-\frac1{2})} \leq \frac{\Gamma(x+\frac1{2})}{\Gamma(x)}$ which implies that the values of $\frac{\Gamma(x+\frac1{2})}{\Gamma(x) \cdot \sqrt{x}}$, with $x$ ranging over half-integers, tend to the same limit $L$ as with $x$ ranging over whole integers. Therefore $L^2 = \underset{n \to \infty}{\lim} \frac{\Gamma(n+1)}{\sqrt{n+\frac1{2}}\cdot \Gamma(n + \frac1{2})} \cdot \frac{\Gamma(n + \frac1{2})}{\sqrt{n}\Gamma(n)} = \underset{n \to \infty}{\lim} \frac{n\Gamma(n)}{\sqrt{(n+\frac1{2})n}\cdot \Gamma(n)} = \underset{n \to \infty}{\lim} \frac{n}{\sqrt{(n+\frac1{2})n}},$ whence $L^2 = 1$, and therefore $L = 1$. References • n-Category Café post • Dan Romik, Stirling’s Approximation for n!: the Ultimate Short Proof?, The American Mathematical Monthly Volume 107 Issue 6 (2000), 556-557. doi.org/10.1080/00029890.2000.12005235 Last revised on April 12, 2024 at 22:24:35. 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https://www.jiskha.com/display.cgi?id=1349363018	1,503,149,569,000,000,000	text/html	crawl-data/CC-MAIN-2017-34/segments/1502886105451.99/warc/CC-MAIN-20170819124333-20170819144333-00645.warc.gz	906,403,350	4,171	# math posted by . Integrate sinx/sinx-cosx..... Plz note sinx-cosx as a whole is in the denominator.. • math - use the good old quotient rule: if y = u/v, y' = (u'v - uv')/v^2 u = sinx v = sinx - cosx y' = [(cosx)(sinx-cosx) - (sinx)(cosx+sinx)]/(sinx-cosx)^2 you can play with that some, but I like 1/(sin(2x)-1) ## Similar Questions 1. ### Math Verify the identity . (cscX-cotX)^2=1-cosX/1+cosX _______ sorry i cant help you (cscX-cotX)=1/sinX - cosX/sinX = (1-cosX)/sinX If you square this you have (1-cosX)^2/(sinX)^2 Now use (sinX)^2 = 1 - (cosX)^2 to get (1-cosX)^2 / 1 - … 2. ### Pre-Calc Trigonometric Identities Prove: (tanx + secx -1)/(tanx - secx + 1)= tanx + secx My work so far: (sinx/cosx + 1/cosx + cosx/cosx)/(sinx/cos x - 1/cosx + cosx/cosx)= tanx + cosx (just working on the left side) ((sinx + 1 - cosx)/cosx)/((sinx … 3. ### Trigonometry. ( tanx/1-cotx )+ (cotx/1-tanx)= (1+secxcscx) Good one! Generally these are done by changing everything to sines and cosines, unless you see some obvious identities. Also generally, it is best to start with the more complicated side … 4. ### Trig........ I need to prove that the following is true. Thanks (cosx / 1-sinx ) = ( 1+sinx / cosx ) I recall this question causing all kinds of problems when I was still teaching. it requires a little "trick" L.S. =cosx/(1-sinx) multiply top and … 5. ### Mathematics - Trigonometric Identities Prove: (tanx)(sinx) / (tanx) + (sinx) = (tanx) - (sinx) / (tanx)(sinx) What I have so far: L.S. = (sinx / cosx) sinx / (sinx / cosx) + sinx = (sin^2x / cosx) / (sinx + (sinx) (cosx) / cosx) = (sin^2x / cosx) / (cosx / sinx + sinxcosx) 6. ### Math - Pre- Clac Prove that each of these equations is an identity. A) (1 + sinx + cos x)/(1 + sinx + cosx)=(1 + cosx)/sinx B) (1 + sinx + cosx)/(1 - sinx + cosx)= (1 + sin x)/cosx Please and thankyou! 7. ### maths - trigonometry I've asked about this same question before, and someone gave me the way to finish, which I understand to some extent. I need help figuring out what they did in the second step though. How they got to the third step from the second. … 8. ### Trigonometry Check Simplify #3: [cosx-sin(90-x)sinx]/[cosx-cos(180-x)tanx] = [cosx-(sin90cosx-cos90sinx)sinx]/[cosx-(cos180cosx+sinx180sinx)tanx] = [cosx-((1)cosx-(0)sinx)sinx]/[cosx-((-1)cosx+(0)sinx)tanx] = [cosx-cosxsinx]/[cosx+cosxtanx] = [cosx(1-sinx]/[cosx(1+tanx] … 9. ### Math Im really struggling with these proving identities problems can somebody please show me how to do these? 10. ### trigonometry can i use factoring to simplify this trig identity? 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https://www.omnicalculator.com/conversion/cubic-meter	1,723,309,152,000,000,000	text/html	crawl-data/CC-MAIN-2024-33/segments/1722640810581.60/warc/CC-MAIN-20240810155525-20240810185525-00771.warc.gz	716,929,930	84,689	Last updated: # Cubic Meter Calculator What is a cubic meter?How to use the cubic meter calculatorHow to calculate the cubic meters of a box or spaceFAQs With the cubic meter calculator, you can convert any volume unit to cubic meters. On top of that, you can also calculate the cubic meters of a box of a known size, regardless of the length units you've got! If you want to convert from cubic ft to cubic meters, or convert cubic meters to cubic feet, or any other combination, this is the fastest and easiest way! So go ahead and use the calculator to calculate cubic meters and convert from and to cubic meters, cubic feet, and whatever other unit of volume you want! ## What is a cubic meter? First things first, we need to make sure we all know what a cubic meter is. A cubic meter is the SI unit for volume. It is the obvious choice for volume, given that the volume is the 3D space taken up by an object. In fact, this is the clearest when we think about a box. In that case, the formula for its volume would be: volume = length × width × height If we use meters for length, width, and height, it makes sense then that the final units for volume would be meter × meter × meter = meter³ That is why we use the cubic meter and its derivative units (cubic kilometer, cubic centimeter, etc.) as the international unit for volume. However, it is not the only one, and in places like the USA, the imperial system is more prevalent. In the imperial system, we find units such as the cubic inch, the cubic mile, and, most similar to cubic meters, the cubic foot. Just like knowing different languages, being able to convert from cubic meters to cubic feet and back from cubic feet to cubic meters is essential. In fact, the more units we can convert from and to, the better! Before we learn how to calculate cubic meters from any other unit, we should take a look at some of the common abbreviations: • Cubic meters: m³ • Cubic centimeters: cm³, cc • Cubic kilometers: km³ • Cubic feet/foot: cu ft, ft³ • Cubic inches: cu in, in³ ## How to use the cubic meter calculator The cubic meter calculator can be thought of as several calculators in one. It has two main modes: conversion and calculation, but they can also be used simultaneously with some clever operation. Assuming you do it correctly, this calculator can: • Calculate the cubic meters of a box; • Calculate the cubic feet of a box; • Calculate the cubic inches of a box; • Calculate the cubic centimeters of a box; • Convert cubic meters to cubic feet; • Convert cubic feet to cubic meters (cubic ft to cubic meters); • Convert cubic inches to liters; • Convert gallons to cubic meters; and • More! Even if it looks like the ultimate conversion tool, this cubic meter calculator is best suited for... well... calculations involving cubic meters, duh! If you are looking for the ultimate converters, check out the volume conversion, the area converter, and the distance converter, all closely related to this one but slightly more focused on pure conversion. It is now time to see how to use the cubic meters calculator by Omni Calculator to compute the volume of a box using any mix of units: 1. Select the units and input the value for the length. 2. Select the units and input the value for the width. 3. Select the units and input the value for the height. 4. At the bottom of the calculator, you will see the resulting volume in 2 different units (by default, cubic ft and cubic meters). Feel free to change if you want to. It's all effortless and very straightforward. You can mix and match units as you see fit, and the results will always be in the units you choose, from cu ft to cubic meters to gallons or liters. But we can make it even easier if you don't need to calculate the cubic meters and just want to convert volume, for example, from cubic feet to cubic meters. If you want to use the cubic meters calculator as a volume converter, you have even fewer steps to follow: 1. Collapse the top section of the calculator. 2. The fields for width, length, and height will be hidden because they are not needed. 3. Select the unit you want to input and enter the value in the top field for volume. 4. Automatically, the tool will calculate the cubic meters equivalent of the volume. It could not be easier! You don't need to remember any conversion factors, do any math or wait. Just select your unit, input the value, and enjoy the result! ## How to calculate the cubic meters of a box or space "Sure, that's fine, but how do you calculate the cubic meters of a box?" You might be saying. Maybe you don't want to use our wonderful calculator. Perhaps you want the knowledge as a backup, or maybe you're preparing for an exam. We won't judge; we only help! So, if you want to learn how to calculate the cubic meters of a box, here is how to do it: 1. Take the values for your box's length, width, and height. If possible, use meters. 2. If you haven't used meters, convert each of them individually to meters. 3. Apply the volume formula for a box: volume = length × width × height 4. Your results should now be in cubic meters. Apply the conversion factor if you want another unit. As you can see, it is not complicated, but it is needlessly time-consuming when you have a nice tool like this one, which we created at Omni Calculator. FAQs ### How many cubic centimeters in a cubic meter? There are 1,000,000 cubic centimeters in a cubic meter. To remember this, you can think about converting meters to centimeters for each of the three dimensions: length, width, and height. One meter has 100 cm, so multiplying that by itself three times, a m³ has 1,000,000 cm³. ### How many cubic feet are in a cubic meter? There are 35.315 cubic feet in a cubic meter. Converting from cubic meters to cubic feet is the same as converting from meters to feet for each of the three dimensions: length, width, and height. One meter has 3.281 feet, so multiplying that by itself three times, one m³ has 35.315 cubic ft. ### How many liters in a cubic meter? There are 1000 liters in a cubic meter. To remember this, you can think about cubic decimeters and liters since they are both the same size. One cubic meter (m³) is 1000 dm³, so since liters and dm³ are the same, there are 1000 liters in a m³. ### How many gallons in a cubic meter? There are 264.17 US gallons in 1 cubic meter, and there are 219.97 UK gallons in 1 cubic meter. You can perform any conversion from cubic meters to gallons by following this instructions: 1. Find out if you want US gallons or UK gallons. 2. Multiply the number of cubic meters by the conversion factor (264.17 for US gallons) and (219.97 for UK gallons). 3. The result will be expressed in your chosen unit. ### What is cubic meter? A cubic meter is a unit of measurement that corresponds to the volume of a square box with 1 meter-long sides. This means a little over 3 ft per side. You can calculate the volume of said box by multiplying all three dimensions together: 1 m × 1 m × 1 m = 1 m³ Another way to visualize a cubic meter is to convert it to other units. One cubic meter is the same as: • 1000 liters; • 264.17 US gallons; and • 35.32 cu ft.	1,673	7,197	{"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	4.0625	4	CC-MAIN-2024-33	latest	en	0.882314	[ 128000, 5966, 6177, 1473, 2, 18521, 292, 53639, 37128, 271, 3923, 374, 264, 41999, 23819, 30, 4438, 311, 1005, 279, 41999, 23819, 31052, 4438, 311, 11294, 279, 41999, 20645, 315, 264, 3830, 477, 3634, 56032, 82, 271, 2409, 279, 41999, 23819, 31052, 11, 499, 649, 5625, 904, 8286, 5089, 311, 41999, 20645, 13, 1952, 1948, 315, 430, 11, 499, 649, 1101, 11294, 279, 41999, 20645, 315, 264, 3830, 315, 264, 3967, 1404, 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https://mathemerize.com/what-is-composition-of-functions/	1,726,214,540,000,000,000	text/html	crawl-data/CC-MAIN-2024-38/segments/1725700651510.65/warc/CC-MAIN-20240913070112-20240913100112-00248.warc.gz	355,481,921	25,573	# What is Composition of Functions ? Here you will learn what is composition of functions with properties and examples. Let’s begin – ## What is Composition of Functions ? Let f : A $$\rightarrow$$ B & g : f : B $$\rightarrow$$ C be two functions. Then the function gof : f : A $$\rightarrow$$ C defined by (gof)(x) = g(f(x)) $$\forall$$ x $$\in$$ A is called the composite of the two function f & g. ## Properties : (a) In general composite of functions is not commutative i.e. gof $$\ne$$ fog. (b) The composition of functions is associative i.e. if f, g, h are three functions such that fo(goh) & (fog)oh are defined, then fo(goh) = (fog)oh. (c) The composition of two bijections is a bijection i.e. if f & g are two bijections such that gof is defined, then gof is also a bijection. Example : If f(x) = $$x^2$$ + 1, g(x) = $$1\over x-1$$, then find (fog)(x). Solution : Now (fog)(x) = f(g(x)) = f($$1\over x-1$$) = f(z), where z = $$1\over x-1$$ = $$z^2 + 1$$ [ $$\because$$ f(x) = $$x^2 + 1$$ ] = $$({1\over x-1})^2$$ + 1 = $$1\over {(x-1)^2}$$ + 1	356	1,065	{"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	4.53125	5	CC-MAIN-2024-38	latest	en	0.802178	[ 128000, 2, 3639, 374, 68206, 315, 24460, 24688, 8586, 499, 690, 4048, 1148, 374, 18528, 315, 5865, 449, 6012, 323, 10507, 382, 10267, 753, 3240, 47896, 567, 3639, 374, 68206, 315, 24460, 24688, 10267, 282, 551, 362, 27199, 59, 20056, 14415, 426, 612, 342, 551, 282, 551, 426, 27199, 59, 20056, 14415, 356, 387, 1403, 5865, 13, 5112, 279, 734, 733, 69, 551, 282, 551, 362, 27199, 59, 20056, 14415, 356, 4613, 555, 320, 70, 1073, 2432, 87, 8, 284, 342, 968, 2120, 595, 27199, 59, 34729, 14415, 865, 27199, 59, 258, 14415, 362, 374, 2663, 279, 28814, 315, 279, 1403, 734, 282, 612, 342, 382, 567, 12094, 14852, 2948, 8, 763, 4689, 28814, 315, 5865, 374, 539, 1081, 332, 1413, 602, 1770, 13, 733, 69, 27199, 59, 818, 14415, 31349, 382, 1921, 8, 578, 18528, 315, 5865, 374, 70003, 602, 1770, 13, 422, 282, 11, 342, 11, 305, 527, 2380, 5865, 1778, 430, 12018, 3348, 2319, 8, 612, 320, 69, 540, 8, 2319, 527, 4613, 11, 1243, 12018, 3348, 2319, 8, 284, 320, 69, 540, 8, 2319, 382, 1361, 8, 578, 18528, 315, 1403, 23232, 38705, 374, 264, 23232, 12181, 602, 1770, 13, 422, 282, 612, 342, 527, 1403, 23232, 38705, 1778, 430, 733, 69, 374, 4613, 11, 1243, 733, 69, 374, 1101, 264, 23232, 12181, 382, 13617, 551, 1442, 282, 2120, 8, 284, 27199, 87, 61, 17, 14415, 489, 220, 16, 11, 342, 2120, 8, 284, 27199, 16, 59, 2017, 865, 12, 16, 14415, 11, 1243, 1505, 320, 69, 540, 2432, 87, 3677, 37942, 551, 4800, 320, 69, 540, 2432, 87, 8, 284, 282, 3348, 2120, 595, 284, 282, 703, 3, 16, 59, 2017, 865, 12, 16, 14415, 8, 284, 282, 13476, 705, 1405, 1167, 284, 27199, 16, 59, 2017, 865, 12, 16, 14415, 271, 28, 27199, 89, 61, 17, 489, 220, 16, 14415, 510, 27199, 59, 28753, 14415, 282, 2120, 8, 284, 27199, 87, 61, 17, 489, 220, 16, 14415, 10661, 28, 27199, 2358, 16, 59, 2017, 865, 12, 16, 5525, 61, 17, 14415, 489, 220, 16, 284, 27199, 16, 59, 2017, 33898, 87, 12, 16, 30876, 17, 92, 14415, 489, 220, 16, 128001 ]	[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ]
https://www.teachoo.com/2195/584/Ex-3.3--23---Prove-tan-4x-=-4-tan-x-(1---tan2-x)---1---6tan2x/category/Ex-3.3/	1,513,374,695,000,000,000	text/html	crawl-data/CC-MAIN-2017-51/segments/1512948579567.73/warc/CC-MAIN-20171215211734-20171215233734-00298.warc.gz	809,186,138	13,753	1. Chapter 3 Class 11 Trigonometric Functions 2. Serial order wise Transcript Ex 3.3, 23 Prove that tanβ‘4π₯ = (4 tanβ‘γπ₯ (1βtan2π₯)γ)/(1 β 6 tan2 π₯+tan4 π₯) Taking L.H.S. tan 4x = (2 tanβ‘2x)/(1 β tan2 2x) tan 4x = 2((2 tanβ‘π₯)/(1 β π‘ππ2 π₯))/(1 β ((2 tanβ‘π₯)/(1 β π‘ππ2 π₯))^2 ) = (((4 tanβ‘π₯)/(1 β π‘ππ2 π₯)))/(1 β((2 tanβ‘π₯ )^2/(1 β π‘ππ2 π₯)^2 ) ) = (((4 tanβ‘π₯)/(1 β π‘ππ2 π₯)))/(1 β((4 γπ‘ππγ^2 π₯)/(1 β π‘ππ2 π₯)^2 ) ) = (((4 tanβ‘π₯)/(1 β π‘ππ2 π₯)))/((((1 β π‘ππ2 π₯)^2 β 4 γπ‘ππγ^2 π₯)/(1 β π‘ππ2 π₯)^2 ) ) = (4 tanβ‘π₯)/(1 β tan2β‘π₯ ) Γ ((1 β tan2β‘γπ₯)2γ)/((1 βπ‘ππ2 π₯)2 β4 tan2β‘π₯ ) = (4 tanβ‘π₯)/1 Γ ((1 β tan2β‘γπ₯)γ)/((1 βπ‘ππ2 π₯)2 β4 tan2β‘π₯ ) = (4 tanβ‘γπ₯ (1 βπ‘ππ2 π₯γ))/((1 βtan2β‘π₯ )2 β4 tanβ‘2π₯ ) Using (a β b)2 = a2 +b2 β 2ab = (4 tanβ‘γπ₯ ( 1 βtan2β‘π₯)γ)/(( (1)2+(π‘ππ2 π₯)2 β2 Γ 1 Γ π‘ππ2 π₯) β4 π‘ππ2 π₯) = (4 tanβ‘γπ₯ ( 1 βtan2β‘π₯)γ)/(( (1)2+(π‘ππ2 π₯)2 β2 Γ 1 Γ π‘ππ2 π₯) β4 π‘ππ2 π₯) = (4 tanβ‘γπ₯ ( 1 βtan2β‘π₯)γ)/(1 + tanβ‘γ4 π₯ β 2 tan2β‘γπ₯ β4 tan2β‘π₯ γ γ ) = (4 tanβ‘γπ₯ ( 1 β tan2β‘γπ₯ )γ γ)/(1 + tan4β‘γπ₯ β6 π‘ππ4 π₯γ ) = R.H.S. Hence L.H.S. = R.H.S. 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https://postvines.com/how-do-you-calculate-payroll-minutes/	1,653,658,219,000,000,000	text/html	crawl-data/CC-MAIN-2022-21/segments/1652662647086.91/warc/CC-MAIN-20220527112418-20220527142418-00591.warc.gz	534,215,249	20,998	# How do you calculate payroll minutes? All you need to do is divide your minutes by 60. For example, say your employee worked 20 hours and 15 minutes during the week. Divide your total minutes by 60 to get your decimal. For this pay period, your employee worked 20.25 hours. ## What is 1/6th of an hour? Each Hour Has 60 Minutes For example, 10 minutes is 10/60 = 1/6 of an hour, and 24 minutes is 24/60 = 6/15 of an hour. ## How many hours is 7am to 5pm with a 30 minute lunch? For example, a 7:30 to 4:30 work day with a 30 minute lunch break means an 8.5 hour work day (9 hours in between, minus 30 minutes or 0.5 hours equals 8.5). If you are working shifts your break may not be a lunch break, and you may also have multiple breaks allowed. ## How do I calculate work? Work can be calculated with the equation: Work = Force × Distance. The SI unit for work is the joule (J), or Newton • meter (N • m). One joule equals the amount of work that is done when 1 N of force moves an object over a distance of 1 m. ## How do you calculate 15 minutes? To convert any fraction to decimal form, we just need to divide its numerator by the denominator. In this case, if we convert 15 minutes to hours we write it as 15/60 because 1 hour = 60 minutes. This means 15 minutes can be written as 0.25 hours in the decimal form. ## How do I calculate my hours per week? Add up the number of hours from each week to get your total. Divide by the total number of weeks. The resulting number is the average hours you would have worked during weeks when you took your previous leave. ## What is .25 on a timesheet? Conversion Chart – Minutes to Hundredths of an Hour Enter time in Oracle Self Service as hundredths of an hour. For example 15 minutes (¼ hour) equals . 25, 30 minutes (½ hour) equals . 5, etc. ## How do you turn 6 7 into a percent? Convert 6/7 to Percentage by Changing Denominator Our percent fraction is 85.714285714286/100, which means that 67 as a percentage is 85.71%. ## What is 0.3 as a percent? To write 0.3 as a percent, multiply 0.3 by 100. Append % symbol in the product obtained. So, 0.3 as a percent is 30 %. ## What is a 85? Letter grade Percentage Grade definition A+ 90-100 Excellent A 85-89 Very good A– 80-84 Very good B+ 75-79 Good B. ## Why does Europe use military time? The 24-hour clock has a long history. It is easier than specifying AM and PM, when there might be doubt. Railways were major users. The military also need to avoid ambiguity. ## How many hours is 9/5 in a day? The traditional American business hours are 9:00 a.m. to 5:00 p.m., Monday to Friday, representing a workweek of five eight-hour days comprising 40 hours in total. These are the origin of the phrase 9-to-5, used to describe a conventional and possibly tedious job. ## How are lunch timesheets calculated? Enter this formula: =SUM((C2-B2)+(E2-D2))*24 into a blank cell beside your time record cells, F2, for instance, see screenshot: Note: In the above formula: C2 is the lunch start time, B2 is the log in time, E2 indicates the log out time and D2 is the lunch end time. ## How do you calculate payroll hours for employees? You do this by dividing the minutes worked by 60. You then have the hours and minutes in numerical form, which you can multiply by the wage rate. For example, if your employee works 38 hours and 27 minutes this week, you divide 27 by 60. This gives you 0.45, for a total of 38.45 hours. ## How do you calculate payroll per hour? First, determine the total number of hours worked by multiplying the hours per week by the number of weeks in a year (52). Next, divide this number from the annual salary. For example, if an employee has a salary of \$50,000 and works 40 hours per week, the hourly rate is \$50,000/2,080 (40 x 52) = \$24.04. ## How do you calculate late minutes? How to calculate Late In: 1. Subtract the scheduled work time from the actual arrival time. Using the sample above: 10:42 – 8:00 = 2.42, or 2 hours and 42 minutes late arrival. 2. ## What percentage of one hour is 45 minutes? Therefore, 1 hour and 45 minutes is 7.291% of the total number of hours in a day. Hence the correct option is C. ## What is a 100 minute clock? A 100-minute clock is typically referred to when talking about the decimal equivalent of the minutes of an hour. In other words, 1 is 60 minutes in a 100-minute clock. ## How do you write time in hours? In general, to write time in the 24-hour system, omit the colon between hours and minutes, and follow the numerals for time with the word “hours.” The invasion began at 0823 hours . Read aloud as “oh-eight-twenty-three hours” or “zero-eight-twenty-three” (military). ## How do you calculate 30 minutes payroll? For example, if an employee works 8:30 minutes, this is 8.5 hours when converted to decimal, multiply it by their hourly wage; this results in a gross wage amount. ## How do you calculate monthly hours? A quick and easy method of calculating monthly hours is to multiply 40 hours per week by 4 weeks, yielding 160 hours for the month. The other method will provide the average number of work hours in a month. ## What is full time UK? There is no specific number of hours that makes someone full or part-time, but a full-time worker will usually work 35 hours or more a week. 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http://www.chegg.com/homework-help/introduction-to-algorithms-3rd-edition-chapter-11.p-solutions-9780262033848	1,472,685,838,000,000,000	text/html	crawl-data/CC-MAIN-2016-36/segments/1471982939756.54/warc/CC-MAIN-20160823200859-00034-ip-10-153-172-175.ec2.internal.warc.gz	350,163,986	17,180	View more editions Solutions Introduction to Algorithms # Introduction to Algorithms (3rd Edition)Solutions for Chapter 11.P • 954 step-by-step solutions • Solved by publishers, professors & experts • iOS, Android, & web Looking for the textbook? Over 90% of students who use Chegg Study report better grades. May 2015 Survey of Chegg Study Users Chapter: Problem: SAMPLE SOLUTION Chapter: Problem: • Step 1 of 5 Longest-probe bound for hashing Open addressing is the hashing technique in which for inserting an element in the table firstly we search for the elements if it is already contained in it, the search returns either the element or the free space in the array to fill that element. In uniform hashing technique sequence of elements to be search can be any of the m! permutation inthat is this technique gives the method of searching which search in the full list for the single element in the generalized way. For the storage of n elements wherein a hash table of capacity m items it uses the open addressing technique. • Step 2 of 5 a . In the process of insertion of an element by using uniform hashing technique if the element is not found in the table then it is inserted into the first slot which do not contain any element. Ifis the load factor that is to search an element which is not found in the table then the total expected probes is maximum of. Assuming that a variable chosenrepresents the probes which do not return the required variable then . We know that then. The probability of insertion of element in a hash table that uses uniform hashing requires not less than k probes so thus, Hence Thus for insertion of element in the hash table by using uniform hashing where the probability of storage is not more thanwhen it searches more than k times. • Step 3 of 5 b . For insertion of element in the hash table by using uniform hashing where the probability of storage is not more thanwhen it searches more than k times that is Here the probes required for the insertion of element in hash table is more than . Thus, . By using formulae of logarithm: Thus Hence the probability of storage for the insertion of element is for when it probes elements in the table. • Step 4 of 5 c. The insertion of element in the table requires searches and represents the total number of searches necessary for the insertion of n elements. Consider an event A for which the number of probes X are such that, and in event the number of probes, for . Then probability of events is as: For n number of events according to the Boolâ€™s inequality: Then for the set A of n events, Hence if the insertion of element in the table requires searches and if represents the total number of searches necessary for the insertion of n elements then: • Step 5 of 5 d . The expected length of the probe sequence for k probes is given by the formulae: Breaking the sequence into two parts one from and the other from then, For the searching of element for which the number of probes less than then and for the search when number of probes are greater thanthen . Thus, Hence, the expected total lengthof the maximum length probe succession is. 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https://www.basic-mathematics.com/ambiguous-case-of-the-law-of-sines.html	1,696,340,867,000,000,000	text/html	crawl-data/CC-MAIN-2023-40/segments/1695233511106.1/warc/CC-MAIN-20231003124522-20231003154522-00144.warc.gz	704,568,016	11,873	# Ambiguous case of the law of sines The ambiguous case of the law of sines happens when two sides and an angle opposite one of the two sides are given. We can shorten this situation with SSA. Since the length of the third side is not known, we don't know if a triangle will be formed or not. That is the reason we call this case ambiguous. In fact, this kind of situation or SSA can give the following 4 scenarios. • No triangle • 1 triangle • 1 right triangle • 2 triangles ## The first scenario of the ambiguous case of the law of sines occurs when a < h. For example, take a look at the triangle below where only two sides are given. These two given sides are a and b. An angle opposite to one side is also given. Angle A is the angle that is opposite to side a or one of the two sides. Note that SSA in this case means side a side b angle A in that order. Because a is shorter than h, a is not long enough to form a triangle. In fact, the number of possible triangles that can be formed in the SSA case depends on the length of the altitude or h. Notice that sin A = h / b Once you multiply both sides of the equation above by b, we get h = b sin A. An example showing that no triangle can be formed Method #1 Suppose A = 74°, a = 51, and b = 72. h = 72 × sin (74°) = 72 × 0.9612 = 68.20 Since 51 or a is less than h or 69.20, no triangle will be formed. Method #2 We can also show that no triangle exists by using the law of sines. a / sin A = b / sin B The ratio a / sin A is known since a / sin A = 51 / sin 74° Since we also know the length of b, the missing quantity in the law of sines is sin B. It is logical then to look for sin B and see what we end up with. 51 / sin 74° = 72 / sin B 51 sin B = 72 sin 74° sin B = (72 sin 74°) / 51 sin B = (72 × 0.9612) / 51 sin B = (69.2064) / 51 sin B = 1.3569 Since the sine of an angle cannot be bigger than 1, angle B does not exist. Therefore, no triangle can be formed with the given measurements. ## The second scenario of the ambiguous case of the law of sines occurs when a = h. When a = h, the resulting triangle will always be a right triangle. An example showing that a right triangle can be formed Method #1 Suppose A = 30°, a = 25, and b = 50. h = 50 × sin (30°) = 50 × 0.5 = 25 Since 25 or a is equal to h or 25, 1 right triangle will be formed. Method #2 Again, we can use the law of sines to show that this time sin B exists and it is equal to 90 degrees. a / sin A = b / sin B 25 / sin 30° = 50 / sin B 25 sin B = 50 sin 30° sin B = (50 sin 30°) / 25 sin B = (50 × 0.5) / 25 sin B = (25) / 25 sin B = 1 B = sin-1(1) = 90 degrees. ## The third scenario of the ambiguous case of the law of sines occurs when a > h and a > b. When a is bigger than h, again a triangle can be formed. However, since a is bigger than b, we can only have one triangle. Try to make a triangle where a is bigger than b, you will notice that there can only be 1 such triangle. An example showing that exactly 1 triangle can be formed Suppose A = 30°, a = 50, and b = 40. h = 40 × sin (30°) = 40 × 0.5 = 20 Since 50 or a is bigger than both h (or 20) and b (or 40), 1 triangle will be formed. ## Last situation: a > h and a < b When a is less than b, 2 triangles can be formed as clearly illustrated below. The two triangles are triangle ACD and triangle AED. An example showing that exactly 2 triangles can be formed Suppose A = 30°, a = 40, and b = 60 h = 60 × sin (30°) = 60 × 0.5 = 30 Since 40 or a is bigger than h and a is smaller than b or 60, 2 triangles will be formed. ## Recent Articles 1. ### How To Find The Factors Of 20: A Simple Way Sep 17, 23 09:46 AM There are many ways to find the factors of 20. A simple way is to... 2. ### The SAT Math Test: How To Be Prepared To Face It And Survive Jun 09, 23 12:04 PM The SAT Math section is known for being difficult. But it doesn’t have to be. 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https://fahrenheittocelsius.org/1471-f-to-c	1,603,737,679,000,000,000	text/html	crawl-data/CC-MAIN-2020-45/segments/1603107891624.95/warc/CC-MAIN-20201026175019-20201026205019-00172.warc.gz	313,248,166	9,541	# 1471 Fahrenheit to Celsius Welcome to 1471 Fahrenheit to Celsius, or 1471 F to C in short. Here you can find what 1471 degrees Fahrenheit to Celsius is, along with a temperature converter and the formula. For 1471 (degrees) Fahrenheit we write 1471 °F, and (degrees) Celsius or centigrades are denoted with the symbol °C. So if you have been looking for 1471 °F to °C, then you are right here, too. Read on below to learn everything about the temperature conversion. You don’t have to press the blue button, unless you want to swap the conversion. Bookmark this temperature converter now. ## Formula The 1471 Fahrenheit to Celsius formula is: [°C] = ([1471] − 32) x 5 ⁄ 9. Therefore, we get: 1471 F to C = 799.444 °C 1471 °F in °C = 799.444 Celsius 1471 F in C = 799.444 degrees Celsius Right above we have given you the result in the common spelling variants of this temperature conversion. Of course, the result is also valid for 1471F to C, 1471F in C etc. Here you can change 1471 Celsius to Fahrenheit. Next, we explain the math. ## Conversion To convert the temperature start by start by deducting 32 from 1471. Then multiply 1439 by 5 over 9 to obtain 799.444 degrees Celsius. Easier, however, is using our converter above. Similar temperature conversions on our website include: Ahead is the wrap-up of our content. ## Summary How much is 1471 degrees Fahrenheit in Celsius? By reading our article so far, or by means of our converter, you already know the answer 1471 Fahrenheit in other temperature units is: • Newton: 263.817 °N • Kelvin: 1072.594 °K • Réaumur: 639.556 °Ré • Rømer: 427.208 °Ro • Delisle: -1049.167 °De • Rankine: 1930.67 °R This ends our posts about 1471 °F to °C. If you have anything to tell or in case you would like to ask something about 1471 F in C, then fill in the form below. And, if this article about 1471 F to Celsius has been helpful to you, then hit the social buttons please. Thanks for visiting fahrenheittocelsius.org. 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https://stats.stackexchange.com/questions/104403/what-is-the-meaning-of-a-large-p-value	1,653,397,240,000,000,000	text/html	crawl-data/CC-MAIN-2022-21/segments/1652662572800.59/warc/CC-MAIN-20220524110236-20220524140236-00225.warc.gz	601,211,009	68,603	What is the meaning of a large p-value? I understand that the $p$-value is the conditional probability of observing the test statistic or something more extreme given that the null hypothesis is true. I have read the great explanation by @user28 in this post: What is the meaning of p values and t values in statistical tests? However, do large $p$-values say anything? Does a larger $p$-value lend greater support to the null hypothesis? If I set rejection region to be $<0.05$, then does it make a difference if I get $p$-value $0.06$ or $0.99$? (After all, $0.05$ is arbitrary, and $0.06$ is so close to being rejected that if I arbitrarily set $0.05$ as $0.1$ instead, the null hypothesis would have been rejected.) Can one make any statistical use of a non-rejecting $p$-value? • One example is mentioned at the end of this answer: Fisher's re-examination of Mendel's pea experiments. – whuber Jun 23, 2014 at 14:37 • You cannot conclude that a test statistic supports the null hypothesis when the calculation of that statistic is conditional on the assumption that the null is true. That is to say, if $E$ and $H_0$ are events, you cannot say $H_0$ is true if $\Pr[E \mid H_0]$ is "large." Jun 23, 2014 at 16:04 How you should 'use' the p-value depends on how you have designed your study with regard to the analyses you will run. I discuss two different philosophies about p-values in my answer here: When to use Fisher and Neyman-Pearson framework? You may find it helpful to read that. If you have, for example, run a power analysis and intend to use the p-value to make a final decision, you should not use close to the line ('marginally significant') as a meaningful category. It is fine to use a different alpha than $0.05$ (such as $0.10$), but once you decided on it and set your study up accordingly, you should stick with it. In addition, you cannot use a large p-value as evidence for the null hypothesis. I discussed that idea in my answer here: Why do statisticians say a non-significant result means "you cannot reject the null" as opposed to accepting the null hypothesis? Reading that answer may be helpful to you as well. In my view, everything boils down to assumptions, that is, how well the models fits them. If it does agree with all of them, treat the p-value as a probability. Then you can compare p-value of 0.06 with 0.99 by concluding which of the two are more likely. Also, a lot depend on circumstances: in some cases, marginal significance shouldn't be ignored, because as you stated, rejection region can be set rather arbitrarily. But if the models satisfies the assumptions, then you should not seek to reject your hypothesis to some arbitrary level but rather investigate, how likely is the outcome that you got. • It is problematic to (mis)interpret a p-value as a probability in this sense. You seem to be applying a Bayesian-like approach but without the usual care needed to consider and defend a prior distribution. What justification can you provide to support your recommendations? – whuber Jun 23, 2014 at 14:40 • well, I was having a linear regression in mind satisfying all the conventional assumptions of the models and as a result regression coefficients having normal distributions. I suppose in this case it would be safe to treat p-value as a probability, wouldn't it? Jun 23, 2014 at 14:51 • No, it would not. There is a lot of debate about p-values, but practically everybody who has entered into it (on either side) has noted that this use of p-values is invalid. Comparing p-values does not determine whether one model is more or less likely than another. – whuber Jun 23, 2014 at 14:54 • Yes, you are right. I started googling and two source made things more clear: Hubbard, R. (2004). Alphabet soup: Blurring the distinctions between p's and α's in psychological research. Theory and Psychology, 14 (3), 295-327; Hubbard, R., & Lindsay, R.M. (2008). Why P values are not a useful measure of evidence in statistical significance testing. Theory and Psychology, 18 (1), 69-88. Obviously, I had an incorrect view of the concept! Jun 23, 2014 at 15:16 It is true that the acceptance range for the p-value of a hypothesis test is rather arbitrary, but nevertheless a lower p-value means that the test result can be accepted with more certainty, because the p-value essentially defines the confidence interval for the estimate, so a narrower confidence interval should be regarded more significant for the test.	1,066	4,472	{"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	3.65625	4	CC-MAIN-2022-21	longest	en	0.937788	[ 128000, 3923, 374, 279, 7438, 315, 264, 3544, 281, 19625, 1980, 40, 3619, 430, 279, 400, 79, 3, 12, 970, 374, 279, 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http://scanraid.com/glossary	1,670,154,904,000,000,000	text/html	crawl-data/CC-MAIN-2022-49/segments/1669446710972.37/warc/CC-MAIN-20221204104311-20221204134311-00498.warc.gz	43,697,308	6,171	Print Version Page Index Solvers Puzzles Basic Strategies Tough Strategies Diabolical Strategies Extreme Strategies Deprecated Strategies Str8ts Other Order Now! Order Now! # Glossary • Almost Locked Set (ALS) – A set of N unsolved cells with N+1 candidates • Bi-location – A unit with only two occurances of a candidate X • Bi-value – A cell with only two candidates remaining • Box – Any 3x3 shaped unit in a normal sudoku, or any squiggly 9-cell shape in Jigsaw sudoku which must contain 1 to 9. There are 9 boxes in 9x9 sudoku. • Cage – A marked off group of cells for which their sum is provided in Killer Sudoku and variants. • Candidate – A possible solution for a cell. Any cell reduced to one candidate is solved. • Cell – A square on the sudoku grid for which there is one single number solution. • Chain – A sequence of links between candidates such that the two ends imply an elimination. All Loops are closed chains. • Conjugate Pair – The two remaining candidates in a bi-location are conjugate. They are used to form strong link within a unit since one must be true and the other must be false. • Contradiction – In Sudoku an illegal and therefore contradictory situation can occur if a) there are no candidates left in a cell, or b) two or more cells claim to be true. The aim of many strategies is to show a contradiction. • Continuous – In a Loop it signifies that the links between nodes strictly alternate between strong and weak links. • Cycle – Another name for a Loop. See Loop. • Discontinuous – In a Loop it signifies that there is one discontinuity where a node is beset by either two weak links or two strong links. • Grouped Node – A set of candidates of the same value (that must share a unit) that can be used to form a group with the effect of being able to treat it as a single cell. • Hinge – In a Y-Wing the middle cell which joins the two pincer cells. • Inference – Valid deductions that can be made between two linked candidates. There are inferences of the weak kind and the strong kind. • Locked Set – A group of N candidates confined to a group of N cells. All candidates that can see a locked set can be removed. Naked Pairs, Triples etc are Locked Sets. • Loop – Another name for a Cycle. See Cycle. A closed chain or sequence of candidates (which are called nodes in the loop) used to make a deduction. Loops are continuous or discontinuous. • Pincer – In a Y-Wing or Y-Wing Chain the two end nodes are pincers which attack a cell for the purposes of elimination. See also Hinge. • Strong Inference – Where two candidates X and Y are linked with strong inference then both cannot be false at the same time. Therefore if X is false Y is true and if Y is false X is true. • Strong Link – A link between two candidates (which can be a single number or a grouped node) whereby if one is deemed false the other must be true. Strong Links can be between two cells (bi-location), within a call (bi-value) or as part of an ALS or more complex structure. Article here • Unit – My preferred term for any row, column or box. Also called a house. • Variant – A variation on an original puzzle, eg Jigsaw Sudoku is a variant of Sudoku. • Weak Inference – Where two candidates X and Y are linked weak inference implies that both cannot be true at the same time. Therefore if X is true Y is false and if Y is true X is false. • Weak Link – A link between one candidate and all others in a unit or cell whereby if the candidate is deemed to be true all the others are false. A strong link can be deemed to be weak merely because "all the others" includes cases where there is only one other. Article here • XY-Wing – Another name for Y-Wing The following terms are not used on this site but may be found out in the wild • Block – Another name for a box • Chute – Three boxes in a column. • Domain – Another term for a unit. • Given – Another name for a clue. • House – Another name for a unit, sometimes also the same as a chute. • Reduction – Another name for elimination • Region – Another name for a box	941	4,036	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	3.75	4	CC-MAIN-2022-49	latest	en	0.915144	[ 128000, 9171, 6207, 5874, 8167, 198, 50, 40535, 198, 47, 9065, 645, 198, 16323, 56619, 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https://www.bankingcareers.in/blog/bankexam/crack-ibps-po-2017-2-recent-types-simplification-problems/	1,555,943,274,000,000,000	text/html	crawl-data/CC-MAIN-2019-18/segments/1555578555187.49/warc/CC-MAIN-20190422135420-20190422161420-00148.warc.gz	623,182,679	11,175	# Crack IBPS PO 2017: 2 Recent Types Of Simplification Problems Dear Reader, IBPS PO exams are starting very soon! (In 5 months tentatively). Below tutorial will help you for sure. You already know simplification is an important topic for bank exams. In this post, you will find two types of problems asked in previous year PO exam. Example For Type 1: Find the number in place of the question mark: ? / 676.03 = 1295.6 / ? Solution: We can approximate 676.03 as 676 and 1295.6 as 1296. Therefore, the given equation can be simplified as: ?2 = 1296 x 676 Or ? = square root of (1296 x 676) = square root of (1296) x square root of (676) = 36 x 26 = 936 Tip: Remember squares of numbers up to 50 to solve such problems easily. Example For Type 2: Find the number in place of the question mark: ? = square root of (0.25 x 0.36) x 1/30 This type was surprisingly easy (in last year exams). Only one thing you have to know to solve such problems is the correct placement of decimal point in square roots. Now let us solve the above problem. square root of (0.25 x 0.36) x 1/30 = square root of (0.25) x square root of (0.36) x 1/30 = 0.5 x 0.6 x 1/30 = 0.30 x 1/30 = 0.01	359	1,177	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	4.34375	4	CC-MAIN-2019-18	latest	en	0.895286	[ 128000, 2, 56045, 12373, 5119, 13228, 220, 679, 22, 25, 220, 17, 35390, 21431, 5046, 62342, 2461, 45635, 271, 31765, 26226, 11, 12373, 5119, 13228, 40786, 527, 6041, 1633, 5246, 0, 320, 644, 220, 20, 4038, 16006, 8046, 570, 21883, 22237, 690, 1520, 499, 369, 2771, 382, 2675, 2736, 1440, 15858, 2461, 374, 459, 3062, 8712, 369, 6201, 40786, 13, 763, 420, 1772, 11, 499, 690, 1505, 1403, 4595, 315, 5435, 4691, 304, 3766, 1060, 13228, 7151, 382, 13617, 1789, 4078, 220, 16, 512, 10086, 279, 1396, 304, 2035, 315, 279, 3488, 1906, 512, 30, 611, 220, 24187, 13, 2839, 284, 220, 9748, 20, 13, 21, 611, 24688, 37942, 512, 1687, 649, 45968, 220, 24187, 13, 2839, 439, 220, 24187, 323, 220, 9748, 20, 13, 21, 439, 220, 9748, 21, 13, 15636, 11, 279, 2728, 24524, 649, 387, 44899, 439, 1473, 30, 17, 284, 220, 9748, 21, 865, 220, 24187, 198, 2244, 949, 284, 9518, 3789, 315, 320, 9748, 21, 865, 220, 24187, 340, 28, 9518, 3789, 315, 320, 9748, 21, 8, 865, 9518, 3789, 315, 320, 24187, 340, 28, 220, 1927, 865, 220, 1627, 198, 28, 220, 25612, 271, 16394, 25, 20474, 32440, 315, 5219, 709, 311, 220, 1135, 311, 11886, 1778, 5435, 6847, 382, 13617, 1789, 4078, 220, 17, 512, 10086, 279, 1396, 304, 2035, 315, 279, 3488, 1906, 512, 30, 284, 9518, 3789, 315, 320, 15, 13, 914, 865, 220, 15, 13, 1927, 8, 865, 220, 16, 14, 966, 271, 2028, 955, 574, 29392, 4228, 320, 258, 1566, 1060, 40786, 570, 8442, 832, 3245, 499, 617, 311, 1440, 311, 11886, 1778, 5435, 374, 279, 4495, 22165, 315, 12395, 1486, 304, 9518, 20282, 382, 7184, 1095, 603, 11886, 279, 3485, 3575, 382, 38576, 3789, 315, 320, 15, 13, 914, 865, 220, 15, 13, 1927, 8, 865, 220, 16, 14, 966, 198, 28, 9518, 3789, 315, 320, 15, 13, 914, 8, 865, 9518, 3789, 315, 320, 15, 13, 1927, 8, 865, 220, 16, 14, 966, 198, 28, 220, 15, 13, 20, 865, 220, 15, 13, 21, 865, 220, 16, 14, 966, 198, 28, 220, 15, 13, 966, 865, 220, 16, 14, 966, 198, 28, 220, 15, 13, 1721, 128001 ]	[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ]
https://metanumbers.com/54194	1,638,875,447,000,000,000	text/html	crawl-data/CC-MAIN-2021-49/segments/1637964363376.49/warc/CC-MAIN-20211207105847-20211207135847-00550.warc.gz	438,811,348	7,396	# 54194 (number) 54,194 (fifty-four thousand one hundred ninety-four) is an even five-digits composite number following 54193 and preceding 54195. In scientific notation, it is written as 5.4194 × 104. The sum of its digits is 23. It has a total of 5 prime factors and 16 positive divisors. There are 22,932 positive integers (up to 54194) that are relatively prime to 54194. ## Basic properties • Is Prime? No • Number parity Even • Number length 5 • Sum of Digits 23 • Digital Root 5 ## Name Short name 54 thousand 194 fifty-four thousand one hundred ninety-four ## Notation Scientific notation 5.4194 × 104 54.194 × 103 ## Prime Factorization of 54194 Prime Factorization 2 × 73 × 79 Composite number Distinct Factors Total Factors Radical ω(n) 3 Total number of distinct prime factors Ω(n) 5 Total number of prime factors rad(n) 1106 Product of the distinct prime numbers λ(n) -1 Returns the parity of Ω(n), such that λ(n) = (-1)Ω(n) μ(n) 0 Returns: 1, if n has an even number of prime factors (and is square free) −1, if n has an odd number of prime factors (and is square free) 0, if n has a squared prime factor Λ(n) 0 Returns log(p) if n is a power pk of any prime p (for any k >= 1), else returns 0 The prime factorization of 54,194 is 2 × 73 × 79. Since it has a total of 5 prime factors, 54,194 is a composite number. ## Divisors of 54194 1, 2, 7, 14, 49, 79, 98, 158, 343, 553, 686, 1106, 3871, 7742, 27097, 54194 16 divisors Even divisors 8 8 4 4 Total Divisors Sum of Divisors Aliquot Sum τ(n) 16 Total number of the positive divisors of n σ(n) 96000 Sum of all the positive divisors of n s(n) 41806 Sum of the proper positive divisors of n A(n) 6000 Returns the sum of divisors (σ(n)) divided by the total number of divisors (τ(n)) G(n) 232.796 Returns the nth root of the product of n divisors H(n) 9.03233 Returns the total number of divisors (τ(n)) divided by the sum of the reciprocal of each divisors The number 54,194 can be divided by 16 positive divisors (out of which 8 are even, and 8 are odd). The sum of these divisors (counting 54,194) is 96,000, the average is 6,000. ## Other Arithmetic Functions (n = 54194) 1 φ(n) n Euler Totient Carmichael Lambda Prime Pi φ(n) 22932 Total number of positive integers not greater than n that are coprime to n λ(n) 3822 Smallest positive number such that aλ(n) ≡ 1 (mod n) for all a coprime to n π(n) ≈ 5515 Total number of primes less than or equal to n r2(n) 0 The number of ways n can be represented as the sum of 2 squares There are 22,932 positive integers (less than 54,194) that are coprime with 54,194. And there are approximately 5,515 prime numbers less than or equal to 54,194. ## Divisibility of 54194 m n mod m 2 3 4 5 6 7 8 9 0 2 2 4 2 0 2 5 The number 54,194 is divisible by 2 and 7. ## Classification of 54194 • Arithmetic • Deficient ### Expressible via specific sums • Polite • Non-hypotenuse • Frugal ## Base conversion (54194) Base System Value 2 Binary 1101001110110010 3 Ternary 2202100012 4 Quaternary 31032302 5 Quinary 3213234 6 Senary 1054522 8 Octal 151662 10 Decimal 54194 12 Duodecimal 27442 20 Vigesimal 6f9e 36 Base36 15te ## Basic calculations (n = 54194) ### Multiplication n×y n×2 108388 162582 216776 270970 ### Division n÷y n÷2 27097 18064.7 13548.5 10838.8 ### Exponentiation ny n2 2936989636 159167216333384 8625908121971412496 467472464762118728808224 ### Nth Root y√n 2√n 232.796 37.8428 15.2577 8.84688 ## 54194 as geometric shapes ### Circle Diameter 108388 340511 9.22683e+09 ### Sphere Volume 6.66718e+14 3.69073e+10 340511 ### Square Length = n Perimeter 216776 2.93699e+09 76641.9 ### Cube Length = n Surface area 1.76219e+10 1.59167e+14 93866.8 ### Equilateral Triangle Length = n Perimeter 162582 1.27175e+09 46933.4 ### Triangular Pyramid Length = n Surface area 5.08702e+09 1.8758e+13 44249.2 ## Cryptographic Hash Functions md5 41e84168c20f59d698f99bf46ded6625 50f95adf879f0d833cdd2dace7977b37c80a0602 f3011286091aae62db60d623c71e8d7286153ce75d695a6c1d68cb8f6432f9b5 fae4bb769cc8372eb3671186735c4967dcb1763d2c2b693df8d771b65f79c177b9a0716e4cdd97d6a051358eaa199aa87d9b78060f95e5c1f5afe32c8066a858 5ee7c08e91cd1146afcd2cb5459f679c85e5ba21	1,505	4,217	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, 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https://nrich.maths.org/public/leg.php?code=6&cl=2&cldcmpid=1130	1,568,625,303,000,000,000	text/html	crawl-data/CC-MAIN-2019-39/segments/1568514572516.46/warc/CC-MAIN-20190916080044-20190916102044-00113.warc.gz	596,288,519	9,072	# Search by Topic #### Resources tagged with Place value similar to Reach 100: Filter by: Content type: Age range: Challenge level: ### There are 75 results Broad Topics > Numbers and the Number System > Place value ### Reach 100 ##### Age 7 to 14 Challenge Level: Choose four different digits from 1-9 and put one in each box so that the resulting four two-digit numbers add to a total of 100. ### Two and Two ##### Age 11 to 14 Challenge Level: How many solutions can you find to this sum? Each of the different letters stands for a different number. ### Cayley ##### Age 11 to 14 Challenge Level: The letters in the following addition sum represent the digits 1 ... 9. If A=3 and D=2, what number is represented by "CAYLEY"? ### Football Sum ##### Age 11 to 14 Challenge Level: Find the values of the nine letters in the sum: FOOT + BALL = GAME ##### Age 11 to 14 Challenge Level: Replace the letters with numbers to make the addition work out correctly. R E A D + T H I S = P A G E ### Eleven ##### Age 11 to 14 Challenge Level: Replace each letter with a digit to make this addition correct. ### Calculator Bingo ##### Age 7 to 11 Challenge Level: A game to be played against the computer, or in groups. Pick a 7-digit number. A random digit is generated. What must you subract to remove the digit from your number? the first to zero wins. ### Nice or Nasty for Two ##### Age 7 to 14 Challenge Level: Some Games That May Be Nice or Nasty for an adult and child. Use your knowledge of place value to beat your opponent. ### All the Digits ##### Age 7 to 11 Challenge Level: This multiplication uses each of the digits 0 - 9 once and once only. Using the information given, can you replace the stars in the calculation with figures? ### Which Scripts? ##### Age 7 to 11 Challenge Level: There are six numbers written in five different scripts. Can you sort out which is which? ### The Thousands Game ##### Age 7 to 11 Challenge Level: Each child in Class 3 took four numbers out of the bag. Who had made the highest even number? ### One Million to Seven ##### Age 7 to 11 Challenge Level: Start by putting one million (1 000 000) into the display of your calculator. Can you reduce this to 7 using just the 7 key and add, subtract, multiply, divide and equals as many times as you like? ### Song Book ##### Age 7 to 11 Challenge Level: A school song book contains 700 songs. The numbers of the songs are displayed by combining special small single-digit cards. What is the minimum number of small cards that is needed? ### Tis Unique ##### Age 11 to 14 Challenge Level: This addition sum uses all ten digits 0, 1, 2...9 exactly once. Find the sum and show that the one you give is the only possibility. ##### Age 5 to 11 Challenge Level: Who said that adding couldn't be fun? ### Being Curious - Primary Number ##### Age 5 to 11 Challenge Level: Number problems for inquiring primary learners. ### Being Resourceful - Primary Number ##### Age 5 to 11 Challenge Level: Number problems at primary level that require careful consideration. ### Round the Dice Decimals 1 ##### Age 7 to 11 Challenge Level: Use two dice to generate two numbers with one decimal place. What happens when you round these numbers to the nearest whole number? ### Diagonal Sums ##### Age 7 to 11 Challenge Level: In this 100 square, look at the green square which contains the numbers 2, 3, 12 and 13. What is the sum of the numbers that are diagonally opposite each other? What do you notice? ### Coded Hundred Square ##### Age 7 to 11 Challenge Level: This 100 square jigsaw is written in code. It starts with 1 and ends with 100. Can you build it up? ### Multiply Multiples 2 ##### Age 7 to 11 Challenge Level: Can you work out some different ways to balance this equation? ### Trebling ##### Age 7 to 11 Challenge Level: Can you replace the letters with numbers? Is there only one solution in each case? ### Multiply Multiples 1 ##### Age 7 to 11 Challenge Level: Can you complete this calculation by filling in the missing numbers? In how many different ways can you do it? ### Dicey Operations in Line for Two ##### Age 7 to 11 Challenge Level: Dicey Operations for an adult and child. Can you get close to 1000 than your partner? ### Digit Sum ##### Age 11 to 14 Challenge Level: What is the sum of all the digits in all the integers from one to one million? ### Round the Dice Decimals 2 ##### Age 7 to 11 Challenge Level: What happens when you round these numbers to the nearest whole number? ### Round the Three Dice ##### Age 7 to 11 Challenge Level: What happens when you round these three-digit numbers to the nearest 100? ### Arrange the Digits ##### Age 11 to 14 Challenge Level: Can you arrange the digits 1,2,3,4,5,6,7,8,9 into three 3-digit numbers such that their total is close to 1500? ### Subtraction Surprise ##### Age 7 to 14 Challenge Level: Try out some calculations. Are you surprised by the results? ### Multiply Multiples 3 ##### Age 7 to 11 Challenge Level: Have a go at balancing this equation. Can you find different ways of doing it? ##### Age 5 to 11 Challenge Level: Try out this number trick. What happens with different starting numbers? What do you notice? ### Basically ##### Age 11 to 14 Challenge Level: The number 3723(in base 10) is written as 123 in another base. What is that base? ### Number Detective ##### Age 5 to 11 Challenge Level: Follow the clues to find the mystery number. ### Chocolate Maths ##### Age 11 to 14 Challenge Level: Pick the number of times a week that you eat chocolate. This number must be more than one but less than ten. Multiply this number by 2. Add 5 (for Sunday). Multiply by 50... Can you explain why it. . . . ### Spell by Numbers ##### Age 7 to 11 Challenge Level: Can you substitute numbers for the letters in these sums? ### Oddly ##### Age 7 to 11 Challenge Level: Find the sum of all three-digit numbers each of whose digits is odd. ### ABC ##### Age 7 to 11 Challenge Level: In the multiplication calculation, some of the digits have been replaced by letters and others by asterisks. Can you reconstruct the original multiplication? ### Six Is the Sum ##### Age 7 to 11 Challenge Level: What do the digits in the number fifteen add up to? How many other numbers have digits with the same total but no zeros? ##### Age 11 to 14 Challenge Level: Powers of numbers behave in surprising ways. Take a look at some of these and try to explain why they are true. ### Seven Up ##### Age 11 to 14 Challenge Level: The number 27 is special because it is three times the sum of its digits 27 = 3 (2 + 7). Find some two digit numbers that are SEVEN times the sum of their digits (seven-up numbers)? ### Legs Eleven ##### Age 11 to 14 Challenge Level: Take any four digit number. Move the first digit to the end and move the rest along. Now add your two numbers. Did you get a multiple of 11? ### Being Collaborative - Primary Number ##### Age 5 to 11 Challenge Level: Number problems at primary level to work on with others. ### Mini-max ##### Age 11 to 14 Challenge Level: Consider all two digit numbers (10, 11, . . . ,99). In writing down all these numbers, which digits occur least often, and which occur most often ? What about three digit numbers, four digit numbers. . . . ### Being Resilient - Primary Number ##### Age 5 to 11 Challenge Level: Number problems at primary level that may require resilience. ### Becky's Number Plumber ##### Age 7 to 11 Challenge Level: Becky created a number plumber which multiplies by 5 and subtracts 4. What do you notice about the numbers that it produces? Can you explain your findings? ### Always a Multiple? ##### Age 11 to 14 Challenge Level: Think of a two digit number, reverse the digits, and add the numbers together. Something special happens... ### The Number Jumbler ##### Age 7 to 14 Challenge Level: The Number Jumbler can always work out your chosen symbol. Can you work out how? ### Alien Counting ##### Age 7 to 11 Challenge Level: Investigate the different ways these aliens count in this challenge. You could start by thinking about how each of them would write our number 7. ### What Do You Need? ##### Age 7 to 11 Challenge Level: Four of these clues are needed to find the chosen number on this grid and four are true but do nothing to help in finding the number. Can you sort out the clues and find the number? ### Which Is Quicker? ##### Age 7 to 11 Challenge Level: Which is quicker, counting up to 30 in ones or counting up to 300 in tens? 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# Resources tagged with: Sine, cosine, tangent Filter by: Content type: Age range: Challenge level: ### There are 35 results Broad Topics > Pythagoras and Trigonometry > Sine, cosine, tangent ##### Age 14 to 16 Challenge Level: The sides of a triangle are 25, 39 and 40 units of length. Find the diameter of the circumscribed circle. ### Circle Scaling ##### Age 14 to 16 Challenge Level: Describe how to construct three circles which have areas in the ratio 1:2:3. ### Where Is the Dot? ##### Age 14 to 16 Challenge Level: A dot starts at the point (1,0) and turns anticlockwise. Can you estimate the height of the dot after it has turned through 45 degrees? Can you calculate its height? ### Figure of Eight ##### Age 14 to 16 Challenge Level: On a nine-point pegboard a band is stretched over 4 pegs in a "figure of 8" arrangement. How many different "figure of 8" arrangements can be made ? ### Squ-areas ##### Age 14 to 16 Challenge Level: Three squares are drawn on the sides of a triangle ABC. Their areas are respectively 18 000, 20 000 and 26 000 square centimetres. If the outer vertices of the squares are joined, three more. . . . ### Sine and Cosine ##### Age 14 to 16 Challenge Level: The sine of an angle is equal to the cosine of its complement. Can you explain why and does this rule extend beyond angles of 90 degrees? ### Inscribed in a Circle ##### Age 14 to 16 Challenge Level: The area of a square inscribed in a circle with a unit radius is, satisfyingly, 2. What is the area of a regular hexagon inscribed in a circle with a unit radius? ### Circle Box ##### Age 14 to 16 Challenge Level: It is obvious that we can fit four circles of diameter 1 unit in a square of side 2 without overlapping. What is the smallest square into which we can fit 3 circles of diameter 1 unit? ### Round and Round ##### Age 14 to 16 Challenge Level: Prove that the shaded area of the semicircle is equal to the area of the inner circle. ### Dodecawhat ##### Age 14 to 16 Challenge Level: Follow instructions to fold sheets of A4 paper into pentagons and assemble them to form a dodecahedron. Calculate the error in the angle of the not perfectly regular pentagons you make. ### Six Discs ##### Age 14 to 16 Challenge Level: Six circular discs are packed in different-shaped boxes so that the discs touch their neighbours and the sides of the box. Can you put the boxes in order according to the areas of their bases? ### From All Corners ##### Age 14 to 16 Challenge Level: Straight lines are drawn from each corner of a square to the mid points of the opposite sides. Express the area of the octagon that is formed at the centre as a fraction of the area of the square. ### Raising the Roof ##### Age 14 to 16 Challenge Level: How far should the roof overhang to shade windows from the mid-day sun? ### Farhan's Poor Square ##### Age 14 to 16 Challenge Level: From the measurements and the clue given find the area of the square that is not covered by the triangle and the circle. ### Round and Round a Circle ##### Age 14 to 16 Challenge Level: Can you explain what is happening and account for the values being displayed? ### History of Trigonometry - Part 3 ##### Age 11 to 18 The third of three articles on the History of Trigonometry. ### A Scale for the Solar System ##### Age 14 to 16 Challenge Level: The Earth is further from the Sun than Venus, but how much further? Twice as far? Ten times? ### Lying and Cheating ##### Age 11 to 14 Challenge Level: Follow the instructions and you can take a rectangle, cut it into 4 pieces, discard two small triangles, put together the remaining two pieces and end up with a rectangle the same size. Try it! ### Cosines Rule ##### Age 14 to 16 Challenge Level: Three points A, B and C lie in this order on a line, and P is any point in the plane. Use the Cosine Rule to prove the following statement. ### At a Glance ##### Age 14 to 16 Challenge Level: The area of a regular pentagon looks about twice as a big as the pentangle star drawn within it. Is it? ### Sine and Cosine for Connected Angles ##### Age 14 to 16 Challenge Level: The length AM can be calculated using trigonometry in two different ways. Create this pair of equivalent calculations for different peg boards, notice a general result, and account for it. ### The History of Trigonometry- Part 1 ##### Age 11 to 18 The first of three articles on the History of Trigonometry. This takes us from the Egyptians to early work on trigonometry in China. ### History of Trigonometry - Part 2 ##### Age 11 to 18 The second of three articles on the History of Trigonometry. ### Making Maths: Clinometer ##### Age 11 to 14 Challenge Level: You can use a clinometer to measure the height of tall things that you can't possibly reach to the top of, Make a clinometer and use it to help you estimate the heights of tall objects. ### Moving Squares ##### Age 14 to 16 Challenge Level: How can you represent the curvature of a cylinder on a flat piece of paper? ### 8 Methods for Three by One ##### Age 14 to 18 Challenge Level: This problem in geometry has been solved in no less than EIGHT ways by a pair of students. How would you solve it? How many of their solutions can you follow? How are they the same or different?. . . . ### Orbiting Billiard Balls ##### Age 14 to 16 Challenge Level: What angle is needed for a ball to do a circuit of the billiard table and then pass through its original position? ### Screen Shot ##### Age 14 to 16 Challenge Level: A moveable screen slides along a mirrored corridor towards a centrally placed light source. A ray of light from that source is directed towards a wall of the corridor, which it strikes at 45 degrees. . . . ##### Age 11 to 18 Logo helps us to understand gradients of lines and why Muggles Magic is not magic but mathematics. See the problem Muggles magic. ### Coke Machine ##### Age 14 to 16 Challenge Level: The coke machine in college takes 50 pence pieces. It also takes a certain foreign coin of traditional design... ##### Age 14 to 16 Challenge Level: In this problem we are faced with an apparently easy area problem, but it has gone horribly wrong! What happened? ### Far Horizon ##### Age 14 to 16 Challenge Level: An observer is on top of a lighthouse. How far from the foot of the lighthouse is the horizon that the observer can see? ### Eight Ratios ##### Age 14 to 16 Challenge Level: Two perpendicular lines lie across each other and the end points are joined to form a quadrilateral. Eight ratios are defined, three are given but five need to be found. ### Trigonometric Protractor ##### Age 14 to 16 Challenge Level: An environment that simulates a protractor carrying a right- angled triangle of unit hypotenuse.

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The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation. Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!) A127330 Begin with the empty sequence and a starting number s = 0. At step k (k >= 1) append the k consecutive numbers s to s+k-1 and change the starting number (for the next step) to 2s+2. 5 0, 2, 3, 6, 7, 8, 14, 15, 16, 17, 30, 31, 32, 33, 34, 62, 63, 64, 65, 66, 67, 126, 127, 128, 129, 130, 131, 132, 254, 255, 256, 257, 258, 259, 260, 261, 510, 511, 512, 513, 514, 515, 516, 517, 518, 1022, 1023, 1024, 1025, 1026, 1027, 1028, 1029, 1030, 1031, 2046 (list; table; graph; refs; listen; history; text; internal format) OFFSET 0,2 COMMENTS From a TV show. A129142 and A129143 are similar, slightly more natural, but for a puzzle perhaps too transparent sequences. Can be seen as a triangle (row by step) read by rows: T(n,k) = T(n-1,k) + 2^n for k < n and T(n,n) = T(n-1,n-1) + 2^n + 1. - Reinhard Zumkeller, Nov 16 2013 LINKS Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened EXAMPLE In step 1 starting number 0 is appended to the empty sequence and the next starting number is 2*0 + 2 = 2. In step 2 the two numbers 2, 3 are appended and the starting number is changed to 2*2 + 2 = 6. MATHEMATICA Table[ Range[2^k-2, 2^k+k-3], {k, 1, 11}] // Flatten (* Jean-François Alcover, Oct 07 2013, after Klaus Brockhaus *) Join[{0}, Flatten[With[{nn=10}, Range[#[[1]], Total[#]]&/@Thread[ {Accumulate[ 2^Range[nn]], Range[nn]}]]]] (* Harvey P. Dale, Nov 05 2017 *) PROG (PARI) {v=[]; s=0; for(k=1, 11, w=vector(k, j, j+s-1); s=2*s+2; v=concat(v, w)); for(n=1, #v, print1(v[n], ", "))} \\ Klaus Brockhaus, Mar 31 2007 (MAGMA) &cat[ [2^k-2..2^k+k-3]: k in [1..11] ]; // Klaus Brockhaus, Mar 31 2007 (Haskell) a127330 n k = a127330_tabl !! n !! k a127330_row n = a127330_tabl !! n a127330_tabl = step 0 1 where step s k = [s .. s + k - 1] : step (2 * s + 2) (k + 1) -- Reinhard Zumkeller, Nov 16 2013 CROSSREFS Cf. A129142, A129143. Cf. A000918 (left edge), A145071 (right edge). Sequence in context: A166458 A189013 A242940 * A035346 A030164 A066646 Adjacent sequences: A127327 A127328 A127329 * A127331 A127332 A127333 KEYWORD easy,nice,nonn,tabl AUTHOR Steven Cartier (steven.cartier(AT)rogers.com), Mar 30 2007 EXTENSIONS Edited and extended by Klaus Brockhaus, Mar 31 2007 Keyword tabl added and offset changed by Reinhard Zumkeller, Nov 16 2013 STATUS approved Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent The OEIS Community | Maintained by The OEIS Foundation Inc. Last modified September 23 19:41 EDT 2020. Contains 337315 sequences. (Running on oeis4.)

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files/ kains -- Blogmeister We have three 6th grade Science classes and two 8th grade Science classes blogging here from the Pacific Northwest in Chimacum, WA! Sixth graders are learning a bit about Mt Saint Helens, environmental science through fresh water ecology, and physical science this year. Eighth graders are learning about life science this year. Please join us as we learn Science by exploring our world. Mr. G's Blog Mr. G's Class Facebook Page by kains teacher: Alfonso Gonzalez Blog Entries List 25, 50, all Article posted May 16, 2012 at 03:29 PM GMT • comment • Reads 1078 here are the blogs I visited (links) 1. http://danteawapuni.blogspot.co.nz/2012/02/wonder-beads.html 2.http://ryanawapuni.blogspot.co.nz/2012/02/this-is-me.html those are the blogs I visited here is some information about them 1.dante plays sports and reads 2.ryan likes superheros Article posted May 16, 2012 at 03:29 PM GMT • comment • Reads 1078 Article posted May 15, 2012 at 03:16 PM GMT • comment • Reads 790 A. I thought that the wide side on a block fo wood would be the side that would reaquier ( i know i spelled that wrong) the most force to pull because it has the most surface area but I found out that the side with the least surface ( the narrow side) took the most force to pull. What I learned about surface area, and weight of blocks I shall type in the next letters starting now. I learned that that if you have more weight on a smaller surface it will take more force to pull the block then if the block has the weight spread out and if you add more blocks the weight will add and it will be even more force needed to pull the block. B.the scale measures the fricton force witch ( yes I spelled that wrong to ( I think)) is caused by pulling on the scale with gravity pulling the block down. C.we would use almost no force to pull a block of wood across ice because it is very smooth so the force needed to pull the block will be about .1 newtons per block on the wide side of the block. Article posted May 15, 2012 at 03:16 PM GMT • comment • Reads 790 Article posted April 26, 2012 at 03:34 PM GMT • comment • Reads 867 You can put a rubber band on 2 sticks then you put it (the rubber band) around the cart then pull back then you let it go to slingshot it across the floor. A way to make it twice as powerful ( the rubber bands elastic force) is to double up the ruber band then reapeat the stuff before. The difference between mass and weight! well mass affects wieght because gravity affects mass makeing wieght Article posted April 26, 2012 at 03:34 PM GMT • comment • Reads 867 Article posted March 13, 2012 at 04:35 PM GMT • comment • Reads 1094 K: I know that without bateries we would not have cars or handheld gaming systems or anything like those things that need bateries. W: Some things that I want to learn about bateries are: 1. how to make one 2. how to see if I could make a way better one and 3. why there is acidic stuff in bateries L: I learned that batterys have strong acid in them to make energy because without that it would not work. I also learned that batterys have copper and zinc in them. Article posted March 13, 2012 at 04:35 PM GMT • comment • Reads 1094 Article posted March 13, 2012 at 04:20 PM GMT • comment (1) • Reads 1495 Hi I am a 6th grader going to chimacum middle school I live in a small house close to port townsend. If I want to I can walk to the beach next to my house. A few things I like are: riding my bike, playing video games and,talking to my friends. The subjects in school I like are science, p.e., and math. I am also interesed in space (yes that big empty thing thats all around us with other planets and stuff). Thats pretty much it so good bye. Article posted March 13, 2012 at 04:20 PM GMT • comment (1) • Reads 1495 Article posted March 5, 2012 at 04:34 PM GMT • comment • Reads 938 @font-face { font-family: "Times New Roman"; }@font-face { font-family: "Wingdings"; }@font-face { font-family: "Modern No. 20"; }@font-face { font-family: "ヒラギノ角ゴ Pro W3"; }p.MsoNormal, li.MsoNormal, div.MsoNormal { margin: 0in 0in 0.0001pt; font-size: 12pt; font-family: "Times New Roman"; }table.MsoNormalTable { font-size: 10pt; font-family: "Times New Roman"; }div.Section1 { page: Section1; } DO: what are these years average DO levels: 8.125. The oxygen in the air makes contact with water to make dissolved oxygen (do). Chimacum creek has been doing well even at a low of about 6 ppm. PH: the average is 6.3 ph for this year. 7 is pure water 1-6 is acidic 8-14 is alkaline. When base and acid come together it makes is a balance. Ph stands for positive hydrogen. Fish like 6.5-7. Turbitity: the average is in the 30’ of ntu’s. n.t.u stands for nephelometric turbitity units somebody might ask what turbitity is well turbitity is the difference between clear water and dirty water Nitrogen (this is my perrmater): nitrates are important because they help the soil and water and creatures and everything(about this).the average for this year in nitrates is 0.3 this is good because fish die in to high of nitrate and so will people mostly babys because to much nitrates can cause brown blood disease or blue baby syndrome(same thing) or Methogloblin (also the same thing). Ammonia: This years average ammonia is 1.6. this is good because it wont kill the fish. Ammonia is very dangeros chemical thing. Ammonia can be found in many cleaning supplies. Pure Ammonia is a very very smelly colorless liquid. Eutrophication is a thing that is very bad. A thing you do not want to do is mix ammonia and bleach ( it can kill you) Flow rate: you x length x width (in a river) to get cubic meters per second (if using that measurement) . flow rate can be important because if fish can not lay there eggs then they will die out. The average for flow rate is 1.5934 m3/s (??????) Temperature: water that is shallow takes shorter to warm up than deep. This years average water temp is 33.5 farenhite. The airs average is 37.5 farenhite. One of the highest temp max for fish is 97º farenhite where the fish will die L. Article posted March 5, 2012 at 04:34 PM GMT • comment • Reads 938 Article posted January 30, 2012 at 04:16 PM GMT • comment • Reads 1111 When I went to the place next to snow creek I was really exited to plant trees. when we started I found out that it was much harder to plant trees then I thought. We also found lots of worms and we put them next to the trees to help the soil. Article posted January 30, 2012 at 04:16 PM GMT • comment • Reads 1111 Article posted November 21, 2011 at 04:34 PM GMT • comment (2) • Reads 1855 this is the better video Click here. Article posted November 21, 2011 at 04:34 PM GMT • comment (2) • Reads 1855 Article posted November 7, 2011 at 04:22 PM GMT • comment (1) • Reads 1727 In some other blogs I showed pictures in this blog I will tell you which creek scored better yellow jacket or chimacum. Well my pictures would mean that chimacum scored 2 points because scuds are worth 2 points. but luckly other people found more bugs like mayflys and stoneflys that are worth more. chimacum scored 21 points total. And yellow jacket creek scored 25 points total. so that means yellow jacket creek is a little bit cleaner. it also means that we have potentially good water quality and yellow jacket creek has potentially exelent water quality. Article posted November 7, 2011 at 04:22 PM GMT • comment (1) • Reads 1727 Article posted November 7, 2009 at 06:00 AM GMT • comment • Reads 1639 this is a picture of a moving scud. this is another moving scud. these are worth 3 points from our out door education program which scored using how tolorent the bug was to pollution. there is 4 different groups 1.intolerent, 2. somewhat intolerent, 3.somewhat tolerent, 4. very tolerent this is another part of my other macro blog which also had a scud but it was dead. Article posted November 7, 2009 at 06:00 AM GMT • comment • Reads 1639 Article posted November 2, 2011 at 04:16 PM GMT • comment • Reads 1696 this is a photo of a dead scud i found in chimacum creek. thanks for reading and looking. Article posted November 2, 2011 at 04:16 PM GMT • comment • Reads 1696 Article posted October 20, 2011 at 04:35 PM GMT • comment (1) • Reads 1859 1.What is water pollution? well it's contamination of water bodies such as lakes, rivers, oceans, ground water, ect. 2.What are some sources of water pollution? Well 14 billion pounds of garbage is being dumped into the oceans every year. Another source is factories, refineries, waste treatment plants, ect. 3.What are the consequences of water pollution? Well it can be disastrous for both humans and the ecosystem .14,000 humans are suffering from waterborne diseases every day! Water pollution effects the chemical and physical properties of water afecting the life present inside the water. 4.What can we do to improve water quality. Well we can drive less because the pollution from the gas is geting into the water. Another way is to stop using chemicals like pesticide and herbicide because they get into our water polluting it. 5.What can you do to prevent water from being polluted? One thing to do is dispose of used oil, antifreeze, paints, ect. properly which means not in sewers or drains. Another thing to do is have your septic system [if you have one] inspected and pumped every three to five years. One last thing to do is purchase household detergents and cleaners that are low in phosphorous to reduce the pollution flowwing into our water. Article posted October 20, 2011 at 04:35 PM GMT • comment (1) • Reads 1859 Article posted October 3, 2011 at 04:27 PM GMT • comment (7) • Reads 2740 1. I Love sushi 2. I Like candy 3. I am a boy 4. My brain is abnormal 5. Most of my friends are smart 6. Some Are stupid(of my friends) 7. I am not very good at coming up with ideas Article posted October 3, 2011 at 04:27 PM GMT • comment (7) • Reads 2740 Previous Entries All Entries All Titles

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+0 0 1 286 1 +4 1.Use the figure to answer the questions. (a) Explain why the triangles are similar. (b) Find the value of y. Show your work 2. Danae is making a candle by pouring melted wax into a mold in the shape of a cone. The radius of the base of the cone is 2 in and the height of the cone is 8 in. To get the wax for the candle, Danae melts blocks of wax that are each 1 in by 1 in by 2 in. How many of the wax cubes will Danae need in order to make the candle? Show your work. StrongJet Jun 16, 2017 Sort: #1 +6954 +2 1. (a) Since ∠C = ∠S , these triangles are similar IF $$\frac{30}{45}=\frac{16}{24} \\~\\ \frac{30\,\div\,15}{45\,\div\,15}=\frac{16\,\div\,8}{24\,\div\,8} \\~\\ \frac23=\frac23\qquad\text{true}$$ So, these triangles are similar. (b) Since these triangles are similar... $$\frac{y}{14}=\frac{30}{16} \\~\\ y=\frac{30}{16}\,*\,14 \\~\\ y=26.25$$ 2. volume of cone = (1/3) * (pi * radius2) * (height) = (1/3) * (pi * 22) * (8) = (1/3) * pi * 4 * 8 = 32pi / 3 cubic inches volume of block = length * width * height = 1 * 1 * 2 = 2 cubic inches How many blocks fit into the cone? How many 2 cubic inches fit into 32pi / 3 cubic inches ? $$\frac{32\pi}{3}\,\div\,2 \,=\,\frac{32\pi}{3}\,*\,\frac12\,=\,\frac{32\pi}{6}\,\approx\,16.755$$ hectictar Jun 16, 2017 ### 11 Online Users We use cookies to personalise content and ads, to provide social media features and to analyse our traffic. We also share information about your use of our site with our social media, advertising and analytics partners. See details

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https://electronics.stackexchange.com/questions/700694/obtaining-transfer-function

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# Obtaining transfer function In the above circuit j(t)=u(t)A, where u(t) is the step function. Z is a component/circuit system with no independent variables and all initial values zero. The goal is to find the transfer function Z(s). So I think I know the transfer function here is the laplace transform of the output divided by the laplace transform of the input. The output being u1(t) and input being j(t). u1(t) is also Z*j(t) and the laplace transform Z(s)*1/s. The laplace transform of the input is 1/s. So using the equation for the transfer function (Z(s)*1/s)/1/s you get Z(s), which does not seem right at all. What am I missing here? The goal is to find the transfer function Z(s). Z is a component/circuit system with no independent variables The transfer function is independent of the input so the fact that it is a current step is irrelevant. So the only information you have to express the transfer function is the output voltage and the input current. The ratio then is an impedance Z(s)=Z. This you have verified as: $$Z(s)=\frac{U1_{out}(s)}{J_{in}(s)}=Z$$ I think you are missing the fact that you have the right answer. Understand that the impedance of any passive element is the transfer function of the element where voltage is the output and current is the input. • Oh that's interesting. Makes sense now lol. Only thing that I don't understand is how you write the equation for the transfer function. I have never seen t(s) or n(s) before, so I'm kind of confused on that. Commented Feb 8 at 20:03 • Just some finger trouble with MathJax. The equation should make more sense now. @JohnnyB Commented Feb 8 at 20:06 • Oh lol. Thanks for the help @RussellH Commented Feb 8 at 22:11

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Question When solutions of silver nitrate and potassium chloride are mixed, silver chloride precipitates out of solution... When solutions of silver nitrate and potassium chloride are mixed, silver chloride precipitates out of solution according to the equation AgNO3(aq)+KCl(aq)→AgCl(s)+KNO3(aq). A) What mass of silver chloride can be produced from 1.90 L of a 0.133 M solution of silver nitrate? B) The reaction described in Part A required 3.67 L of potassium chloride. What is the concentration of this potassium chloride solution? A) number of moles of AgNO3 = M(AgNO3)*V(AgNO3) = 0.133 M * 1.90 L = 0.2527 mol from reaction, mol of AgCl formed = mol of AgNO3 reacted = 0.2527 mol Molar mass of AgCl, MM = 1*MM(Ag) + 1*MM(Cl) = 1*107.9 + 1*35.45 = 143.35 g/mol use: mass of AgCl, m = number of mol * molar mass = 0.2527 mol * 1.434*10^2 g/mol = 36.22 g B) From reaction, mol of KCl reacted = mol of AgNO3 formed = 0.2527 mol Now use: mol of KCl = M(KCl)*V(KCl) 0.2527 mol = M(KCl) * 3.67 L M(KCl) = 0.0689 M

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The OEIS is supported by the many generous donors to the OEIS Foundation. Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!) A006345 Linus sequence: a(n) "breaks the pattern" by avoiding the longest doubled suffix. (Formerly M0074) 9 1, 2, 1, 1, 2, 2, 1, 2, 1, 1, 2, 1, 2, 2, 1, 1, 2, 1, 1, 1, 2, 2, 1, 2, 1, 1, 2, 2, 1, 1, 1, 2, 1, 1, 2, 2, 1, 2, 1, 1, 2, 1, 2, 2, 1, 1, 2, 1, 1, 1, 2, 2, 1, 2, 1, 1, 2, 2, 1, 2, 2, 2, 1, 1, 2, 1, 2, 2, 1, 1, 2, 2, 2, 1, 2, 2, 1, 1, 2, 1, 2, 2, 1, 2, 1, 1, 2, 2, 1, 2, 2, 2, 1, 1, 2, 1, 2, 2, 1, 1, 2, 2, 2, 1, 2 (list; graph; refs; listen; history; text; internal format) OFFSET 1,2 COMMENTS To find a(n), consider either a 1 or a 2. For each, find the longest repeated suffix, that is, for each of a(n)=1,2, find the longest sequence s with the property that the sequence a(1),...,a(n) ends with ss. Use the digit that results in the shorter such suffix. a(1) = 1. The empty sequence of length 0 is the shortest possible suffix and is trivially doubled. Note that this doesn't result in exactly Linus's choices. - K. Ramsey, kramsey(AT)aol.com On average, it seems that (# of 1s up to n) - (# of 2s up to n) -> infinity as n -> infinity (as O(log n)?), while the asymptotic density of either 1s or 2s appears to be 1/2. - Daniel Forgues, Mar 01 2017 REFERENCES N. S. Hellerstein, Letter to N. J. A. Sloane (1978). N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS T. D. Noe, Robert Israel and Hugo van der Sanden, Table of n, a(n) for n = 1..50000 (1..1000 from T. D. Noe, 1001..20000 from Robert Israel) P. Balister, S. Kalikow, A. Sarkar, The Linus sequence, Preprint May 2007; Combinatorics, Probability and Computing, Volume 19, Issue 1 January 2010 , pp. 21-46.. N. Hellerstein, Letter to N. J. A. Sloane, 1978 N. Hellerstein, M. Gardner, & S. Kim, Correspondence related to the Linus and Sally sequences, 1977 N. J. A. Sloane, Illustration of initial terms Eric Weisstein's World of Mathematics, Linus Sequence. EXAMPLE After 1,2,1,1,2,2,1,2, if we put a 1, the suffix {2,1} repeats, but if we put a 2 the longer suffix {1,2,2} repeats, so the next term is 1. MAPLE LDS:= proc(L) local Cands, r, m; Cands:= {\$1..floor(nops(L)/2)}; r:= 0; for m from 1 while nops(Cands) > 0 do Cands:= select(c -> L[-m] = L[-c-m], Cands); if min(Cands) = m then r:= m; Cands:= subs(m=NULL, Cands); fi od; r end proc: A:= 1: for n from 2 to 10^3 do if LDS([A, 1]) < LDS([A, 2]) then A:= A, 1 else A:= A, 2 fi; od: seq(A[i], i=1..10^3); # Robert Israel, Jun 22 2015 MATHEMATICA a[1]=1; a[2]=2; suffix[lst_] := If[MatchQ[lst, {___, b__, b__}], lst /. {___, b__, b__} :> {b}, {}]; a[n_] := a[n] = Module[{aa, lg1, lg2}, aa = Array[a, n-1]; lg1 = suffix[Append[aa, 1]] // Length; lg2 = suffix[Append[aa, 2]] // Length; If[lg1 <= lg2, 1, 2]]; Table[a[n], {n, 1, 105}] (* Jean-François Alcover, Dec 11 2014 *) PROG (Perl) -le 'print\$_.=3**/(.*)(.)\1\$/-\$2for(\$_)x99' (Ton Hospel/Phil Carmody) [An example of Perl golfing: use as few (key)strokes as possible] (PARI) {a(n)=local(A, t); if(n<2, n>0, A=[1]; for(i=2, n, forstep(j=i\2-1, 0, -1, for(k=1, j, if(A[i-j-k-1]!=A[i-k], next(2))); t=j; break); A=concat(A, [3-A[i-t-1]])); A[n])} /* Michael Somos, May 04 2006 */ Comment on calculating this sequence and A006346 with Perl, from Hugo van der Sanden, Jun 23 2015: (Start) The approach I used was to take advantage of Perl's regular expression capabilities, coupled with the realization that Perl can optimize patterns anchored to the start far better than those anchored to the end - reversing the string to allow that gave a speedup of several orders of magnitude: my \$string = ""; digit('1', 0); for (2 .. \$limit) { my(\$repeat, \$digit) = (\$string =~ m{ ^ (.*) ([12]) \1 }x) or die; digit(\$digit eq '1' ? '2' : '1', length(\$repeat) + 1); } sub digit { my(\$digit, \$repeat) = @_; \$string = \$digit . \$string; # n A6345(n) A6346(n) printf "%s %s %s\n", length(\$string), \$digit, \$repeat; } This takes about 45s to calculate 50000 terms of both sequences. (End) CROSSREFS Cf. A006346, A094840, A157238 (A006345(n) - 1). Sequence in context: A202340 A049705 A060236 * A122497 A350330 A154402 Adjacent sequences: A006342 A006343 A006344 * A006346 A006347 A006348 KEYWORD nonn,easy,nice AUTHOR N. J. A. Sloane EXTENSIONS More terms from Naohiro Nomoto, May 21 2001 Additional comments from Mitch Harris, Dec 31 2003 STATUS approved Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents The OEIS Community | Maintained by The OEIS Foundation Inc. Last modified June 3 15:36 EDT 2023. Contains 363116 sequences. (Running on oeis4.)

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## RNG147 ### March 28, 2017 We have looked at random number generators in several previous exercises, but most of them returned integers. In today’s exercise we look at a simple random number generator that returns floating-point numbers. The generator is simple: Given a seed between zero and one, the next number in the sequence is the fractional portion of 147 times the seed. Your task is to implement the random number generator described above, and to assess its suitability. When you are finished, you are welcome to read or run a suggested solution, or to post your own solution or discuss the exercise in the comments below. Pages: 1 2 ### 4 Responses to “RNG147” 1. Rutger said In python 3 ```def rng147(seed): while True: yield seed seed = (147.0 * seed) % 1.0 r = rng147(0.1234) for v in range(10): print(next(r)) ``` 2. Graham said Your sine trick only moves the problem elsewhere; try seeding with pi/6 (arcsin of 1/2). Solution in Haskell, using the Stream package for explicitly infinite lists: ```module Main where import Data.Fixed (mod') import Data.Stream (Stream) import qualified Data.Stream as S rng147 :: Double -> Stream Double rng147 = S.iterate (\x -> (147 * x) `mod'` 1) main :: IO () main = mapM_ print . S.take 10 . rng147 \$ pi / 10 ``` 3. Globules said Here’s another Haskell version. With an integer argument, n, it prints the first n random numbers. With no arguments it converts the stream of random Doubles into a stream of 32-bit unsigned integers. The purpose is to feed that binary data to the dieharder program, which evaluates its randomness. We don’t make use of all the bits in the Double; it’s just an easy way of getting the data to dieharder in a form that it likes. ```{-# LANGUAGE ScopedTypeVariables #-} import Data.Word (Word32) import Foreign.Marshal.Alloc (alloca) import Foreign.Storable (poke, sizeOf) import System.Environment (getArgs) import System.IO (hPutBuf, stdout) -- A stream of "random" numbers. (We take the tail to exclude the seed from the -- values.) rng147 :: RealFrac b => b -> [b] rng147 = tail . iterate step where step x = let (_ :: Int, f) = properFraction (147 * x) in f main :: IO () main = do let rs = rng147 (0.1234567 :: Double) ns <- fmap (map read) getArgs :: IO [Int] case ns of -- Output an endless stream of 32-bit binary data for dieharder. [] -> put \$ map cvt rs -- Print the first n random values. [n] -> mapM_ print \$ take n rs -- Invalid argument(s). _ -> putStrLn "Usage: rng147 [n]" -- Convert a value to an 32-bit unsigned integer, possibly throwing away the -- least significant bits of the argument. cvt :: RealFrac a => a -> Word32 cvt x = round \$ fromIntegral (maxBound :: Word32) * x -- Output the unsigned 32-bit values as binary data. put :: [Word32] -> IO () put is = let n = sizeOf (0 :: Word32) in alloca (\p -> void (mapM_ (\i -> poke p i >> hPutBuf stdout p n) is)) ``` ```\$ ./rng147 10 0.14813489999999874 0.7758302999998143 4.705409997271204e-2 0.9169526959886696 0.7920463103344275 0.430807619160845 0.3287200166442119 0.32184244669915074 0.310839664775159 0.6934307219483742 ``` Here’s the result of feeding the output to dieharder. I won’t try to interpret the results, here. (The -a argument tells it to run all the tests it has; -g 200 says that stdin is a series of 32-bit unsigned integers.) The summary is that 82 tests passed, 27 failed and 5 were weak. ```\$ ./rng147 | dieharder -a -g 200 #=============================================================================# # dieharder version 3.31.1 Copyright 2003 Robert G. Brown # #=============================================================================# rng_name |rands/second| Seed | stdin_input_raw| 1.01e+07 |2776686247| #=============================================================================# test_name |ntup| tsamples |psamples| p-value |Assessment #=============================================================================# diehard_birthdays| 0| 100| 100|0.64044752| PASSED diehard_operm5| 0| 1000000| 100|0.00000000| FAILED diehard_rank_32x32| 0| 40000| 100|0.45366134| PASSED diehard_rank_6x8| 0| 100000| 100|0.97658733| PASSED diehard_bitstream| 0| 2097152| 100|0.72508491| PASSED diehard_opso| 0| 2097152| 100|0.00000000| FAILED diehard_oqso| 0| 2097152| 100|0.00000000| FAILED diehard_dna| 0| 2097152| 100|0.00000000| FAILED diehard_count_1s_str| 0| 256000| 100|0.00000000| FAILED diehard_count_1s_byt| 0| 256000| 100|0.00000000| FAILED diehard_parking_lot| 0| 12000| 100|0.00000000| FAILED diehard_2dsphere| 2| 8000| 100|0.00000000| FAILED diehard_3dsphere| 3| 4000| 100|0.00000000| FAILED diehard_squeeze| 0| 100000| 100|0.00000000| FAILED diehard_sums| 0| 100| 100|0.06709806| PASSED diehard_runs| 0| 100000| 100|0.00000000| FAILED diehard_runs| 0| 100000| 100|0.00000000| FAILED diehard_craps| 0| 200000| 100|0.13428702| PASSED diehard_craps| 0| 200000| 100|0.00096831| WEAK marsaglia_tsang_gcd| 0| 10000000| 100|0.00000000| FAILED marsaglia_tsang_gcd| 0| 10000000| 100|0.00000000| FAILED sts_monobit| 1| 100000| 100|0.03994929| PASSED sts_runs| 2| 100000| 100|0.22979976| PASSED sts_serial| 1| 100000| 100|0.98471576| PASSED sts_serial| 2| 100000| 100|0.31624292| PASSED sts_serial| 3| 100000| 100|0.13274265| PASSED sts_serial| 3| 100000| 100|0.06274970| PASSED sts_serial| 4| 100000| 100|0.10906298| PASSED sts_serial| 4| 100000| 100|0.71359846| PASSED sts_serial| 5| 100000| 100|0.00790811| PASSED sts_serial| 5| 100000| 100|0.07231695| PASSED sts_serial| 6| 100000| 100|0.17659272| PASSED sts_serial| 6| 100000| 100|0.18513720| PASSED sts_serial| 7| 100000| 100|0.04315090| PASSED sts_serial| 7| 100000| 100|0.45634930| PASSED sts_serial| 8| 100000| 100|0.14826946| PASSED sts_serial| 8| 100000| 100|0.42520026| PASSED sts_serial| 9| 100000| 100|0.71305937| PASSED sts_serial| 9| 100000| 100|0.20822407| PASSED sts_serial| 10| 100000| 100|0.97154428| PASSED sts_serial| 10| 100000| 100|0.69812982| PASSED sts_serial| 11| 100000| 100|0.53454147| PASSED sts_serial| 11| 100000| 100|0.08444014| PASSED sts_serial| 12| 100000| 100|0.22453592| PASSED sts_serial| 12| 100000| 100|0.72236320| PASSED sts_serial| 13| 100000| 100|0.78366045| PASSED sts_serial| 13| 100000| 100|0.37616667| PASSED sts_serial| 14| 100000| 100|0.95764809| PASSED sts_serial| 14| 100000| 100|0.14472891| PASSED sts_serial| 15| 100000| 100|0.43901688| PASSED sts_serial| 15| 100000| 100|0.76714556| PASSED sts_serial| 16| 100000| 100|0.58587727| PASSED sts_serial| 16| 100000| 100|0.67425978| PASSED rgb_bitdist| 1| 100000| 100|0.05846675| PASSED rgb_bitdist| 2| 100000| 100|0.00004623| WEAK rgb_bitdist| 3| 100000| 100|0.66589418| PASSED rgb_bitdist| 4| 100000| 100|0.00026565| WEAK rgb_bitdist| 5| 100000| 100|0.07969442| PASSED rgb_bitdist| 6| 100000| 100|0.17850326| PASSED rgb_bitdist| 7| 100000| 100|0.25911992| PASSED rgb_bitdist| 8| 100000| 100|0.00045592| WEAK rgb_bitdist| 9| 100000| 100|0.00642740| PASSED rgb_bitdist| 10| 100000| 100|0.27514527| PASSED rgb_bitdist| 11| 100000| 100|0.91420132| PASSED rgb_bitdist| 12| 100000| 100|0.09738292| PASSED rgb_minimum_distance| 2| 10000| 1000|0.00000000| FAILED rgb_minimum_distance| 3| 10000| 1000|0.00000000| FAILED rgb_minimum_distance| 4| 10000| 1000|0.00000000| FAILED rgb_minimum_distance| 5| 10000| 1000|0.01232138| PASSED rgb_permutations| 2| 100000| 100|0.55323173| PASSED rgb_permutations| 3| 100000| 100|0.00000000| FAILED rgb_permutations| 4| 100000| 100|0.00000000| FAILED rgb_permutations| 5| 100000| 100|0.00000000| FAILED rgb_lagged_sum| 0| 1000000| 100|0.76792249| PASSED rgb_lagged_sum| 1| 1000000| 100|0.59838659| PASSED rgb_lagged_sum| 2| 1000000| 100|0.95607401| PASSED rgb_lagged_sum| 3| 1000000| 100|0.28565715| PASSED rgb_lagged_sum| 4| 1000000| 100|0.95085961| PASSED rgb_lagged_sum| 5| 1000000| 100|0.45582064| PASSED rgb_lagged_sum| 6| 1000000| 100|0.60565985| PASSED rgb_lagged_sum| 7| 1000000| 100|0.04195406| PASSED rgb_lagged_sum| 8| 1000000| 100|0.30831976| PASSED rgb_lagged_sum| 9| 1000000| 100|0.36546234| PASSED rgb_lagged_sum| 10| 1000000| 100|0.19936994| PASSED rgb_lagged_sum| 11| 1000000| 100|0.07303233| PASSED rgb_lagged_sum| 12| 1000000| 100|0.34645127| PASSED rgb_lagged_sum| 13| 1000000| 100|0.29195838| PASSED rgb_lagged_sum| 14| 1000000| 100|0.30094975| PASSED rgb_lagged_sum| 15| 1000000| 100|0.99790005| WEAK rgb_lagged_sum| 16| 1000000| 100|0.75196523| PASSED rgb_lagged_sum| 17| 1000000| 100|0.38809626| PASSED rgb_lagged_sum| 18| 1000000| 100|0.61180091| PASSED rgb_lagged_sum| 19| 1000000| 100|0.19440479| PASSED rgb_lagged_sum| 20| 1000000| 100|0.57917094| PASSED rgb_lagged_sum| 21| 1000000| 100|0.83129601| PASSED rgb_lagged_sum| 22| 1000000| 100|0.57292407| PASSED rgb_lagged_sum| 23| 1000000| 100|0.10381929| PASSED rgb_lagged_sum| 24| 1000000| 100|0.93651195| PASSED rgb_lagged_sum| 25| 1000000| 100|0.50848729| PASSED rgb_lagged_sum| 26| 1000000| 100|0.81756629| PASSED rgb_lagged_sum| 27| 1000000| 100|0.29194323| PASSED rgb_lagged_sum| 28| 1000000| 100|0.14934239| PASSED rgb_lagged_sum| 29| 1000000| 100|0.96722837| PASSED rgb_lagged_sum| 30| 1000000| 100|0.41045537| PASSED rgb_lagged_sum| 31| 1000000| 100|0.33379469| PASSED rgb_lagged_sum| 32| 1000000| 100|0.67528985| PASSED rgb_kstest_test| 0| 10000| 1000|0.12422629| PASSED dab_bytedistrib| 0| 51200000| 1|1.00000000| FAILED dab_dct| 256| 50000| 1|0.00000000| FAILED Preparing to run test 207. ntuple = 0 dab_filltree| 32| 15000000| 1|0.00000000| FAILED dab_filltree| 32| 15000000| 1|0.00000000| FAILED Preparing to run test 208. ntuple = 0 dab_filltree2| 0| 5000000| 1|0.00000000| FAILED dab_filltree2| 1| 5000000| 1|0.00000000| FAILED Preparing to run test 209. ntuple = 0 dab_monobit2| 12| 65000000| 1|1.00000000| FAILED \$ ``` 4. […] built several random number generators: [1], [2], [3], [4], [5], [6], [7], [8], [9] (I didn’t realize it was so many until I went back and looked). In today’s exercise we […]

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# The Chain Rule The Chain Rule – The topic for today’s article is the chain rule. So you all might have come across this particular equation or this particular term in your lower standards it is when you were in your Mathematics High School or somewhere. So but in deep learning is the most important concept when we have to deal with the backpropagation, So backpropagation is one such algorithm or technique where you try to backtrack to individual layers and you try to come to that particular unit or neuron in order to adjust the weights. So there you basically use this equation or this concept from mathematics. That is the chain rule. So we’ll be having a quick revision of what the chain rule is in this particular video. So say for example Simple you are given an equation like you have some exponential equation E raised to the sine of some function say x square. So now you have this particular equation and it’s been asked that you have to take the total derivative of with respect to X say, this is some function say Y and Y is equal to is given like this and you want to take the derivative of this particular function. So how would you basically do this? So for in order to perform this, you need to have to learn the equation so the derivatives of certain functions like implicit functions explicit function derivative of many things so that you come across when you do certain types of equations or when you have certain kinds of activation functions, especially their this particular equation comes into the picture. So now how would you take the derivative? So whenever if you want to differentiate y with respect to X now, you want to check whether where the X comes in this particular equation. So now this works in the format of Of derivative of an outer function multiplied by derivative of inner function So it is a kind of nested functions. So these are essentially called as composite functions. That is function inside function. So how do you take the derivative so for the derivative for E raised to X now this particular one particular equation that is you can consider it as e raised to t. So when you differentiate you first write it as e raised to sine x square. So for E raise to X or E raise to D. You have the same form. And then you need to take the derivative of that is d by dX of sine X Square. So till you reach the derivative of x squared e to differentiate it this so that is the each individual particular unit. You are trying to differentiate so that is you have e raised to the sine of x square into if you take the derivative of sine of x squared that becomes cosine of x square now, you should not stop that is you have to take the derivative with respect to x square also. So now this is pure in terms of X. So finally your derivative would be for this you have 2X. So now if you just observe you have each and everything as individual units, which are just multiplying. So these are essentially you’re seeing of different equations that you are multiplying. So essentially these are used for Composite functions so Whenever say you have a function like you are giving Y is equal to x square. So now say I’m representing this as Y. So we have taken x square as y now. We are taking Z is equal to sine y. So now this became sine of Y. So we are taking this sine of y as z and say we have W is equal to e raise to z. So now if we are asked to find the derivative, of course, we are considering here the partial derivative of with respect to say we want to take the derivative of w with respect to Z. Now in order to do this this W should have something that is expressed in terms of x square. So in order to differentiate this, there is some term there should be something in blue that is representative of X then only you can apply the chain rule else. You cannot go back and connect to that particular thing. So essentially when you take the derivative so first it will be dou W by dou z, so that is with respect to Z. Then you have dou z by dou Y and then you finally have dou y by dou X. So if you see just this particular equation this consists of this dou Z and dou Y so basically you can just assume like this cancels out, but in mathematically we cannot cancel this but just for Simplicity, you can consider this. So how you represent. Are you can find the chain of different equations that are coming across has given with the help of this chain rule now for the general format say this is represented as F of G of x. So if you want to take the derivative that is we have y Prime is equal to f’ G of X into the derivative of G of X. So that becomes a composite functions. Now this not only goes with this exponential functions. Like you have some functions. So 3x plus 1 raise to 7. So are you using the derivative? Also if you have some logarithmic functions like 5X Or if you have some square root functions like this, but we are familiar with these kinds of equations. Right? So these are nothing but your loss functions that you encounter. So if you have this y & y predicted, you may have some kind of equations where you have y log y cap plus 1 minus y into log 1 minus y cap so you can differentiate this and similarly. This is nothing but your x square plus y Square which is nothing but your L2 norm and so all these are basically convex functions. So this is the beauty of the chain rule comes into for differentiating in Neural Networks. So you can essentially take the derivative of this and you can backtrack and to update each and every particular individual unit. So well that was all regarding the chain rule in deep learning for backpropagation.

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# ONE STEP EQUATIONS MULTIPLICATION AND DIVISION WORKSHEET Problem 1 : When we multiply a number by 4, we get 124. Find the number. Problem 2 : When we divide a number by 7, we get 14. Find the number. Problem 3 : Alex borrowed some money from Jose. After 3 years, Alex returned 2 times of borrowed money to Jose. If the returned money is \$226, how much money did Alex borrow from Jose ? Problem 4 : David has some money. He gave one fourth of the money to Lily. If Lily gets \$8 from David, how much money did David have initially ? Problem 5 : In a deposit, invested money will become 4 times itself in 5 years. If Rosy receives \$3280 after five years, how much money did Rosy invest ? Problem 6 : Jacob has some number of candies and Michael has 35 candies. If Michael has candies 5 times as Jacob, how many candies does Jacob have ? Problem 7 : Daniel had some hot dogs and he gave one third of the hot dogs to Alex. If Alex gets 8 hot dogs from Daniel, how many hot dogs did have initially ? Problem 8 : Between the hours of 10 P.M. and 6 A.M., the temperature decreases an average of of a degree per hour. How long, in hours and minutes, will it take for the temperature to decrease by 5 °F ? Problem 1 : When we multiply a number by 4, we get 124. Find the number. Solution : Let x be the number. Then, 4x = 124 Divide each side by 4. x = 31 So, the number is 31. Problem 2 : When we divide a number by 7, we get 14. Find the number. Solution : Let m be the number. Then, m/7 = 14 Multiply each side by 7. m = 98 So, the number is 98. Problem 3 : Alex borrowed some money from Jose. After 3 years, Alex returned 2 times of borrowed money to Jose. If the returned money is \$226, how much money did Alex borrow from Jose ? Solution : Let x be the borrowed money. Then, 2x = 226 Divide each side by 2. x = 113 So, the borrowed money is \$113. Problem 4 : David has some money. He gave one fourth of the money to Lily. If Lily gets \$8 from David, how much money did David have initially ? Solution : Let m be the money that David had initially. Then, m/4 = 32 Multiply each side by 4. m = 128 Problem 5 : In a deposit, invested money will become 4 times itself in 5 years. If Rosy receives \$3280 after five years, how much money did Rosy invest ? Solution : Let x be the money that Rosy invested. Then, 4x = 3280 Divide each side by 4. x = 820 So, Rosy invested \$820. Problem 6 : Jacob has some number of candies and Michael has 35 candies. If Michael has candies 5 times as Jacob, how many candies does Jacob have ? Solution : Let p be the number of candies that Jacob has. Then, 5p = 35 Divide each side by 5. p = 7 So, Jacob has 5 candies. Problem 7 : Daniel had some hot dogs and he gave one third of the hot dogs to Alex. If Alex gets 8 hot dogs from Daniel, how many hot dogs did have initially ? Solution : Let h be the number of hot dogs that Daniel had initially. Then, m/3 = 8 Multiply each side by 3. m = 24 So, Daniel had 24 hot dogs initially. Problem 8 : Between the hours of 10 P.M. and 6 A.M., the temperature decreases an average of of a degree per hour. How long, in hours and minutes, will it take for the temperature to decrease by 5 °F ? Solution : Let x represent the number of hours it takes for the temperature to decrease by 5 °F. Write an equation (-3/4)x = -5 3x/4 = -5 Solve the equation using an inverse operation. Multiply each side by - 4/3. x = 20/3 x = 6 2/3 It takes 6 2/3 hours. Convert 2/3 hours to minutes. 2/3 hours = (2/3) ⋅ 60 minutes 2/3 hours = 40 minutes So, it takes 6 hours and 40 minutes for the temperature to decrease by 5 °F. Apart from the stuff given aboveif you need any other stuff in math, please use our google custom search here. You can also visit the following web pages on different stuff in math. WORD PROBLEMS Word problems on simple equations Word problems on linear equations Algebra word problems Word problems on trains Area and perimeter word problems Word problems on direct variation and inverse variation Word problems on unit price Word problems on unit rate Word problems on comparing rates Converting customary units word problems Converting metric units word problems Word problems on simple interest Word problems on compound interest Word problems on types of angles Complementary and supplementary angles word problems Double facts word problems Trigonometry word problems Percentage word problems Profit and loss word problems Markup and markdown word problems Decimal word problems Word problems on fractions Word problems on mixed fractrions One step equation word problems Linear inequalities word problems Ratio and proportion word problems Time and work word problems Word problems on sets and venn diagrams Word problems on ages Pythagorean theorem word problems Percent of a number word problems Word problems on constant speed Word problems on average speed Word problems on sum of the angles of a triangle is 180 degree OTHER TOPICS Profit and loss shortcuts Percentage shortcuts Times table shortcuts Time, speed and distance shortcuts Ratio and proportion shortcuts Domain and range of rational functions Domain and range of rational functions with holes Graphing rational functions Graphing rational functions with holes Converting repeating decimals in to fractions Decimal representation of rational numbers Finding square root using long division L.C.M method to solve time and work problems Translating the word problems in to algebraic expressions Remainder when 2 power 256 is divided by 17 Remainder when 17 power 23 is divided by 16 Sum of all three digit numbers divisible by 6 Sum of all three digit numbers divisible by 7 Sum of all three digit numbers divisible by 8 Sum of all three digit numbers formed using 1, 3, 4 Sum of all three four digit numbers formed with non zero digits Sum of all three four digit numbers formed using 0, 1, 2, 3 Sum of all three four digit numbers formed using 1, 2, 5, 6

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https://socratic.org/questions/how-do-you-graph-x-2-2-y-1-2-32-1

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text/html

crawl-data/CC-MAIN-2020-50/segments/1606141195929.39/warc/CC-MAIN-20201128214643-20201129004643-00536.warc.gz

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5,969

# How do you graph (x+2)^2 + (y+1)^2 =32? Feb 5, 2016 Circle with the center $\left(- 2 , - 1\right)$ and the radius $4 \sqrt{2}$. #### Explanation: What you have here is a circle equation. The standard form of a circle equation is ${\left(x - {x}_{m}\right)}^{2} + {\left(y - {y}_{m}\right)}^{2} = {r}^{2}$ which describes a circle with the center point $\left({x}_{m} , {y}_{m}\right)$ and the radius $r$. In your case, ${x}_{m} = - 2$, ${y}_{m} = - 1$ and $r = \sqrt{32} = \sqrt{16 \cdot 2} = 4 \sqrt{2} \approx 5.66$ Thus, you can graph a circle with the center point $\left(- 2 , - 1\right)$ and the radius $\approx 5.66$: graph{(x+2)^2 + (y+1)^2 = 32 [-15.54, 16.49, -8.36, 7.66]}

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A point source of light S is placed on the major optical axis of concave mirror at a distance of 60 cm. At what distance from the concave mirror should a flat mirror be placed for the rays to converge again at the point S having been reflected from the concave mirror and then from the flat one? Will the position of the point where the rays meet change if they are first reflected from the flat mirror? The radius of the concave mirror is 80 cm. : Kaysons Education # A Point Source Of Light S Is Placed On The Major Optical Axis Of Concave Mirror At A Distance Of 60 Cm. At What Distance From The Concave Mirror Should A Flat Mirror Be Placed For The Rays To Converge Again At The Point S Having Been Reflected From The Concave Mirror And Then From The Flat One? Will The Position Of The Point Where The Rays Meet Change If They Are First Reflected From The Flat Mirror? The Radius Of The Concave Mirror Is 80 Cm. #### Video lectures Access over 500+ hours of video lectures 24*7, covering complete syllabus for JEE preparation. #### Online Support Practice over 30000+ questions starting from basic level to JEE advance level. #### National Mock Tests Give tests to analyze your progress and evaluate where you stand in terms of your JEE preparation. #### Organized Learning Proper planning to complete syllabus is the key to get a decent rank in JEE. #### Test Series/Daily assignments Give tests to analyze your progress and evaluate where you stand in terms of your JEE preparation. ## Question ### Solution Correct option is 90 cm For concave mirror v = –120 cm Now, x = 120 – (x + 60) x = 30 cm So the distance is 60 + 30 = 90 cm. #### SIMILAR QUESTIONS Q1 A point object is placed at a distance of 30 cm from a convex mirror of focal length 30 cm. the image will form at Q2 Figure shows two rays A and B being reflected by a mirror and going as A` and B`. The mirror Q3 Figure shows three transparent media of refractive indices µ1, µ2 and µ3. A point object O is placed in the medium µ2. If the entire medium on the right of the spherical surface has refractive index µ1, the image forms at O`. If this entire medium has refractive index µ3, the image forms at O``. In the situation shown, Q4 An object of length 2.5 cm is placed at a distance of 1.5f from a concave mirror where f is the magnitude of the focal length of the mirror. The length of the object is perpendicular to the principal axis. Find the length of the image. Is the image erect or inverted? Q5 A convex mirror has its radius of curvature 20 cm. Find the position of the image of an object placed at a distance of 12 cm from the mirror. Q6 which mirror should a boy use, if he stands straight in front of a mirror at a distance of 30cm away from it and sees his erect image whose height is 1/6 th of his real height, is Q7 A hair dresser stands with her nose 20 cm in front of a plane mirror for what distance must she focus her eyes in order to see her nose in the mirror? Q8 An object of height 5 cm is placed in midway between a concave mirror of radius of curvature 30 cm and a convex mirror of radius of curvature 30 cm. The mirrors are placed opposite to each other and are 60 cm apart. The position of the image formed by reflection at convex mirror is. Q9 A convex lens of focal length 15 cm is placed in front of a convex mirror. Both are coaxial and the lens is 5 cm from the apex of the mirror. When an object is placed on the axis at a distance of 20 cm from the lens, if is found that image coincides with the object is 40 cm. Consider only two steps. Calculate the radius of curvature of mirror Q10 A small object is placed on the principal axis of a concave spherical mirror of radius 20 cm at a distance of 30 cm. By how much will the position and size of the image alter, when a parallel – sided slab of glass of thickness 6 cm and refractive index 1.5 is introduced between the centre of curvature and the object? The parallel sides are perpendicular to the principal axis

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# How Much Is My Bond Worth? By A bond's current market value depends on its own interest rate, or coupon rate, along with its face value and the current market interest rate. There is a mathematical formula to calculate how much your bond is worth, but simply put, rising interest rates cause bond values to drop while falling interest rates cause bond values to rise. ## The basics of bond pricing When a bond is first issued, it is sold at a certain par value (also known as face value), which is typically \$1,000 but other amounts are possible. It also has a stated interest rate known as the coupon rate, which is the amount of interest the bond pays as a percentage of its par value. For example, a bond with a par value of \$1,000 and a coupon rate of 5% would pay \$50 in interest per year. Interest rate fluctuations can have a big impact on your bond's value. Image source: Getty Images. ## How interest rates affect bond values The market value of a bond depends on two factors -- the current market interest rate, and the bond's par value. In mathematical terms, the price of a bond is the sum of the present values of all of its future interest payments, plus its par value at maturity. If interest rates rise, the present value of future payments is less. I'll spare you the complex formula used to calculate this, but here's an overview of how this works in practice: For example, let's say that you buy a 30-year Treasury bond for \$1,000 with a coupon rate of 4%, so it pays \$40 per year. If the 30-year Treasury rate jumps to 5%, investors will now expect this yield from their Treasury bonds, or \$50 per \$1,000 invested. Since yours is only paying \$40, the market value must fall in order to make your bond's yield more attractive to new investors. At a 5% interest rate, a bond that pays \$40 per year would be worth \$800. However, keep in mind that the par value of the bond is \$1,000, so an investor would get back considerably more than they pay upon maturity. This adds to the bond's yield to maturity and needs to be taken into consideration. In reality, a 30-year Treasury bond with a 4% coupon rate in a 5% rate environment would be worth about \$845, with the extra \$45 to compensate for the future profit expected from the bond's higher par value. ## So, how much is your bond worth? With all of that in mind, here's a quick calculator that can help you determine your bond's value or predict what it would be worth if interest rates were to change. A couple of notes: • "Desired yield to maturity" refers to the theoretical interest rate you'd like to evaluate. See my example below the calculator. • If your bond is callable, it may affect the bond's value, based on the call date and market interest rate. In this case, the bond's yield to call needs to be considered as well. * Calculator is for estimation purposes only, and is not financial planning or advice. As with any tool, it is only as accurate as the assumptions it makes and the data it has, and should not be relied on as a substitute for a financial advisor or a tax professional. For example, let's say that I own a bond with a \$1,000 par value and a 5% coupon rate, with 12 years to maturity. We'll say that the current market interest rate for bonds with this maturity length is 4%. And I want to know what will happen to the value of my bond if the market interest rate spikes to 6%. Well, based on today's market rate, my bond is worth \$1,095, or \$95 more than its par value. However, if the market rate were to rise to 6%, my bond's market value would drop to just \$915. The \$16,122 Social Security bonus most retirees completely overlook If you're like most Americans, you're a few years (or more) behind on your retirement savings. But a handful of little-known "Social Security secrets" could help ensure a boost in your retirement income. For example: one easy trick could pay you as much as \$16,122 more... each year! Once you learn how to maximize your Social Security benefits, we think you could retire confidently with the peace of mind we're all after.Simply click here to discover how to learn more about these strategies. The Motley Fool has a disclosure policy.

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## Motivation for write-up This is the 4th part of a multi-part series blog post on modeling in epidemiology. The COVID-19 pandemic has brought a lot of attention to study of epidemiology and more specifically to the various mathematical models that are used to inform public health policies. Everyone has been trying to understand the growth or slowing of new cases and trying to predict the necessary sanitary resources. This blog post attempts to explain the foundations for some of the most used models and enlighten the reader on two key points. After introducing the concepts of compartmentalization and disease dynamics in the first blog post, the second part looked at a deterministic numerical solution for the SEIR model discussed, and the effects of the parameters $\beta$, $\sigma$, and $\gamma$ in parts 1 and 2. Part 3 made the argument that most models ignore individual-level disease dynamics in favor of averaging population-level $\sigma$ and $\gamma$ parameters and showed some big discrepancies between actual COVID-19 probability distributions for those parameters and those used in research. This 4th part is where I build a numerical SEIR model that takes into account these probability distributions in order to tweak the model as close to COVID-19 data as possible. ## Building a stochastic model As opposed to the deterministic model from Part 2, this model is going to focus on individual level disease dynamics to model the disease propagation. The basic idea of this model is to have a dataframe with the number of rows equal to the population size (each individual is a row) and two columns: • State column to describe the state of each individual (S, E, I, or R) • Day column to save the day of transition of the individual into that state However, the population-level rates of transmission still apply here i.e. a person goes from S → E following three points: 1. the number of contacts the person has per unit time (given by $r$) 2. the chance a given contact is with an I - infectious individual (the higher thenumber of I, the higher the chance) 3. the chance of an S contracting the disease from a contact with an I (given by $\rho$) This is done stochastically. Once a person becomes E, their progression is unique to them. This progression is calculated in advance for computational reason, but it allows to use the time ditributions we want. #collapse_hide !pip install plotly==4.14.3 import pandas as pd import numpy as np import math import plotly.graph_objects as go import plotly.express as px from scipy.stats import expon from scipy.stats import gamma from scipy.stats import weibull_min from numpy.random import default_rng rng = default_rng() # Let's build a numerical solution def seir_model(init, parms, days): S_0, E_0, I_0, R_0 = init Epd, Ipd, Rpd = [0], [0], [0] S, E, I, R = [S_0], [E_0], [I_0], [R_0] dt=0.1 t = np.linspace(0,days,int(days/dt)) sigma, beta, gam = parms for _ in t[1:]: next_S = S[-1] - beta*S[-1]*I[-1]*dt Epd.append(beta*S[-1]*I[-1]*dt) next_E = E[-1] + (beta*S[-1]*I[-1] - sigma*E[-1])*dt Ipd.append(sigma*E[-1]*dt) next_I = I[-1] + (sigma*E[-1] - gam*I[-1])*dt Rpd.append(gam*I[-1]*dt) next_R = R[-1] + (gam*I[-1])*dt S.append(next_S) E.append(next_E) I.append(next_I) R.append(next_R) return np.stack([S, E, I, R, Epd, Ipd, Rpd]).T Collecting plotly==4.14.3 |████████████████████████████████| 13.2MB 303kB/s Requirement already satisfied: retrying>=1.3.3 in /usr/local/lib/python3.7/dist-packages (from plotly==4.14.3) (1.3.3) Requirement already satisfied: six in /usr/local/lib/python3.7/dist-packages (from plotly==4.14.3) (1.15.0) Installing collected packages: plotly Found existing installation: plotly 4.4.1 Uninstalling plotly-4.4.1: Successfully uninstalled plotly-4.4.1 Successfully installed plotly-4.14.3 ### Creating the initial population dataframe Below is a function to create the initial population dataframe: • $p$ is the population number • $num_E$ is the number of people exposed on day 0 • $num_I$ is the number of infectious on day 0 • $num_R$ is the number of people recovered on day 0 #collapse_hide # Need this new function for model below: def make_df(p, num_I, num_R): df = pd.DataFrame(np.full((p,1), 'S').T[0], columns=['State']) df['Year'] = 0 df['Age'] = (np.random.random(p)*35+15).astype(int) tochange=df.loc[rng.choice(p, size=num_I+num_R, replace=False),'State'].index df.loc[tochange[0:num_I],'State'] = 'I' df.loc[tochange[num_I:num_I+num_R],'State'] = 'R' return df ### Building the model #np.random.random(size=(p,days)) #np.log(4) j=11 over = 10 #10/np.cumsum(np.ones(100)) 0.05 + (0.3/np.exp((j+1-over)/10)) 0.29561922592339457 #collapse_hide def seir_model_stoch(beta, beta2, p, num_I, num_R, years, T_Infectious, ART, control): # Initialize population dataframe with data given by user df = make_df(p, num_I, num_R) # This variable is used to track daily value of beta if it varies over time xxbeta=np.array([],dtype=float) # Initialize the arrays to return # Below are numbers of S, I, R total S=np.array([],dtype=int) I=np.array([],dtype=int) R=np.array([],dtype=int) # Below are the daily additions in S, I, R Spd=np.array([],dtype=int) Ipd=np.array([],dtype=int) Rpd=np.array([],dtype=int) b=beta #b2=beta[0] b2=np.array([],dtype=float) b1=b # signal diminshing beta over = 0 # signal end of deaths due to ART art1 = 0 art2 = 0 # Stochastic model so use random values to decide on progression rand = np.random.random(size=(p,years)) # Depending if you want exponential, gamma, or Weibull distribution for T_Infectious # Uses distributions found on blog part 3 if T_Infectious == 'expon': ItoR = expon.rvs(loc=0,scale=10,size=p) elif T_Infectious == 'gamma': ItoR = gamma.rvs(4,loc=3,scale=2,size=p) else: ItoR = weibull_min.rvs(2.3, loc=2, scale=20.11, size=p) # Iterate over every day the simulation is run for j in range(0,years-1): # Record daily beta values xxbeta=np.append(xxbeta, b[j]) # First we get the index of the individuals that will change state today: # Random number tells you which 'S' have been exposed on this day #StoE_index = df.loc[(df.State == 'S') & (rand[:,j] < b[j]*len(np.where(df.State=='I')[0])/p)].index if ART < 2: StoI_index = df.loc[(df.State == 'S') & (df.Age < 49) & (rand[:,j] < b[j]*len(np.where(df.State=='I')[0])/(len(np.where(df.State=='I')[0])+len(np.where(df.State=='S')[0])))].index StoS_index = df.loc[(df.State == 'S') & (df.Age < 49) & (rand[:,j] < b[j]*len(np.where(df.State=='I')[0])/(len(np.where(df.State=='I')[0])+len(np.where(df.State=='S')[0])))].index elif ART == 2: if art2 == 0: StoI_index = df.loc[(df.State == 'S') & (df.Age < 49) & (rand[:,j] < b[j]*len(np.where(df.State=='I')[0])/(len(np.where(df.State=='I')[0])+len(np.where(df.State=='S')[0])))].index StoS_index = df.loc[(df.State == 'S') & (df.Age < 49) & (rand[:,j] < b[j]*len(np.where(df.State=='I')[0])/(len(np.where(df.State=='I')[0])+len(np.where(df.State=='S')[0])))].index elif art2 == 1: StoI_index = df.loc[(df.State == 'S') & (df.Age > 55)].index StoS_index = df.loc[(df.State == 'S') & (df.Age < 49)].index StoRem_index = df.loc[(df.State == 'S') & (df.Age == 49)].index # For each row, if a person has been a certain number of days in E, they will go to I # This follows EtoI variable which is either exponential or gamma distributed according to above #EtoI_index = df.loc[(df.State == 'E') & (j-df.Day >= EtoI)].index # Similaraly as above # For each row, if a person has been a certain number of days in I, they will go to R # This follows EtoI variable which is either exponential or gamma distributed according to above ItoRem_index = df.loc[(df.State == 'I') & (df.Age == 49)].index if ART == 0: #don't use ART ItoR_index = df.loc[(df.State == 'I') & (j-df.Year >= ItoR) & (df.Age < 49)].index ItoI_index = df.loc[(df.State == 'I') & (j-df.Year < ItoR) & (df.Age < 49)].index elif ART > 0: if art2 == 0: ItoR_index = df.loc[(df.State == 'I') & (j-df.Year >= ItoR) & (df.Age < 49)].index ItoI_index = df.loc[(df.State == 'I') & (j-df.Year < ItoR) & (df.Age < 49)].index elif art2 ==1: ItoR_index = df.loc[(df.State == 'I') & (df.Age > 49)].index ItoI_index = df.loc[(df.State == 'I') & (df.Age < 49)].index RtoRem_index = df.loc[(df.State == 'R') & (df.Age == 49)].index RtoR_index = df.loc[(df.State == 'R') & (df.Age < 49)].index # Use indexes collected above to populate per day values #Epd = np.append(Epd,len(StoE_index)) #Ipd = np.append(Ipd,len(EtoI_index)) Ipd = np.append(Ipd,len(StoI_index)) Rpd = np.append(Rpd,len(ItoR_index)) # Now we use the indexes collected above randomly to change the actual population dataframe to the new states df.iloc[ItoRem_index] = ['S', j, 15] df.loc[ItoR_index, ['State','Year']] = ['S', j] df.loc[ItoR_index, 'Age'] = df.loc[ItoR_index, 'Age'] + 1 df.loc[ItoI_index, 'Age'] = df.loc[ItoI_index, 'Age'] + 1 df.iloc[StoRem_index] = ['S', j, 15] df.loc[StoI_index, ['State','Year']] = ['I', j] df.loc[StoI_index, 'Age'] = df.loc[StoI_index, 'Age'] + 1 df.loc[StoS_index, 'Age'] = df.loc[StoS_index, 'Age'] + 1 df.iloc[RtoRem_index] = ['S', j, 15] df.loc[RtoR_index, 'Age'] = df.loc[RtoR_index, 'Age'] + 1 # Append the S, I, and R arrays S=np.append(S,len(np.where(df.State=='S')[0])) I=np.append(I,len(np.where(df.State=='I')[0])) R=np.append(R,len(np.where(df.State=='R')[0])) # Code below for control measures to reduce beta values if control == 1: if (I[-1]/p > 0.015): art1 = 1 if over == 0: over = j if art1 == 1: if j > over + 15: #if Ipd[-2] > Ipd[-1]: art2 = 1 if over != 0: #b = beta2+(b1/np.exp((j+3-over)/15)) b = beta2+(b1/np.exp((j+1-over)/10)) if control == 2: if (I[-1]/p > 0.3): art1 = 1 if over == 0: over = j #print(over) if art1 == 1: if j > over + 15: #if Ipd[-2] > Ipd[-1]: art2 = 1 if over != 0: #b = beta2+(b1/np.exp((j+3-over)/15)) b = beta2+(b1/np.exp((j+1-over)/10)) xxbeta2 = ((S[j-1]+I[j-1])/I[j-1])*Ipd[j]/S[j-1] #xxbeta2 = 0.5 #print(xxbeta2) b2 = np.append(b2, xxbeta2) #Epd[0]+=num_E Ipd[0]+=num_I Rpd[0]+=num_R #return S,E,I,R, Epd, Ipd, Rpd, xxbeta return S, I, R, Spd, Ipd, Rpd, xxbeta, b2, over ## Testing the model #collapse_hide # Define parameters for stochastic model days = 200 p = 10000 num_E = 0 num_I = 1 num_R = 0 beta_stoch = 0.3*np.ones(days) beta_stoch2 = 0.05 # Run 3 stochastic simulations results_stoch1 = seir_model_stoch(beta_stoch,beta_stoch2, p, num_I, num_R, years, 'gamma', 0, 1) results_stoch2 = seir_model_stoch(beta_stoch, beta_stoch2, p, num_I, num_R, years, 'gamma', 0, 1) results_stoch3 = seir_model_stoch(beta_stoch, beta_stoch2, p, num_I, num_R, years, 'gamma', 0, 2) results_stoch4 = seir_model_stoch(beta_stoch, beta_stoch2, p, num_I, num_R, years, 'gamma', 0, 2) results_stoch1[8] 26 #collapse_hide fig = go.Figure(data=[ go.Scatter(name='Beta_stoch1', x=np.arange(len(results_stoch1[0])), y=results_stoch1[6], line={'dash':'dot','color':'yellow'}, legendgroup="stoch1"), go.Scatter(name='Beta_meas1', x=np.arange(len(results_stoch1[0])), y=results_stoch1[7], line={'dash':'dot','color':'yellow'}, legendgroup="stoch1"), go.Scatter(name='I_stoch1', x=np.arange(len(results_stoch1[0])), y=results_stoch1[1]/p, line={'dash':'dot', 'color':'red'}, legendgroup="stoch1"), go.Bar(name='Ip_stoch1', x=np.arange(len(results_stoch1[0])), y=results_stoch1[4]*10/p, legendgroup="stoch1"), go.Scatter(name='R_stoch1', x=np.arange(len(results_stoch1[0])), y=results_stoch1[2]/p, line={'dash':'dot', 'color':'green'}, legendgroup="stoch1"), go.Scatter(name='Beta_stoch2', x=np.arange(len(results_stoch2[0])), y=results_stoch2[6], line={'dash':'dot','color':'yellow'}, legendgroup="stoch2"), go.Scatter(name='Beta_meas2', x=np.arange(len(results_stoch2[0])), y=results_stoch2[7], line={'dash':'dot','color':'yellow'}, legendgroup="stoch2"), go.Scatter(name='I_stoch2', x=np.arange(len(results_stoch2[0])), y=results_stoch2[1]/p, line={'dash':'dot', 'color':'red'}, legendgroup="stoch2"), go.Bar(name='Ip_stoch2', x=np.arange(len(results_stoch2[0])), y=results_stoch2[4]*10/p, legendgroup="stoch2"), go.Scatter(name='R_stoch2', x=np.arange(len(results_stoch2[0])), y=results_stoch2[2]/p, line={'dash':'dot', 'color':'green'}, legendgroup="stoch2"), go.Scatter(name='Beta_stoch3', x=np.arange(len(results_stoch3[0])), y=results_stoch3[6], line={'dash':'dot', 'color':'yellow'}, legendgroup="stoch3"), go.Scatter(name='Beta_meas3', x=np.arange(len(results_stoch3[0])), y=results_stoch3[7], line={'dash':'dot','color':'yellow'}, legendgroup="stoch3"), go.Scatter(name='I_stoch3', x=np.arange(len(results_stoch3[0])), y=results_stoch3[1]/p, line={'dash':'dot', 'color':'red'}, legendgroup="stoch3"), go.Bar(name='Ip_stoch3', x=np.arange(len(results_stoch3[0])), y=results_stoch3[4]*10/p, legendgroup="stoch3"), go.Scatter(name='R_stoch3', x=np.arange(len(results_stoch3[0])), y=results_stoch3[2]/p, line={'dash':'dot', 'color':'green'}, legendgroup="stoch3"), go.Scatter(name='Beta_stoch4', x=np.arange(len(results_stoch4[0])), y=results_stoch4[6], line={'dash':'dot', 'color':'yellow'}, legendgroup="stoch4"), go.Scatter(name='Beta_meas4', x=np.arange(len(results_stoch4[0])), y=results_stoch4[7], line={'dash':'dot','color':'yellow'}, legendgroup="stoch4"), go.Scatter(name='I_stoch4', x=np.arange(len(results_stoch4[0])), y=results_stoch4[1]/p, line={'dash':'dot', 'color':'red'}, legendgroup="stoch4"), go.Bar(name='Ip_stoch4', x=np.arange(len(results_stoch4[0])), y=results_stoch4[4]*10/p, legendgroup="stoch4"), go.Scatter(name='R_stoch4', x=np.arange(len(results_stoch4[0])), y=results_stoch4[2]/p, line={'dash':'dot', 'color':'green'}, legendgroup="stoch4") ]) fig.update_layout( xaxis_title = 'Day', yaxis_title = 'Proportion of population', title={ 'text':r'$\text{Effect of stochasticity on Deterministic SEIR model}$', 'x':0.5, 'xanchor':'center' } ) fig.show()

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Question # A rectangular swimming pool has a length of 14 feet, a width of 26 feet and a depth of 5 feet. Round answers to the nearest hundredth as needed. $a$ How many cubic feet of water can the pool hold? cubic feet $b$ The manufacturer suggests filling the pool to 95% capacity. How many cubic feet of water is this? cubic feet 220 likes 1102 views ## Answer to a math question A rectangular swimming pool has a length of 14 feet, a width of 26 feet and a depth of 5 feet. Round answers to the nearest hundredth as needed. $a$ How many cubic feet of water can the pool hold? cubic feet $b$ The manufacturer suggests filling the pool to 95% capacity. How many cubic feet of water is this? cubic feet Eliseo 4.6 $a$ To find the volume of the rectangular swimming pool, we need to multiply its length, width, and depth. The volume of a rectangular prism can be found using the formula: \text{{Volume}} = \text{{length}} \times \text{{width}} \times \text{{depth}} Given: Length $L$ = 14 feet Width $W$ = 26 feet Depth $D$ = 5 feet Substituting the values into the formula: \text{{Volume}} = 14 \times 26 \times 5 Calculating this: \text{{Volume}} = 1820 \text{{ cubic feet}} Therefore, the pool can hold 1820 cubic feet of water. $b$ To find the amount of water to fill the pool to 95% capacity, we need to multiply the volume of the pool by 95%. The amount of water is given by: \text{{Amount of water}} = \text{{Volume of pool}} \times \left$\frac{{95}}{{100}} \right$ Substituting the value of the volume of the pool: \text{{Amount of water}} = 1820 \times \left$\frac{{95}}{{100}} \right$ Calculating this: \text{{Amount of water}} = 1729 \text{{ cubic feet}} Therefore, filling the pool to 95% capacity would require 1729 cubic feet of water. Frequently asked questions $FAQs$ Question: What is the value of f$x$ = 3x² - 4x + 2 when x = 5?

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# Section 14 1 Practice Problems Shared by: Categories Tags - Stats views: 65 posted: 6/16/2012 language: pages: 3 Document Sample ``` Section 14.1 Practice Problems 1. A teacher asked her 8 introductory statistics students to record the total amount of time they spent studying for a particular test. The amounts of study time x (in hours) and the resulting test grades y are given below. x 2 1 1.5 0.5 1 3 0 2 y 92 81 84 68 85 96 48 74 a. Make a scatterplot of the data. b. Use your TI-83 to obtain the equation of the least-squares regression line and the correlation. c. Explain in words what the slope  of the true regression line says about d. What is the estimate of  from the data? What is your estimate of the intercept  of the true regression line? e. Use your calculator to calculate the residuals. Report the sum of the residuals and the sum of the squares of the residuals. Then use these results to estimate the standard deviation  in the regression model.  SE b = s f. The standard error of the slope SEb is defined as  (x – x) 2 Calculate SEb. g. Suppose we want to find out if the number of hours studied helps predict grade awarded on this statistics test. Formulate null and alternative hypotheses about the slope of the true regression line. State a two-sided alternative. h. Determine the test statistic, the degrees of freedom, and the P-value of t against the alternative. 2. Ideal proportions Once upon a time, a class like yours made measurements of their arm span and height. They entered their results into a Minitab worksheet, requested least squares regression of height on arm span (both in inches) and obtained the following output: Predictor Coef Stdev t-ratio p Constant 11.547 5.600 2.06 0.056 Arm span 0.84042 0.08091 10.39 0.000 s = 1.613 R-sq = 87.1% R-sq(adj) = 86.3% A residual plot for the data looks like this: o o o o o o o o 0.0 o o o o o o RESI1 o o o o -3.0 64.0 68.0 72.0 76.0 armspan a. Determine the equation of the least squares regression line from the printout. b. In your opinion, is the least squares line an appropriate model for the data? Would you be willing to predict a student’s height, knowing that his arm span is 76 inches? Explain. Then do it – use this model to predict the height of a student whose arm span is 76 inches. c. Estimate the parameters , and . d. Construct a 95% confidence interval for the true slope of the regression line. e. Would you reject the null hypothesis at the 1% significance level? Explain briefly. f. Write your conclusions in plain language. g. Compute a 95% confidence interval for the slope ß of the true regression line. ``` Related docs Other docs by P1OBWG

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# What is an Absolute Value? An error occurred trying to load this video. Try refreshing the page, or contact customer support. Coming up next: How to Evaluate Absolute Value Expressions ### You're on a roll. Keep up the good work! Replay Your next lesson will play in 10 seconds • 0:10 Distance from Zero • 2:16 Absolute Value Notation • 2:55 Solving Equations Want to watch this again later? Timeline Autoplay Autoplay #### Recommended Lessons and Courses for You Lesson Transcript Instructor: Jeff Calareso Jeff teaches high school English, math and other subjects. He has a master's degree in writing and literature. When we're talking and comparing numbers, we often don't care whether its positive or negative, just how big it is. This is often called the magnitude of a number and we find it by taking the absolute value. Learn all about it here! ## The Distance from Zero One February, during Super Bowl XLV, which was the Packers vs. the Steelers, I made a bet with a friend over how much money it costs to get a 30-second commercial on during one of the commercial breaks. Now I'm a huge Packers fan and he was a Steelers fan, so we decided that the loser would have to wear the opposing team's jersey for the rest of the game. I thought it would cost around \$1,250,000 and my friend was convinced that it cost \$5,500,000. I wasn't sure if I was exactly right or not, but I was pretty confident that he was way off, so I decided to make the bet. Well, a few internet searches later, we discovered that a 30-second commercial that year cost \$3,000,000. So who won? Well, a simple calculation of \$3,000,000 minus \$1,250,000 gave us the answer that I was \$1,750,000 off the actual answer. Doing the same for him, \$3,000,000 minus \$5,500,000 told us that he was \$2,500,000 off. This means that I was definitely closer, so I won and my Packer pride stayed intact. But a closer inspection into this reveals a situation that actually comes up a lot in mathematics. What my friend and I cared about was purely how far away our guess was from the actual answer. We didn't care about whether we were over or under, or whether the number we did after the little subtraction problem was positive or negative. All that we cared about was the magnitude of the answer, or how large the number was regardless of the sign in front of it. In mathematics, this operation of taking the number and only looking at the size of it and ignoring the sign, is called the absolute value. We often use absolute values when talking about distance because there is no such thing as negative distance. For example, just because the store you're going to is a block behind you doesn't mean it's negative one blocks away. It's just one block, right? ## Notation of Absolute Values Just like how the winner of our bet was the one whose guess was less far away from the answer, the absolute value of any number can be thought of as how far away that number is from zero. For example, the absolute value 7 (written as |7|)? Still 7. It's 7 units away from zero. The absolute value of -7 (written as |-7|)? 7, because it's still 7 units away from zero. To unlock this lesson you must be a Study.com Member. ### Register for a free trial Are you a student or a teacher? #### See for yourself why 30 million people use Study.com ##### Become a Study.com member and start learning now. Back What teachers are saying about Study.com ### Earning College Credit Did you know… We have over 160 college courses that prepare you to earn credit by exam that is accepted by over 1,500 colleges and universities. You can test out of the first two years of college and save thousands off your degree. Anyone can earn credit-by-exam regardless of age or education level.

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# Thread: Inverse of 3X3 Matrix 1. ## Inverse of 3X3 Matrix Any hints? Got everything down with the "cofactor method" for finding the inverse of a 3X3 Matrix, except this part: 2. ## Re: Inverse of 3X3 Matrix First of all, you have calculated the matrix of minors wrong, it should be: $\displaystyle \begin{bmatrix}2&2&2\\-2&3&3\\0&-10&0 \end{bmatrix}$. Now to create the adjugate (or adjoint) matrix, we multiply each ij-entry by $\displaystyle (-1)^{i+j}$, that is, alternate signs, starting with postive in the upper-left corner, to get: $\displaystyle \begin{bmatrix}2&-2&2\\2&3&-3\\0&10&0 \end{bmatrix}$. We will want the transpose of this matrix, which is: $\displaystyle \begin{bmatrix}2&2&0\\-2&3&10\\2&-3&0 \end{bmatrix}$. We need to now calculate the determinant of the original matrix, to get a suitable scalar factor: The easiest way is to expand by minors along the second column, since it has only one non-zero entry. Since this is in the 3,2 position, we take the negative of this determinant (since 3+2 = 5 is odd), the cofactor of the other 2 subdeterminants will be 0, so we don't have to calculate the matrix of the other minors (although we've already done it, so meh...). that is, the determinant of the original matrix is: $\displaystyle \det(A) = (-1)\ast 1 \ast \begin{vmatrix}3&2\\2&-2 \end{vmatrix} = (-1)(-10) = 10$. So our scaling factor will be 1/10. This gives the inverse: $\displaystyle A^{-1} = \frac{1}{\det(A)}\text{adj}(A) = \begin{bmatrix} \frac{1}{5}&\frac{1}{5}&0\\ -\frac{1}{5}&\frac{3}{10}&1\\ \frac{1}{5}&-\frac{3}{10}&0 \end{bmatrix}$ 3. ## Re: Inverse of 3X3 Matrix Ok, corrected now. 4. ## Re: Inverse of 3X3 Matrix Ok here are the two charts. I have one last question on the 2nd one. As far as understanding the question in the first chart, I understand it. Wherever the two lines meet (in a cross-out pattern), that place is where the determinant goes for the new "matrix of determinant answers", so to speak. 5. ## Re: Inverse of 3X3 Matrix Go entry-by-entry. Draw one horizontal and one vertical line that intersect at the entry you are on. You should wind up with 9 "charts" as you call them: one cross-out for each entry in the matrix.

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Home > blog > How Do You Find A Unit Vector That Is Orthogonal To Both U = 1, 0, V = 0, 1, ? # How Do You Find A Unit Vector That Is Orthogonal To Both U = 1, 0, V = 0, 1, ? We can formalize this result into a theorem regarding orthogonal vectors. Find the net force Express the answer using standard unit vectors. Find a vector of magnitude that points in the opposite direction than vector where and Express the answer in component form. Assume the quarterback and the receiver are in the same place as in the previous example. Therefore, two vectors are parallel if they have the same or opposite directions. We will see the first application of this in the next chapter. Let’s start off by supposing that we wanted the rate of change of $$f$$ at a particular point, say $$\left( , \right)$$. Let’s also suppose that both $$x$$ and $$y$$ are increasing and that, in this case, $$x$$ is increasing twice as fast as $$y$$ is increasing. So, as $$y$$ increases one unit of measure $$x$$ will increase two units of measure. To this point we’ve only looked at the two partial derivatives $$\left( \right)$$ and $$\left( \right)$$. Its magnitude is obtained by multiplying their magnitudes by the sine of the angle between them. The direction of the vector product can be determined by the corkscrew right-hand rule. The vector product of two either parallel or antiparallel vectors vanishes. The magnitude of the vector product is largest for orthogonal vectors. The distributive property is applied frequently when vectors are expressed in their component forms, in terms of unit vectors of Cartesian axes. The corkscrew right-hand rule is a common mnemonic used to determine the direction of the vector product. Determine the requested vectors and express each of them a. Vector uu is a unit vector that forms an angle of 60°60° with the positive x-axis. Determine all three-dimensional vectors u orthogonal to vector Express the answer by using standard unit vectors. The boat’s motor generates a force in one direction, and the current of the river generates a force in another direction. We must take both the magnitude and direction of each force into account if we want to know where the boat will go. Each arrow has the same length and direction. A closely related concept is the idea of parallel vectors. When finding the cross product, in practice, we can use either or , depending on which one of them seems to be less complex computationally. One way to make sure if the final result is correct is to use them both. Is a well-known indicator that they are perpendicular. Where $$\theta$$ is the angle between the gradient and $$\vec u$$. You appear to be on a device with a “narrow” screen width (i.e. you are probably on a mobile phone). Due to the nature of the mathematics on this site it is best views in landscape mode. For the following exercises, find the center and radius of the sphere with an equation in general form that is given. Write the equation of the plane passing through point that is parallel to the xz-plane. Write the equation of the plane passing through point that is parallel to the xy-plane. what is the maximum frame size for ethernet ii frames on a vlan? To add vectors in three dimensions, we follow the same procedures we learned for two dimensions. Think about what happens if you plot this equation in two dimensions in the xz-plane. Describe the set of points in three dimensional space that satisfies and graph the surface. /r/cheatatmathhomework is FREE math homework help sub. Asking for or offering payment will result in a permanent ban. The triangle looks to be equilateral and equiangular. Α is the internal angle of the equilateral triangle.

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Simple double integration of square wave question Finding Fourier coefficients for square wave Finding Fourier coefficients for square wave • #1 864 17 Hi, Simple question, sort of: I see that according to the internet the mathematical description of a triangular wave is rather complex, so I’ll try to stay as far away from that as I can, because I’m a bit rusty. I understand that if you integrate a square wave you get a triangular wave on the x-axis. But If you integrate that triangular wave you get something resembling a sineusoid. (something about it being the first harmonic of the triangular wave I believe) My questions are: How close is that to a sine wave? What does it look like graphically? (I don’t have matlab) Secondly, are there different slopes of triangular waves, including asymmetrical triangular waves, which are closer to a sine wave? I am so rusty on Fourier transforms / series, it’s not funny. But regarding this: http://mathworld.wolfram.com/FourierSeriesTriangleWave.html I see the asymmetrical figure, which is kind of what I have in mind, but I’m not sure what the red sine wave is indicating. Cheers Simple question, sort of: I see that according to the internet the mathematical description of a triangular wave is rather complex, so I’ll try to stay as far away from that as I can, because I’m a bit rusty. I understand that if you integrate a square wave you get a triangular wave on the x-axis. But If you integrate that triangular wave you get something resembling a sineusoid. (something about it being the first harmonic of the triangular wave I believe) My questions are: How close is that to a sine wave? What does it look like graphically? (I don’t have matlab) Secondly, are there different slopes of triangular waves, including asymmetrical triangular waves, which are closer to a sine wave? I am so rusty on Fourier transforms / series, it’s not funny. But regarding this: http://mathworld.wolfram.com/FourierSeriesTriangleWave.html I see the asymmetrical figure, which is kind of what I have in mind, but I’m not sure what the red sine wave is indicating. Cheers You are watching: Simple double integration of square wave question. Info created by THVinhTuy selection and synthesis along with other related topics. Rate this post

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# A number has two digits whose sum is 8. If 18 is added to the number, its digits are reversed. What is that number.Help me please. 2 2014-12-03T20:36:29+05:30 Let the no be 10x + y given, x + y = 8 ------------ (1) 10x +y + 18 = 10y + x 9x - 9y + 18 =0 x - y +2 = 0 x - y = -2 ----- (2) from (1) & (2) x + y = 8 x - y = -2 ----------------- 2x = 6 x = 3 sub x = 3 in (1) 3 + y = 8 y = 5 number 10x + y = 10(3) + 5 = 30+5 = 35 required number is 35. 2014-12-03T20:37:03+05:30 The number is 62. 6+2=8 +18= 26 hence the number is reversed

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https://math.stackexchange.com/questions/545474/a-weird-infinity-problem?noredirect=1

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A weird infinity problem. [duplicate] A weird infinity problem. I saw this on youtube but could not understand it: Let us add 1 + 2 + 4 + 8 + 16 + ... up to infinity x=(1+2+4+8+...) = 1(1+2+4+8+...) = (2-1)(1+2+4+8+...) = (2+4+8+16+...)-(1+2+4+8+...) = -1 x=-1 Now that is weird since we are adding up to infinity towards the positive side yet we end up with a negative number. Somebody please explain me why. • As long as only sums are involved, everything is fine. But to substract $+\infty$ from $+\infty$ (which you do as soon as a minus sign appears) can lead to chaos. This has been explained several times on the site. – Did Oct 30 '13 at 12:09 • can u provide me a link? – Shaurya Gupta Oct 30 '13 at 12:18 • – Gerry Myerson Oct 30 '13 at 12:34

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# What is 2/3 cup in ounces? There are 5 1/3 ounces in 2/3 of a cup. One cup is the equivalent of 8 ounces, and 2/3 of 8 is 5 1/3. Because 1 tablespoon equals 1/2 ounce, 5 1/3 tablespoons equal 1/3 of a cup. This also means that 10 2/3 tablespoons equal 2/3 of a cup. Since there are 3 teaspoons in 1 tablespoon, 2/3 of a cup contains 32 teaspoons and because 2 cups equal 1 pint, 2/3 of a cup is the equivalent of 1/3 of a pint. The measurement of 2/3 cup is also equal to 1/6 of a quart and 1/24 of a gallon. Reference:

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https://www.objectivistliving.com/topic/14133-is-using-someones-reason-against-them-fraud/page/2/

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# Is Using Someone's Reason Against Them Fraud? ## Recommended Posts Why isn't it reasonable to suppose that they could have predicted it? My 9-year-old son could have predicted it, and he's only just now learning long division. If a third grader's basic math skills could work that out, I don't see why I couldn't expect the same of Bob and Alice. I think you don't give poor Bob and Alice nearly enough credit for being thinking folks. I explained why in my response to selene. But I'll explain why in more detail here: At the beginning of the auction, Alice bids \$1, because she thinks that there is a 50% chance that Bob will drop out and she will get \$19. There is also a 50% chance that Bob will bid \$2 and win the auction, in which case she loses \$1. Her expected gains for bidding on the first round is 0.5*19 - 0.5*1 = \$9, while her expected gains for not bidding at all is \$0. Thus, she should bid \$1 on the first round. Similarly for Bob. Since Alice has already bid \$1, he needs to bid \$2 to win. If he does, his expected gains are 0.5*18 - 0.5*2 = \$8, which is more than if he doesn't, i.e. \$0. This goes on and on until the bid is at \$10. At this point, Alice reasons, well if Bob bets more than \$10, then his expected gains would be negative, so he won't bid more than \$10, and Alice can certainly win the auction by betting \$11 and getting \$9. Bob, of course, is thinking the same thing and so he tries to bet more than \$10. Once the bid reaches \$20, the same reasoning applies as in the first paragraph, except in this case each player is trying to minimize his losses. But by that point, Carl has already won. That doesn't explain anything except that you think Bob and Alice are lacking in basic math skills. If you can do the math (and if a third-grader can do the math), why would I not expect Bob and Alice to be able to do the math? • Replies 208 • Created #### Popular Days Any endeavor which depends entirely on 'knowing' the content of another person's consciousness is irrational, so the premise of your game is irrational as are the actions of the 'players'. ##### Share on other sites Thing is, you are buying into the notion that people should be treated as beings that require protection both from things that they cannot predict and from things that they can predict but don't. Objectivism treats people as 1) beings that are entirely capable of protecting themselves, and 2) beings that can survive (learn, recover, thrive) even when they have failed to protect themselves. For the sake of argument, go ahead and tell us what specifically it is that you think Carl has done wrong and what you would suggest to do about it. Again, you're misunderstanding the situation. The auction was not something that simply happened to Alice and Bob, like a natural disaster. It was all Carl's doing. Unlike a natural disaster or some kind of accident, there is someone who can be held responsible for their actions. I can't tell you exactly what Carl has done wrong since, as I said, it has not yet been identified as a kind of wrong. I think Carl should be tried and punished if found guilty, just like with any other crime. That doesn't explain anything except that you think Bob and Alice are lacking in basic math skills. If you can do the math (and if a third-grader can do the math), why would I not expect Bob and Alice to be able to do the math? As I've shown, Bob and Alice already did do the math. They did it perfectly. And yet, they were still screwed. ##### Share on other sites Any endeavor which depends entirely on 'knowing' the content of another person's consciousness is irrational, so the premise of your game is irrational as are the actions of the 'players'. This is absolutely ridiculous. If it is irrational to try to predict what someone else will do, then any attempt to deal with a reality with more than one person in it would be irrational. ##### Share on other sites Alice and Bob can be held responsible for their actions. We can agree to disagree. ##### Share on other sites Alice and Bob can be held responsible for their actions. We can agree to disagree. If you believe that someone can act against their own interests by being rational, then you need to check your premises. Reason cannot fail, unless someone tries to use force or fraud or this third thing, apparently. ##### Share on other sites Any endeavor which depends entirely on 'knowing' the content of another person's consciousness is irrational, so the premise of your game is irrational as are the actions of the 'players'. This is absolutely ridiculous. If it is irrational to try to predict what someone else will do, then any attempt to deal with a reality with more than one person in it would be irrational. Nope, your OP described the participants as acting "in their rational self-interest all the way". Do you understand rational self-interest? It is not rational when the outcome turns completely on another's unpredictable acts. It's irrational and unself-interested to guess and gamble on anything significant, it's instead mystical whimsy. Notwithstanding those mathematical calculations, which prove - what? So the game can be dismissed out of hand as ridiculous. ##### Share on other sites Nope, your OP described the participants as acting "in their rational self-interest all the way". Do you understand rational self-interest? It is not rational when the outcome turns completely on another's unpredictable acts. It's irrational to guess and gamble on anything significant, it's instead mystical whimsy. Notwithstanding those mathematical calculations, which prove - what? So the game can be dismissed out of hand as ridiculous. In what sense is it "not rational"? When you do something rationally, that means that you've done the best you can given your goals and information. Sometimes, you have no information about something, but reality still demands that you make a decision. ##### Share on other sites Here's my issue with that line of reasoning. If it is not fraud, then Alice and Bob were being entirely rational. Non sequitur. You seem to be laboring under the mistaken opinion that any decision absent fraud must, by definition, be "rational." Not so. "Rational" doesn't mean "in the absence of fraud." J ##### Share on other sites Nope, your OP described the participants as acting "in their rational self-interest all the way". Do you understand rational self-interest? It is not rational when the outcome turns completely on another's unpredictable acts. It's irrational to guess and gamble on anything significant, it's instead mystical whimsy. Notwithstanding those mathematical calculations, which prove - what? So the game can be dismissed out of hand as ridiculous. In what sense is it "not rational"? When you do something rationally, that means that you've done the best you can given your goals and information. Sometimes, you have no information about something, but reality still demands that you make a decision. SoMad: Traders trade value for value where both sides feel they're getting the greater value. It is irrational to suppose someone would not pursue their self interest and trade a greater value for a lesser one (i.e. \$20 for \$1). The persons who thought they could accomplish this deserved the lesson. Everyone pursues, rightfully, their self interest. ##### Share on other sites As some of you have said, I don't think that Carl did anything fraudulent, in the technical sense. However, I disagree that Alice and Bob's choice to participate was rational because they were entertained. Are you claiming that any and all forms of entertainment that people choose must be rational?!!! First of all, they were not entertained. I'd say they were the exact opposite of entertained. We may have been entertained, Carl may have been very very entertained, but Alice and Bob certainly weren't. Alice and Bob appeared to be entertaining themselves with the act of competing with each other. If not, then your hypothetical seems to be something constructed out of imaginary psychology and behavior that has no relevance to reality. Secondly, they were not expecting to purchase entertainment in any case. They were expecting to buy money. Yeah, but then time continued on, and they discovered an opportunity for entertainment, and took it. Their "not expecting it" earlier has no bearing. If they knew that they were going to be "entertained" they may have considered the price too high, which would mean that Carl's plan would fail. Think about it this way. Who would go to a store where every item was some unknown price that you couldn't know until you agreed to buy the item? No one, obviously. Yeah, so why invent fictional people who would do what real people wouldn't? As for gambling, gambling is irrational. If you are a rational person, then you would know you can't beat the odds in a casino. Wrong. If you're a rational person, you know that you can beat the odds, but that doing so is unlikely. J ##### Share on other sites Here's my issue with that line of reasoning. If it is not fraud, then Alice and Bob were being entirely rational. Non sequitur. You seem to be laboring under the mistaken opinion that any decision absent fraud must, by definition, be "rational." Not so. "Rational" doesn't mean "in the absence of fraud." J Sorry, in that post I meant to say that Bob and Alice were capable of acting in their rational self-interest. If this was fraud, then you're right they could still be rational even if they could not act in their rational self-interest. ##### Share on other sites SoMad: Traders trade value for value where both sides feel they're getting the greater value. It is irrational to suppose someone would not pursue their self interest and trade a greater value for a lesser one (i.e. \$20 for \$1). The persons who thought they could accomplish this deserved the lesson. Everyone pursues, rightfully, their self interest. False. Immoral people certainly don't. ##### Share on other sites Not false. Explain yourself. ##### Share on other sites As some of you have said, I don't think that Carl did anything fraudulent, in the technical sense. However, I disagree that Alice and Bob's choice to participate was rational because they were entertained. Are you claiming that any and all forms of entertainment that people choose must be rational?!!! The context of that quote is that some people were arguing that Alice and Bob were acting rationally because they may have been entertained by the outcome. But I was pointing out that just because one is entertained by the outcome, that does not mean that their behavior was rational. Alice and Bob appeared to be entertaining themselves with the act of competing with each other. If not, then your hypothetical seems to be something constructed out of imaginary psychology and behavior that has no relevance to reality. Non-sequitur. It's entirely possible that someone might do something even if they were not entertained by the act. Yeah, but then time continued on, and they discovered an opportunity for entertainment, and took it. Their "not expecting it" earlier has no bearing. It does matter because if you don't get the product you expected to buy and have paid for, then you have been defrauded. Yeah, so why invent fictional people who would do what real people wouldn't? That case only considers what would happen if they knew the outcome. The whole point is that they didn't. Wrong. If you're a rational person, you know that you can beat the odds, but that doing so is unlikely. You know what I mean. ##### Share on other sites Not false. Explain yourself. Morality is acting in your own rational-self interest. Immoral people, by definition, don't do this. ##### Share on other sites I think it is time to send Doll Head to the Doll Hospital and do some serious interventions: Do some testing... write some meds.... and keep her under observation... ##### Share on other sites And what did Bob and Alice do with Ted and Carol? ...inquiring kinksters need to know... ##### Share on other sites Not false. Explain yourself. Morality is acting in your own rational-self interest. Immoral people, by definition, don't do this. I did not say rational self interest. Value is subjective (Mises). People pursue what they perceive is their self interest in trying to trade what to them is a lessor value for a greater one. In this case it would be hard to argue that a lesser amount of money is subjectively of more value than a greater one (unless Carl was nuts or a member of some strange cult). The trader principle is trading value for value, it is understood by both sides that a trade will take place when both parties perceive gain. That is a rational exchange. Therefore Alice and Bob were irrational to suppose they could gain while Carl lost. By your definition they were acting immorally and deserved what they got. ##### Share on other sites I did not say rational self interest. Value is subjective (Mises). People pursue what they perceive is their self interest in trying to trade what to them is a lessor value for a greater one. In this case it would be hard to argue that a lesser amount of money is subjectively of more value than a greater one (unless Carl was nuts or a member of some strange cult). The trader principle is trading value for value, it is understood by both sides that a trade will take place when both parties perceive gain. That is a rational exchange. Therefore Alice and Bob were irrational to suppose they could gain while Carl lost. By your definition they were acting immorally and deserved what they got. Is this objectively true? Because if value is subjective, then Alice and Bob cannot say that Carl is acting irrationally just because his values don't conform to their own. ##### Share on other sites I think that the kind of thing that Carl pulled is called a "gambit": A device, action, or opening remark, typically one entailing a degree of risk, that is calculated to gain an advantage. ##### Share on other sites I did not say rational self interest. Value is subjective (Mises). People pursue what they perceive is their self interest in trying to trade what to them is a lessor value for a greater one. In this case it would be hard to argue that a lesser amount of money is subjectively of more value than a greater one (unless Carl was nuts or a member of some strange cult). The trader principle is trading value for value, it is understood by both sides that a trade will take place when both parties perceive gain. That is a rational exchange. Therefore Alice and Bob were irrational to suppose they could gain while Carl lost. By your definition they were acting immorally and deserved what they got. Is this objectively true? Because if value is subjective, then Alice and Bob cannot say that Carl is acting irrationally just because his values don't conform to their own. SoMad: I believe so. If you haven't read Mises he is such a pleasure to read. The exemplar reasoning human being. When I first read Mises many years ago I filled a notebook with words and their definitions. I reread many times before not needing (mostly) my notebook. I recommend him for years of enjoyment and enlightenment if you'd like to put off your deathwish for awhile. I started this habit reading Ayn Rand when I read the word "epistemology" for the first time and I was so delighted when I looked it up and found out what it meant. I found a home. Carl is running a con but it is so obvious and the price is cheap the lesson does Alice and Bob a service. ##### Share on other sites I started this habit reading Ayn Rand when I read the word "epistemology" for the first time and I was so delighted when I looked it up and found out what it meant. I found a home. Very nicely stated. James Joyce was wrong on that because Ayn showed me that same home and I can live in it each and every day that I conscously choose too. Thanks A... ##### Share on other sites Carl is running a con but it is so obvious and the price is cheap the lesson does Alice and Bob a service. Carl is running a con, but the question is, is he doing anything wrong? Do there exist non-coercive and non-fraudulent unethical actions? ##### Share on other sites Carl is running a con but it is so obvious and the price is cheap the lesson does Alice and Bob a service. Carl is running a con, but the question is, is he doing anything wrong? Do there exist non-coercive and non-fraudulent unethical actions? Carl is perpetrating a fraud but an obvious one with not serious consequences. Are you familiar with the term hormesis? A low level toxin can have a positive effect on an organism. Carl's con game does not rise to the level of being prosecuted for a crime. I wouldn't put anything past him though, and wouldn't invite him to dinner.

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https://www.gamedev.net/forums/topic/474905-how-many-vertices-can-result-from-clipping-a-a-triangle-to-a-view-frustum/

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# How many vertices can result from clipping a a triangle to a view frustum. This topic is 3997 days old which is more than the 365 day threshold we allow for new replies. Please post a new topic. ## Recommended Posts If I have a triangle and a frustum how many vertices can result from clipping that triangle to the frustum. I can only come up with 7 points. Is this really the maximum? I think that a triangle can only pass through more than 4 planes of a frustum if it passes exactly through the corners in which case it only creates one additional vertex anyways. ##### Share on other sites A triangle (or plane) can pass through all 6 planes of the frustum...but I only see 6 vertices being created there ##### Share on other sites In the worst case, 6*4 , assuming the frustum has 6 planes and the triangle and its resulting part intersect the other 6-1 planes in turn. This is a rare case, but it could happen, More general cases depend on triangle positions , generally the formula is P= ( original triangle points - clipped points ) * intersected frustum planes ##### Share on other sites The triangle could potentially intersect all of the six frustum planes. You would have to test one plane after the other. For each plane two new vertices could be generated if an intersection occurs, so in the end it would be possible that you would have to generate 6*2=12 new vertices in total of which not all have to actually be used by the resulting, clipped polygon. ##### Share on other sites Seven is the maximum for clipping to the X and Y planes, but you can get one more each from clipping to the near and far planes as well. Think of it this way - each plane can turn a single vertex of your (already partially clipped) polygon into an edge - so it takes away one vertex and adds two. So you start with a triangle = 3 vertices, and each of the 6 planes +/-X, +/-Y, +/-Z can add a single vertex each. Total = 9. 1. 1 Rutin 38 2. 2 3. 3 4. 4 5. 5 • 11 • 9 • 12 • 14 • 9 • ### Forum Statistics • Total Topics 633350 • Total Posts 3011470 • ### Who's Online (See full list) There are no registered users currently online ×

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Sections: # Rounding Numbers Test #3 About: Additional Resources: In this section, we learn about rounding numbers. Specifically, we will focus on how to round with whole numbers. Often, an exact value is not needed. Let’s suppose we are looking at cars. We might find that a model X car costs $29,755.78, whereas a model G car costs$40,226.38. To make matters simple and easy to remember, we may just say the model X car costs about $30,000 and the model G car costs about$40,000. These values of $30,000 and$40,000 represent approximations. An approximation is an estimate or value that is close, but not exactly the same as the original. Rounding a whole number is a way to give an approximation. To round a whole number, first we must identify how precise we want our approximation to be. This stems from giving a round off place. Once this is known, we begin our procedure: 1. Identify the digit in the round off place 2. Move one digit right: • If this digit is 4 or less, we keep the digit in the round off place as is and change every digit that follows into a zero • If this digit is 5 or greater, we add one to the digit in the round off place, and change every digit that follows into a zero. Example 1: Round 326 to the nearest ten 326 - The 2 is the digit in the round off place 326 - The 6 is one digit to the right of the 2, this digit is in the category of 5 or greater. For this scenario we add one to the digit in the round off place (2). So 2 + 1 = 3. The digit in the tens place will be changed into a 3 and the digit that follows is changed into a 0. 326 Rounded to the Nearest Ten is 330 + Show More +

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https://www.calculatoratoz.com/en/power-on-exponential-of-temperature-time-relation-calculator/Calc-30978

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crawl-data/CC-MAIN-2022-05/segments/1642320301863.7/warc/CC-MAIN-20220120130236-20220120160236-00063.warc.gz

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Credits Shri Govindram Seksaria Institute of Technology and Science (SGSITS), Indore Ravi Khiyani has created this Calculator and 100+ more calculators! National Institute Of Technology (NIT), Hamirpur Anshika Arya has verified this Calculator and 1600+ more calculators! Power on exponential of temperature-time relation Solution STEP 0: Pre-Calculation Summary Formula Used constantt = -(Convection heat transfer coefficient*Surface Area*Time elapsed)/(Density*Volume*Specific Heat Capacity) a = -(h*SA*t)/(ρ*V*c) This formula uses 6 Variables Variables Used Convection heat transfer coefficient - Convection heat transfer coefficient is the rate of heat transfer between a solid surface and a fluid per unit surface area per unit kellvin. (Measured in Watt per Meter² per K) Surface Area - The Surface Area of a three-dimensional shape is the sum of all of the surface areas of each of the sides. (Measured in Square Meter) Time elapsed - Time elapsed after a particular task is started. (Measured in Second) Density - Density is the degree of compactness of a substance. (Measured in Kilogram per Meter³) Volume - Volume is the amount of space that a substance or object occupies or that is enclosed within a container. (Measured in Cubic Meter) Specific Heat Capacity - Specific Heat Capacity is the heat required to raise the temperature of the unit mass of a given substance by a given amount. (Measured in Kilojoule per Kilogram per K) STEP 1: Convert Input(s) to Base Unit Convection heat transfer coefficient: 1 Watt per Meter² per K --> 1 Watt per Meter² per K No Conversion Required Surface Area: 50 Square Meter --> 50 Square Meter No Conversion Required Time elapsed: 1 Second --> 1 Second No Conversion Required Density: 5.51 Kilogram per Meter³ --> 5.51 Kilogram per Meter³ No Conversion Required Volume: 63 Cubic Meter --> 63 Cubic Meter No Conversion Required Specific Heat Capacity: 4.184 Kilojoule per Kilogram per K --> 4184 Joule per Kilogram per K (Check conversion here) STEP 2: Evaluate Formula Substituting Input Values in Formula a = -(h*SA*t)/(ρ*V*c) --> -(1*50*1)/(5.51*63*4184) Evaluating ... ... a = -3.44259695413343E-05 STEP 3: Convert Result to Output's Unit -3.44259695413343E-05 --> No Conversion Required -3.44259695413343E-05 <-- Constant (Calculation completed in 00.016 seconds) < 10+ Transient Heat Conduction Calculators Instantaneous heat transfer rate heat_rate = Convection heat transfer coefficient*Surface Area*(Initial Temperature-Fluid temperature)*(exp(-(Convection heat transfer coefficient*Surface Area*Time elapsed)/(Density*Volume*Specific Heat Capacity))) Go Temperature after given time elapsed temperature = ((Initial Temperature-Fluid temperature)*(exp(-(Convection heat transfer coefficient*Surface Area*Time elapsed)/(Density*Volume*Specific Heat Capacity))))+Fluid temperature Go Total heat transfer during a time interval heat_transfer_KJ = Density*Specific heat*Volume*(Initial Temperature-Fluid temperature)*(1-(exp(-(Biot number*Fourier Number)))) Go Ratio of temperature difference for given time elapsed temperature_ratio = exp(-(Convection heat transfer coefficient*Surface Area*Time elapsed)/(Density*Volume*Specific Heat Capacity)) Go Power on exponential of temperature-time relation constantt = -(Convection heat transfer coefficient*Surface Area*Time elapsed)/(Density*Volume*Specific Heat Capacity) Go Product of Biot and Fourier Number in terms of system properties constantt = (Convection heat transfer coefficient*Surface Area*Time elapsed)/(Density*Volume*Specific Heat Capacity) Go Time Constant in unsteady state heat transfer tau_mi = (Density*Specific Heat Capacity*Volume)/(Convection heat transfer coefficient*Surface Area) Go Thermal Capacitance thermal_capacitance = Density*Specific Heat Capacity*Volume Go Ratio of temperature difference for given time elapsed in terms of Biot and Fourier Number temperature_ratio = exp(-(Biot number*Fourier Number)) Go Power on exponential of temperature-time relation in terms of Biot and Fourier Number constantt = -(Biot number*Fourier Number) Go Power on exponential of temperature-time relation Formula constantt = -(Convection heat transfer coefficient*Surface Area*Time elapsed)/(Density*Volume*Specific Heat Capacity) a = -(h*SA*t)/(ρ*V*c) What is Temperature-Time relation? The temperature-time relationship of unsteady-state heat transfer helps to determine the rate of heat transfer that has been conducted in the lumped system in a given time period. How to Calculate Power on exponential of temperature-time relation? Power on exponential of temperature-time relation calculator uses constantt = -(Convection heat transfer coefficient*Surface Area*Time elapsed)/(Density*Volume*Specific Heat Capacity) to calculate the Constant, Power on exponential of temperature-time relation formula calculates the value of power on exponential in the equation which makes the further calculation of lumped system easy. Constant is denoted by a symbol. How to calculate Power on exponential of temperature-time relation using this online calculator? To use this online calculator for Power on exponential of temperature-time relation, enter Convection heat transfer coefficient (h), Surface Area (SA), Time elapsed (t), Density (ρ), Volume (V) & Specific Heat Capacity (c) and hit the calculate button. Here is how the Power on exponential of temperature-time relation calculation can be explained with given input values -> -3.443E-5 = -(1*50*1)/(5.51*63*4184). FAQ What is Power on exponential of temperature-time relation? Power on exponential of temperature-time relation formula calculates the value of power on exponential in the equation which makes the further calculation of lumped system easy and is represented as a = -(h*SA*t)/(ρ*V*c) or constantt = -(Convection heat transfer coefficient*Surface Area*Time elapsed)/(Density*Volume*Specific Heat Capacity). Convection heat transfer coefficient is the rate of heat transfer between a solid surface and a fluid per unit surface area per unit kellvin, The Surface Area of a three-dimensional shape is the sum of all of the surface areas of each of the sides, Time elapsed after a particular task is started, Density is the degree of compactness of a substance, Volume is the amount of space that a substance or object occupies or that is enclosed within a container & Specific Heat Capacity is the heat required to raise the temperature of the unit mass of a given substance by a given amount. How to calculate Power on exponential of temperature-time relation? Power on exponential of temperature-time relation formula calculates the value of power on exponential in the equation which makes the further calculation of lumped system easy is calculated using constantt = -(Convection heat transfer coefficient*Surface Area*Time elapsed)/(Density*Volume*Specific Heat Capacity). To calculate Power on exponential of temperature-time relation, you need Convection heat transfer coefficient (h), Surface Area (SA), Time elapsed (t), Density (ρ), Volume (V) & Specific Heat Capacity (c). With our tool, you need to enter the respective value for Convection heat transfer coefficient, Surface Area, Time elapsed, Density, Volume & Specific Heat Capacity and hit the calculate button. You can also select the units (if any) for Input(s) and the Output as well. How many ways are there to calculate Constant? In this formula, Constant uses Convection heat transfer coefficient, Surface Area, Time elapsed, Density, Volume & Specific Heat Capacity. We can use 10 other way(s) to calculate the same, which is/are as follows - • heat_rate = Convection heat transfer coefficient*Surface Area*(Initial Temperature-Fluid temperature)*(exp(-(Convection heat transfer coefficient*Surface Area*Time elapsed)/(Density*Volume*Specific Heat Capacity))) • heat_transfer_KJ = Density*Specific heat*Volume*(Initial Temperature-Fluid temperature)*(1-(exp(-(Biot number*Fourier Number)))) • tau_mi = (Density*Specific Heat Capacity*Volume)/(Convection heat transfer coefficient*Surface Area) • constantt = -(Convection heat transfer coefficient*Surface Area*Time elapsed)/(Density*Volume*Specific Heat Capacity) • constantt = -(Biot number*Fourier Number) • temperature_ratio = exp(-(Convection heat transfer coefficient*Surface Area*Time elapsed)/(Density*Volume*Specific Heat Capacity)) • temperature_ratio = exp(-(Biot number*Fourier Number)) • temperature = ((Initial Temperature-Fluid temperature)*(exp(-(Convection heat transfer coefficient*Surface Area*Time elapsed)/(Density*Volume*Specific Heat Capacity))))+Fluid temperature • constantt = (Convection heat transfer coefficient*Surface Area*Time elapsed)/(Density*Volume*Specific Heat Capacity) • thermal_capacitance = Density*Specific Heat Capacity*Volume Let Others Know

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# How do you rationalize the denominator and simplify sqrt2/sqrt5? Apr 4, 2016 $\frac{\sqrt{10}}{5}$ #### Explanation: Multiply by 1 and you do not change the inherent value. 1 can come in many forms. Foe example: $1 = \frac{\sqrt{5}}{\sqrt{5}}$ Multiply by 1 but in the form of $1 = \frac{\sqrt{5}}{\sqrt{5}}$ giving $\frac{\sqrt{2}}{\sqrt{5}} \times \frac{\sqrt{5}}{\sqrt{5}}$ $= \frac{\sqrt{2} \sqrt{5}}{5}$ $= \frac{\sqrt{2 \times 5}}{5}$ $\frac{\sqrt{10}}{5}$

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# Re: how to encrypt the 10-digit values into encrypted 10-digit values? On 26 Sep., 10:08, makissy <maki...@xxxxxxxxx> wrote: Hi All, I have an assignment about increasing the security of the system by encrypting the identification numbers in database. how to encrypt the 10-digit values into encrypted 10-digit values? i don't have any idea yet. Can anyone give me any idea. pls.. requirements • Each plain text 10-digit value with digits in the interval [0..9] should be encrypted to a 10-digit number with digits in the interval [0..9]. • Each encrypted value of 10 digits should be possible to decrypt back to a plain text 10-digit value. Thanks for any guidance. Hello, I would do it like this (this method is easy on a piece of paper): 1) split the number into 2 rows of 5 digits 2) shift the digits in the first row 1 place the the left, and 2 places to the left in the second row - appending the leftmost digits to the end 3) mod10 add each column's digits 4) mod10 subtract each column's calculated value from that column's digits 5) repeat steps 2-4 (at least once) It is very fast and very easy, and has the benefit over just simply to the number - that it adds diffusion, a very important property in encryption (changing the data in the smallest amount results in changes in the entire cipher). Let me show an example: Number to be encrypted: 4685197643 Secret key (that you have created somehow): 5216940876 1) split number into 2 rows 46851 97643 2) shift places to the left 68514 64397 3) mod10 add each column's digits 68514 64397 -------- 22801 (modulo 10 addition of each column's digits) 4) mod10 subtract each column's calculated value from that column's digits 46713 42596 Let me explain this step. First column: 6 6 -- 2 and becomes: 4 4 because: 6-2=4, 6-2=4 Let's take the 3rd column: 5 3 -- 8 becomes: 7 5 because: 5-8 = -3. We cant have a negative number, so we add 10 - and it then becomes 7. And the last digit: 3-8 = -5. -5 + 10 = 5 So as you can see it is pretty easy. 5) repeat steps 2-4 (at least once) 46713 42596 ....shifted: 67134 59642 6) mod10 add each column's digits 67134 59642 -------- 16776 7) mod10 subtract each column's calculated value from that column's digits 51468 43976 5146843976 (straightened out) 5216940876 (key) ----------------- 0352783742 = cipher (modulo 10 addition of the number and the key) And the encryption is done. To decrypt, just do everything in reverse: 0352783742 (cipher) 5216940876 (key) ----------------- 5146843976 (modulo 10 subtraction) 51468 43976 -------- .... 94334 mod10 subtracted 67134 59642 .... unshift 46713 42596 .... repeat these steps once more 46713 42596 ----- .... 88209 mod10 subtracted 68514 64397 .... unshift 46851 97643 .... straighten 4685197643 And it's decrypted! If you can't decrypt it, you made an error somewhere (errors happen quite easily, so you should always either: doublecheck the calculations or try to decrypt it). .

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# Computational Thinking Computational Thinking by Laura Schonfeld Get Started. It's Free Computational Thinking ## 2. Problem: Laura wants to go see band, but can’t take off work and only have \$375 of funds. ### 2.1. Decomposition: Laura needs to determine all the locations and times that band is playing where she would not need to take off work. Laura needs to figure out the cost of plane tickets, concert tickets, and accommodations for all of those times and dates. She will need to add up all the costs and determine which locations would be within her budget. Then, she would make a decision of where to see band from the options she had left within her budget. 2.1.1. Rationale: This is an example of the computational step, decomposition. Decomposition is breaking down a problem into smaller parts (“Google computational thinking for educators”, n.d.) Because the problem included determining where band was playing on days she could see them without taking off work, calculating multiple different costs and then adding up those costs to determine the total costs, each part of the process was separated to make it easier to solve the whole problem of wanting to go to the concert on a budget. 2.1.2. Pattern Recognition- Laura researches the tour dates to see what days and places band plays. She learns that band only plays on Saturdays (the only day when she could see them without taking off work) this year in Chicago, New York, and Red Rocks. She begins researching flights to these locations and determines that the cost for flying in on Fridays is \$100 for both Red Rocks and New York while the cost for flying to Chicago is \$275. She also finds that flying in on Saturdays is \$200 for Red Rocks and \$175 for both Chicago and New York. Next, she must research the costs of accommodations in each of those places. The cost for accommodations in Red Rocks is \$75 a night. The cost for accommodations for New York is \$150 a night. There is no cost for accommodations in Chicago because she can stay with her friend Lauren. Next, Laura must research the costs of each concert ticket in their respective cities. She discovers that Chicago has the most expensive ticket at \$125 followed by \$75 for Red Rocks and \$60 for New York. 2.1.2.1. Rationale: This is an example of the next step, pattern recognition. In this step, the computational thinker must look at the data and find patterns (“Google computational thinking for educators”, n.d.). Here, Laura has looked at all of the data including tour schedules, airplane tickets, and accommodations to determine the pattern of lowest cost options as well as options that fit within her time constraint. 2.1.2.2. Algorithm Design- Laura must write down and calculate all of the possible combinations of cost by adding up the cost for flying into New York on Friday and Saturday, the accommodations cost for staying both one and two nights depending on what day she flies in, and the cost of the ticket. She must also add up the cost for flying to Chicago on both Friday and Saturday as well as the cost of the ticket, but she does not need to add anything for accommodations since she would stay with her friend. Laura must add up the cost for flying to Red Rocks on both Friday and Saturday, the cost of accommodations for staying both one and two nights, as well as the cost for the concert ticket. She learns that it would be within her budget to see band at Red Rocks if she flys in on either Friday or Saturday or Chicago if she flys in on Saturday. She decides to buy a ticket to see band on Saturday in Chicago, books her Saturday plane ticket, and has a blast! 2.1.2.2.1. Rationale: In this step of computational thinking, algorithm design, the thinker must come up with a step by step process that will solve this problem and others like it (“Google computational thinking for educators”, n.d.). Because Laura needed to determine the total cost of each trip, she considered each different scenario (Friday New York, Friday Red Rocks, Friday Chicago, Saturday New York, Saturday Red Rocks, Saturday Chicago), calculated the costs, and compared the costs to her budget of \$375 to determine the possible options. From her possible options, she makes a decision on which show to attend. This is an example of computational thinking because of the steps that have just been described. These same steps could also be used to figure out attending future concerts and could thus be applied to similar problems. 2.1.2.2.2. Abstraction- Laura decides that other concert going lovers would benefit from being able to streamline this process using technology. She decides she could create an app using her friend Walton’s computer science skills that automatically scours the internet for concert tickets for a specific band, and combines it with the cost of plane tickets and accommodation all in one place! The user simply inputs their budget as well as the artists they are interested in seeing. They can also input their hometowns or places where they would not need accommodations. The app quickly searches possible combinations that would allow the user to see the artist within their budget. She becomes rich off of the success of her app and she never has to worry about not being able to spend money on seeing concerts ever again.

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Date: Thu, 4 Jun 1998 11:18:21 -0700 Dilbert's Salary Law: Engineers Vs. Managers -------------------------------------------- Engineers and scientists will never make as much money as business managers. Now we have a mathematical proof that explains why this is true: Postulate 1: Knowledge is Power. Postulate 2: Time is Money. As every engineer knows, Work ---------- = Power Time Since Knowledge = Power, and Time = Money, we have: Work ---------- = Knowledge Money Solving for Money, we get: Work ---------- = Money Knowledge Thus, as Knowledge approaches zero, Money approaches infinity regardless of the amount of Work done. Conclusion: The Less you Know, the More you Make. Note: It has been speculated that the reason why Bill Gates dropped out of Harvard's math program was because he stumbled upon this proof as an undergraduate, and dedicated the rest of his career to the pursuit of ignorance. [1997.jul.30] After applying some simple algebra to some trite phrases and cliches a new understanding can be reached of the secret to wealth and success. > > Here it goes. > Knowledge is Power > Time is Money and as every engineer knows, Power is Work over Time. > > So, substituting algebraic equations for these time worn bits of > wisdom, we get: > > K = P (1) > T = M (2) > P = W/T (3) > > Now, do a few simple substitutions: > > Put W/T in for P in equation (1), which yields: > K = W/T (4) > > Put M in for T into equation (4), which yields: > > K = W/M (5) > Now we've got something. Expanding back into English, we get: > > Knowledge equals Work over Money. > > What this MEANS is that: > > 1. The More You Know, the More Work You Do, and > 2. The More You Know, the Less Money You Make. > > Solving for Money, we get: > > M = W/K (6) > > Money equals Work Over Knowledge. > > From equation (6) we see that Money approaches infinity as Knowledge approaches 0, regardless of the Work done. > > What THIS MEANS is: > > The More you Make, the Less you Know. > > Solving for Work, we get > > W = M K (7) > > Work equals Money times Knowledge > > From equation (7) we see that Work approaches 0 as Knowledge > approaches 0. > > What THIS MEANS is: > The stupid rich do little or no work. > > Working out the socioeconomic implications of this breakthrough is > left as an exercise for the reader > > >------------------------------------------------------------------+ > Hard work has a future payoff. Laziness pays off now.

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Search a number 15761 is a prime number BaseRepresentation bin11110110010001 3210121202 43312101 51001021 6200545 763644 oct36621 923552 1015761 1110929 129155 137235 145a5b 154a0b hex3d91 15761 has 2 divisors, whose sum is σ = 15762. Its totient is φ = 15760. The previous prime is 15749. The next prime is 15767. The reversal of 15761 is 16751. Subtracting from 15761 its product of digits (210), we obtain a palindrome (15551). Subtracting 15761 from its reverse (16751), we obtain a triangular number (990 = T44). It is a strong prime. It can be written as a sum of positive squares in only one way, i.e., 14161 + 1600 = 119^2 + 40^2 . It is a cyclic number. It is not a de Polignac number, because 15761 - 210 = 14737 is a prime. It is a Chen prime. It is equal to p1838 and since 15761 and 1838 have the same sum of digits, it is a Honaker prime. It is a congruent number. It is not a weakly prime, because it can be changed into another prime (15767) by changing a digit. It is a polite number, since it can be written as a sum of consecutive naturals, namely, 7880 + 7881. It is an arithmetic number, because the mean of its divisors is an integer number (7881). 215761 is an apocalyptic number. It is an amenable number. 15761 is a deficient number, since it is larger than the sum of its proper divisors (1). 15761 is an equidigital number, since it uses as much as digits as its factorization. 15761 is an evil number, because the sum of its binary digits is even. The product of its digits is 210, while the sum is 20. The square root of 15761 is about 125.5428213798. The cubic root of 15761 is about 25.0723239017. The spelling of 15761 in words is "fifteen thousand, seven hundred sixty-one".

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## Introduction to Set Theory ### Subjects to be Learned • set • set membership --- "belong to" ### Contents The concept of set is fundamental to mathematics and computer science. Everything mathematical starts with sets. For example, relationships between two objects are represented as a set of ordered pairs of objects, the concept of ordered pair is defined using sets, natural numbers, which are the basis of other numbers, are also defined using sets, the concept of function, being a special type of relation, is based on sets, and graphs and digraphs consisting of lines and points are described as an ordered pair of sets. Though the concept of set is fundamental to mathematics, it is not defined rigorously here. Instead we rely on everyone's notion of "set" as a collection of objects or a container of objects. In that sense "set" is an undefined concept here. Similarly we say an object "belongs to " or "is a member of" a set without rigorously defining what it means. "An object(element) x belongs to a set A" is symbolically represented by "x A" . It is also assumed that sets have certain (obvious) properties usually asssociated with a collection of objects such as the union of sets exists, for any pair of sets there is a set that contains them etc. This approach to set theory is called "naive set theory " as opposed to more rigorous "axiomatic set theory". It was first developed by the German mathematician Georg Cantor at the end of the 19th century. For more on naive and axiomatic set theories click here which is not required for this course. Though the naive set theory is not rigorous, it is simpler and practically all the results we need can be derived within the naive set theory. Thus we shall be following this naive set theory in this course. Next -- Representation of Set Back to Schedule

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https://algebra-class-ecourse.com/question/please-help-i-give-brainliest-sal-s-sandwich-shop-sells-wraps-and-sandwiches-as-part-of-its-lunc-11535266-59/

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## PLEASE HELP!!! I GIVE BRAINLIEST!!!! Sal’s Sandwich Shop sells wraps and sandwiches as part of its lunch specials The pro Question PLEASE HELP!!! I GIVE BRAINLIEST!!!! Sal’s Sandwich Shop sells wraps and sandwiches as part of its lunch specials The profit on every sandwich is \$2 and The profit on every wrap is \$3. Sal made a profit of \$1,470 from lunch specials last month. The equation 2x + 3y = 1,470 represents Sal’s profits last month, where x is the number of sandwich lunch specials sold and y is the number of wrap lunch specials sold. 1. Change the equation into slope-intercept form. Identify the slope and y-intercept of the equation. Be sure to show all of your work below: 0 ## Answers ( No ) 1. 2x+3y =1470 slope intercept form y=mx+b 2x+3y =1470 subtract 2x from each side 3y = -2x+1470 divide by 3 y = -2/3 x +1470/3 y = -2/3x +490 slope = -2/3 y intercept = 490

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# Euler Problem 60 The primes 3, 7, 109, and 673, are quite remarkable. By taking any two primes and concatenating them in any order the result will always be prime. For example, taking 7 and 109, both 7109 and 1097 are prime. The sum of these four primes, 792, represents the lowest sum for a set of four primes with this property. Find the lowest sum for a set of five primes for which any two primes concatenate to produce another prime. `````` In [2]: from time import time start = time() from sympy import primerange, isprime primes = list(primerange(2, 50000)) N = len(primes) minsum = 1e100 def concat(p, q): return int(f'{p}{q}') def edge(p, q): if (p,q) in E: return E[p,q] cliques = [([], 0)] for p in primes: E = {} if p >= minsum: break new_cliques = [] for clique, weight in cliques: if all(edge(p,q) for q in clique): new_clique = clique + [p] if len(new_clique) == 5: minsum = min(minsum, weight + p) new_cliques.append((new_clique, weight + p)) cliques.extend(new_cliques) print(minsum) print("Time: %.2f seconds" % (time() - start)) `````` `````` 26033 Time: 293.43 seconds `````` `````` In [ ]: ``````

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Paul's Online Notes Home / Calculus I / Review / Trig Equations with Calculators, Part II Show Mobile Notice Show All Notes Hide All Notes Mobile Notice You appear to be on a device with a "narrow" screen width (i.e. you are probably on a mobile phone). Due to the nature of the mathematics on this site it is best views in landscape mode. If your device is not in landscape mode many of the equations will run off the side of your device (should be able to scroll to see them) and some of the menu items will be cut off due to the narrow screen width. ### Section 1.6 : Solving Trig Equations with Calculators, Part II 7. Find all the solutions to $$4{\csc ^2}\left( {1 - t} \right) + 6 = 25\csc \left( {1 - t} \right)$$. Use at least 4 decimal places in your work. Show All Steps Hide All Steps Hint : Factor the equation and using basic algebraic properties get two equations that can be dealt with using known techniques. If you’re not sure how to factor this think about how you would factor $$4{x^2} - 25x + 6 = 0$$. Start Solution This equation may look very different from anything that we’ve ever been asked to factor, however it is something that we can factor. First think about factoring the following, $4{x^2} + 6 = 25x\,\,\,\,\,\,\,\,\,\,\, \to \,\,\,\,\,\,\,\,\,\,4{x^2} - 25x + 6 = \left( {4x - 1} \right)\left( {x - 6} \right) = 0$ If we can factor this algebraic equation then we can factor the given equation in exactly the same manner. \begin{align*}4{\csc ^2}\left( {1 - t} \right) + 6 & = 25\csc \left( {1 - t} \right)\\ 4{\csc ^2}\left( {1 - t} \right) - 25\csc \left( {1 - t} \right) + 6 & = 0\\ \left( {4\csc \left( {1 - t} \right) - 1} \right)\left( {\csc \left( {1 - t} \right) - 6} \right) & = 0\end{align*} Now, we have a product of two factors that equals zero and so by basic algebraic properties we know that we must have, $4\csc \left( {1 - t} \right) - 1 = 0\hspace{0.25in}{\rm{OR}}\hspace{0.25in}\csc \left( {1 - t} \right) - 6 = 0$ Hint : Solve each of these two equations to attain all the solutions to the original equation. Show Step 2 Each of these equations are similar to equations solved in the previous section and earlier in this section. Therefore, we will be assuming that you can recall the solution process for each and we will not be putting in as many details. If you are unsure of the process you should go back to the previous section and work some of the problems there before proceeding with the solution to this problem. We’ll start with the first equation, isolate the cosecant and convert to an equation in terms of sine for easier solving. Doing this gives, $\csc \left( {1 - t} \right) = \frac{1}{4}\hspace{0.25in}\hspace{0.25in} \to \hspace{0.25in}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\sin \left( {1 - t} \right) = 4 > 1$ We now know that there are now solutions to the first equation because we know $$- 1 \le \sin \theta \le 1$$. Now, let’s solve the second equation. $\csc \left( {1 - t} \right) = 6\hspace{0.25in}\hspace{0.25in} \to \hspace{0.25in}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\sin \left( {1 - t} \right) = \frac{1}{6}$ Using our calculator we get, $1 - t = {\sin ^{ - 1}}\left( {\frac{1}{6}} \right) = 0.1674$ A quick glance at a unit circle shows us that the second angle in the range $$\left[ {0,2\pi } \right]$$ is $$\pi - 0.1674 = 2.9742$$. All the solutions to the second equation are then, \begin{align*}1 - t & = 0.1674 + 2\pi n & \hspace{0.25in}{\rm{OR}}\hspace{0.25in}1 - t & = 2.9742 + 2\pi n & \hspace{0.25in} n = 0, \pm 1, \pm 2, \ldots \\t & = 0.8326 - 2\pi n & \hspace{0.25in}{\rm{OR}}\hspace{0.55in}t & = - 1.9742 - 2\pi n & \hspace{0.25in} n = 0, \pm 1, \pm 2, \ldots \end{align*} Because we had not solutions to the first equation all the solutions to the original equation are then, $\require{bbox} \bbox[2pt,border:1px solid black]{{t = 0.8326 - 2\pi n \hspace{0.25in} {\rm{OR}} \hspace{0.25in} t = - 1.9742 - 2\pi n \hspace{0.25in} n = 0, \pm 1, \pm 2, \ldots }}$ Do get too excited about the fact that we only got solutions from one of the two equations we got after factoring. This will happen on occasion and so we need to be ready for these cases when they happen. If an interval had been given we would next proceed with plugging in values of $$n$$ to determine which solutions fall in that interval. Since we were not given an interval this is as far as we can go. Note that depending upon the amount of decimals you used here your answers may vary slightly from these due to round off error. Any differences should be slight and only appear around the 4th decimal place or so however.

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• Dec 17th 2012, 04:28 PM skinsdomination09 integral from 1 to infiinity of ln x / (x^2) equals 1. How?! (IMPROPER INTEGRATION) I integrated the function to become ln x / (x^2) u = lnx du = 1/x dv = 1/x^2 v = -1/x UV - int VDU lnx * -1/x - - int of 1/x^2 becomes -1/x so - lnx/x - lnx (or close to this) • Dec 17th 2012, 04:58 PM Plato Re: integral from 1 to infiinity of ln x / (x^2) equals 1. How?! (IMPROPER INTEGRATIO Quote: Originally Posted by skinsdomination09 I integrated the function to become ln x / (x^2) u = lnx du = 1/x dv = 1/x^2 v = -1/x UV - int VDU lnx * -1/x - - int of 1/x^2 becomes -1/x so - lnx/x - lnx (or close to this) Look at this webpage. • Dec 17th 2012, 05:08 PM skeeter $\displaystyle \int_1^\infty \frac{\ln{x}}{x^2} \, dx$ $\displaystyle u = \ln{x}$ ... $\displaystyle du = \frac{1}{x} \, dx$ $\displaystyle dv = \frac{1}{x^2} dx$ ... $\displaystyle v = -\frac{1}{x}$ $\displaystyle \lim_{b \to \infty} \int_1^b \frac{\ln{x}}{x^2} \, dx = \lim_{b \to \infty} \left[-\frac{\ln{x}}{x}\right]_1^b - \int_1^b -\frac{1}{x^2} \, dx$ $\displaystyle \lim_{b \to \infty} \left[-\frac{\ln{x}}{x}\right]_1^b - \left[\frac{1}{x} \right]_1^b$ $\displaystyle \lim_{b \to \infty} \left[-\frac{\ln{b}}{b}\right] - \left[\frac{1}{b} - 1 \right]_1^b = 1$ • Dec 17th 2012, 05:09 PM skinsdomination09 Re: integral from 1 to infiinity of ln x / (x^2) equals 1. How?! (IMPROPER INTEGRATIO Quote: Originally Posted by Plato delete • Dec 17th 2012, 05:13 PM skinsdomination09 Quote: Originally Posted by skeeter $\displaystyle \int_1^\infty \frac{\ln{x}}{x^2} \, dx$ $\displaystyle u = \ln{x}$ ... $\displaystyle du = \frac{1}{x} \, dx$ $\displaystyle dv = \frac{1}{x^2} dx$ ... $\displaystyle v = -\frac{1}{x}$ $\displaystyle \lim_{b \to \infty} \int_1^b \frac{\ln{x}}{x^2} \, dx = \lim_{b \to \infty} \left[-\frac{\ln{x}}{x}\right]_1^b - \int_1^b -\frac{1}{x^2} \, dx$ $\displaystyle \lim_{b \to \infty} \left[-\frac{\ln{x}}{x}\right]_1^b - \left[\frac{1}{x} \right]_1^b$ $\displaystyle \lim_{b \to \infty} \left[-\frac{\ln{b}}{b}\right] - \left[\frac{1}{b} - 1 \right]_1^b = 1$ Makes more sense, Thanks But wouldn't the ln b / b (since it's approaching infinity) be indetirminate so it would be divergant. My original mistake was that I integrated the second half twice by accident, but I still don't get how you can just "ignore" ln b /b since ln(infinity) doesn't equal 0 or anything. Oh... is it bc the top is growing slower than the bottom so its just a really small number over a big one and thus goes to zero? Simple Alg. + early Calc skills holding me back(Angry) • Dec 17th 2012, 05:17 PM skeeter L'Hopital ... $\displaystyle \displaystyle \lim_{b \to \infty} \frac{\ln{b}}{b} = \lim_{b \to \infty} \frac{1}{b} = 0$ • Dec 17th 2012, 05:19 PM skinsdomination09 $\displaystyle \displaystyle \lim_{b \to \infty} \frac{\ln{b}}{b} = \lim_{b \to \infty} \frac{1}{b} = 0$

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# Excercise 14 problems For each exercise, a link to a possible solution is provided. Each solution includes a discussion of how a programmer might approach the problem and interesting points raised by the problem or its solution, as well as complete source code of the solution. Trigometric Ratios We can use Pythagoras' Theorem see above to calculate the length of an unknown side in a right-angled triangle when we are given information about the lengths of 2 other sides. However, if we are given information about an angle other than the 90 degrees and one side and need to calculate the length of another side then we use the trig ratios. This first tutorial takes you through naming the sides of a triangle and an introduction to the trig ratios. ## Best Types of Exercise for Asthma - Health This tutorial is suitable for students in Year 10 or We can use trig ratios to calulate the length of a side on a triangle if we know the length of one side and the size of one angle. You will need a scientific calculator for this tutorial This tutorial is suitable for students in Year 11 or Calculating angles using the trigometric ratios. OK, so probably saw this coming. If you know the length of the two sides of a right-angle triangle then you can caculate the size of the other angles inside the triangle. We're really getting to know stuff about right-angle triangles.Physical Geography Laboratory Manual for McKnightâ€™s Physical Geography: A Landscape Appreciation, Eleventh Edition offers a comprehensive set of lab exercises to accompany any physical geography class. The manual is organized to meet your needs, providing the flexibility to pick and choose a. Student Resources For more information on how to order these items, Chapter Exercise Exercise Exercise Exercise Problem A. Working Papers Plus for Select Exercises and Problems Chapters ISBN: Shop ProForm online. ProForm is a world leader in home fitness equipment. ## Solutions from 0 to 116, not including 113 which is probably not findable by the rules given. Shop professional-grade treadmills, training cycles, and ellipticals! Example Exercise Dilution of a Solution Concentrated hydrochloric acid is available commercially as a 12 M solution. What is the molarity of an HCl solution. Math []. 1. Write a function to calculate if a number is prime. Return 1 if it is prime and 0 if it is not a prime. 2. Write a function to determine the number of prime numbers below n. C programming Exercises, Practice, Solution: C is a general-purpose, imperative computer programming language, supporting structured programming, lexical variable scope and recursion, while a static type system prevents many unintended operations. 14 Leg Exercises for Men - Elite Men's Guide

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# algebra posted by . The circular path of cars on a ferris wheel can be modeled with the equation x^2-14x+y^2-150y=-49, measured in feet. What is the maximum height above ground of the riders. • algebra - write the circle equation in standard form by completing the square x^2 - 14x + 48 + y^2 - 150y + 5625 = -49 + 49 + 5625 (x-7)^2 + (y - 75)^2 = 75^2 so the centre is at (7, 75) and the radius is 75 after making a sketch you should be able to answer the question ## Similar Questions 1. ### trig A ferris wheel with a diameter of 100 feet rotates at a constant rate of 4 revolutions per minute. Let the center of the ferris wheel be at the origin. 1. Each of the ferris wheel's cars travels around a cirlce. a) Write an equation … 2. ### math a ferris wheel has the diameter of 240 feet and the bottom of the ferris wheel is 9 feet above the ground. find the equation of the wheel if the origin is placed on the ground directly below the center of the wheel 3. ### College Algebra A ferris wheel has a diameter of 320 feet and the bottom of the Ferris wheel is 9 feet above the ground. Find the equation of the wheel if the origin is placed on the ground directly below the center of the wheel. 4. ### Trig As you ride a ferris wheel, your distance from the ground varies sinusoidally with time. When the last passenger boards the ferris wheel and the ride starts moving, let your position be modeled by the diagram provided. Let t be the … 5. ### MATH A Ferris wheel completes one revolution every 90 s. The cars reach a maximum of 55 m above the ground and a minimum of 5 m above the ground. The height, h, in metres, above the ground can be modeled using a sine function, where t represents … 6. ### math A Ferris wheel has a maximum height of 245 feet and a wheel diameter of 230 feet. Find an equation for the wheel if the center of the wheel is on the y-axis and y represents the height above the ground. equation is ? 7. ### math trig A Ferris wheel with a diameter of 37 meters rotates at a rate of 4 minutes per revolution. Riders board the Ferris wheel 4 meters above the ground at the bottom of the wheel. A couple boards the Ferris wheel and the ride starts. Write … 8. ### math The height of a person above the ground on a Ferris wheel is given by the function h(t)=9 sin {pi/20 (t- pi/2))+ 12 where h(t) is the height in meters of person t seconds after getting on a) how long does it take the Ferris wheel to … 9. ### Math-Trigonometry Can someone help me with this problem. -A Ferris wheel has a radius of 37.8 feet. The bottom of the Ferris wheel sits 0.7 feet above the ground. You board the Ferris wheel at the 6 o'clock position and rotate counter-clockwise. A)Define … 10. ### precalculus A Ferris wheel has a radius of 40 feet. The bottom of the Ferris wheel sits 0.5 feet above the ground. You board the Ferris wheel at the 6 o'clock position and rotate counter-clockwise. Define a function,f that gives your height above … More Similar Questions

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# D’Alembert Betting System The D’Alembert is among the easiest betting systems that are applicable to the game of roulette. In fact, this system ranks second in terms of popularity after the Martingale. It bears many of the characteristics of the Martingale, which is not surprising considering the fact that both systems are based on negative betting progressions. Therefore, the D’Alembert dictates that players should increase the size of their stakes after a loss and decrease it after each winning bet. The origins of the D’Alembert system can be traced back to the 18th century, when the French mathematician Jean le Rond D’Alembert began his work on a statement of the fundamental laws of motion. The mathematician came to the conclusion that the external forces acting on a body and the inertial forces are a system which exists in equilibrium. This conclusion laid the foundations of Newton’s Second Law and became known as the D’Alembert Principle. ## How Does the D’Alembert Betting System Work When applied to games of chance like roulette, D’Alembert’s law of equilibrium leads to the idea that future outcomes of wheel spins can balance less likely results in the past. According to D’Alembert, the chances of a coin landing on tails increase if it has already landed on heads several consecutive times. In other words, if the ball lands in red pockets several times in a row, black is due to be spun and becomes a more likely outcome. This concept is also known as the Gambler’s fallacy. However, this serves as an indication that the D’Alembert system is based on the idea that players, who place even-money bets, will generate a profit if they win as often as they lose or more. When the flip of a coin is taken into consideration, 50% of the time, the result that will come up will be tails, and 50% of the time, it will be heads. Although this is what the figures dictate, probabilities are not the same in the gambling realm. What gambling enthusiasts should remember is that when they lay any of the roulette bets that pay even money, the chances of winning are not the same as with the flip of a coin. This is so because of the presence of the zero and the double-zero pockets in American-style roulette variants. The D’Alembert system is quite simple to master even by roulette novices, which comes to explain why it is so hugely popular among casino enthusiasts. If you are keen on the idea to give the betting system a trial run, the first thing you need to do is set a base betting unit depending on the overall size of your bankroll. It is important to mention that your betting unit should not exceed 5% of your bankroll for the session. A betting unit of 2% is considered the safest option, especially for gambling aficionados who are reluctant to take risks or do not have that bountiful bankroll. As it seems, some experienced roulette players advise going for a base bet unit that does not overstep the 1% threshold. Others go even further and recommend putting on the line an amount that is 0.50% of the funds you intend on using during your betting session. Since the base bet unit you will settle on will be the backbone of this roulette betting method, it is of crucial importance to pick it wisely due to the fact that it willdetermine the net profit you are expected to accumulate during your betting session. The D’Alembert system dictates that you should start by placing an even-money bet of precisely one betting unit. If luck is not on your side on the first spin and you lose, you need to increase your next wager with one betting unit. What you need to do is to keep on increasing your wagers with a single betting unit after each loss. After each winning bet, you are required to reduce your next stake with one betting unit. If your first bet is a winning one, you continue wagering with the same base betting unit until you lose. The idea here is that if a given player wins and loses roughly the same number of times, they will eventually turn a profit. It would be best to demonstrate how the D’Alembert betting method works in practice by providing an example. Let’s assume your bankroll is £250 and your betting unit is £5 or 2% of the overall sum you intend to join the roulette table with. You bet £5 on Red and lose. You increase your next wager with one betting unit, so you bet £10 on Red and lose again. At this point, you have lost £15. You increase your next wager on Red to £15 and win this time, thus collecting £15 in net profit. After this success, you are supposed to reduce the next wager with one betting unit to £10. You bet on Black and win again, generating a net profit of another £10. It becomes clear you have won £25 with your last two bets and have lost only £15 with your two losing wagers, so your net profit for this betting session is £10. As you can see, the D’Alembert system works efficiently as long as the number of winning bets coincides with or exceeds that of losing bets. Of course, there is the option to set a limit at which you will stop increasing the stakes after a loss and reduce the betting unit to its initial size h. This modification can help you minimise your losses if you happen to enter a longer losing streak. The D’Alembert System Spin Bet (units) Outcome Total Profit 1 5 LOSS -5 2 10 LOSS -15 3 15 WIN 0 4 10 WIN 10 5 5 WIN 15 6 5 LOSS 10 7 10 LOSS 0 8 15 WIN 15 9 10 WIN 25 10 5 LOSS 20 11 10 WIN 30 12 5 WIN 35 13 5 WIN 40 ## The Reverse D’Alembert As it can be expected from the name, it functions in the opposite way but is suitable for even-money bets in roulette. Players are again recommended to choose a base betting unit which should range between 2% and 5% of their overall bankrolls. Their first even-money bet should be one betting unit. According to the Reverse D’Alembert, players should increase their stakes by one betting unit after each winning bet and decrease it after they see a loss. So if your first bet of £5 on Red wins, your second bet should be no more than £10. If you succeed in predicting the winning number again, your next wager needs to be £15, and so on. When players lose a bet, they are required to decrease their next stake by one betting unit. So, if you lose your third bet of £15, you reduce the next wager to £10. One major advantage of the Reverse D’Alembert is that it enables players to minimise their losses. Those, who implement the Reverse D’Alembert are less likely to exhaust their bankrolls when they experience a lengthy losing streak. Besides, implementing this variation may prove to be rather profitable if the player wins several bets in a row. What gambling aficionados should take into account is that while employing the Reverse D’Alembert betting system, the results they will ultimately enjoy might not be that bountiful as the ones delivered by its original version. Yet, one of the biggest benefits roulette lovers will enjoy if they decide to give this betting method a shot is that they will be able to put it through its paces even if their bankroll is rather humble. Furthermore, if fortune does not smile on you and you end up on a sustained losing streak, you will not find yourself with no other solution but to risk a bigger chunk of your funds so as to make up for the losses you have seen in the previous rounds. As with the original version of the D’Alembert betting system, while making use of its reversed version gambling enthusiasts are advised to settle on a stopping point before they kick it off. As it seems, the better part of the gambling aficionados recommends starting the betting progression all over as soon as you augment the staked amount five times. Although taking the betting progression further might seem like an alluring idea because the potential payoffs will be more generous, players should bear in mind that the risk they will need to take will increase significantly. The Reverse D’Alembert System Spin Bet (units) Outcome Total Profit 1 5 WIN 5 2 10 WIN 15 3 15 LOSS 0 4 10 WIN 10 5 15 WIN 25 6 20 LOSS 5 7 15 LOSS -10 8 10 WIN 0 9 15 WIN 15 10 20 LOSS -5 11 15 WIN 10 12 20 WIN 30 13 25 WIN 55 ## How to Tinker the D’Alembert Betting System If you plan on putting the D’Alembert betting system to the proof, you should be aware that there are few more modifications that can be made to it. The results players might ultimately enjoy can turn out to be rather interesting, given that they do not increase the staked amount by one betting unit each time their wagers are settled as losing ones and instead bring the betting progression back to square one. The reason why many roulette mavens are unlikely to frown upon this variant of the betting system is that even if they place several losing wagers, this will not have such a major impact on their bankroll. If gambling enthusiasts decide to take advantage of this modified version of the roulette betting method, the winnings they are expected to generate should make up for the losses they have seen during their betting session. One factor gambling enthusiasts should not forget is the edge the casino gains over them whenever they play. In spite of the fact that the usage of this betting method will help players reduce the losses to a minimum, they should remember that ultimately, they will invariably lose to the casino. Still, the fact that the casino invariably holds the upper hand over players should not put you off from the betting system. In essence, if luck is on your side that day, you might enjoy a long winning streak that can bring you a massive payoff. ## Advantages of the D’Alembert Betting System Without question, the biggest advantage the D’Alembert system has to offer is its very simplicity. It does not require players to take notes like some other systems do. On the contrary, the system is quite easy to learn and incorporate at the roulette table. Another advantage results from the fact that while using this betting method, there is no steep increase in the bet size with this negative progression system. As wagers are increased with a single betting unit after each loss, a longer losing streak is less likely to drain your bankroll completely. On some occasions, the D’Alembert system also enables those who follow it to generate a consistent profit. The size of the wager after each loss is increased slowly which renders the system suitable for players who do not have that many funds to play with. Besides, the risk of reaching the table limit on a lengthy losing streak is much smaller if you implement the D’Alembert, which does not apply to some of the other betting methods. ## Disadvantages of the D’Alembert Betting System Similarly to the other betting systems we have discussed so far, the D’Alembert has a few faults. The system is most effective when it comes to generating short-term profits. Though consistent, the winnings will be far from life-changing which is to be expected from a system that involves a relatively low degree of risk. Then again, in order to turn a profit at all, players are expected to win and lose roughly the same number of times. Unfortunately, there is no guarantee that this will happen since the chances of winning and losing with even-money bets in roulette are not exactly 50%. The scales are tipped in favour of the casino because of the introduction of the zero pocket on the wheel. Even more so, if one plays American roulette, where there is an additional double-zero pocket, which further increases the house edge. Statistically speaking, players lose more often than they win which would make it more difficult for them to recover from their losses. Entering a longer losing streak is not that unlikely, either. If a given player suffers five or six losing bets in a row, there is no guarantee they will succeed in winning the same number of times to offset the losses. Essentially, players need to remember that neither the D’Alembert nor any other betting system can reduce the house edge or influence the spins. This is so because the outcome of each spin is not affected by the previous numbers which had been spun on the wheel. ## Dangers of Using the D’Alembert Betting System As we mentioned above, the D’Alembert betting system revolves around the idea that ultimately, the number of losing and winning wagers gambling enthusiasts have laid will balance. In spite of the fact that this claim makes perfect sense to roulette lovers, what they fail to take into consideration is that in order for the results to come out even, the period that should be taken into consideration will be dramatically prolonged. In essence, this is exactly where the main lacking point of the betting system stems from because the frequencies of the opposite results are unlikely to balance during the betting session of players unless they do not spend a huge amount of time at the roulette table. That is the reason why this idea is unworkable when the game of roulette is concerned. In the event that the number of winning and losing wagers does not balance during the time casino enthusiasts wager, they might end up with no winnings at all, which is certainly not a desirable scenario. That being said, what makes many roulette players give the betting system a chance is that unlike the Martingale betting method that is exceptionally aggressive, this is not the case with the D’Alembert betting system for the simple reason that gambling aficionados will not be prompted to double the staked amount whenever a loss incurs. As it was already mentioned, with the D’Alembert betting system, the amount you need to put on the line following a loss will be increased by a single bet unit, which undoubtedly adds up to a big difference. Although the D’Alembert betting system appeals to gambling aficionados because of its inherent simplicity, roulette lovers should not forget that its usage will not have any influence on their chances of win. The same applies to the house edge as no matter if you are bound to take advantage of the D’Alembert betting method or any of the other roulette betting systems available out there, losing to the house is unavoidable. ## Conclusion At the end of the day, many gambling aficionados are eager to try their hand at the D’Alembert betting system because there are not any complex calculations that should be made while adapting their stakes to the outcome of the previous round. Another thing that makes the betting method so immensely popular among casino enthusiasts is that it is a much safer alternative when compared to the rest of the betting systems that can be applied to the game of roulette because the amount they will need to risk will not soar. With that in mind, gambling aficionados will not stand such a good chance to walk away with a heftier payout. In a similar vein, making up for the losses they have seen might only be possible after placing several successful wagers, which can sometimes be hard. As you can see, using the D’Alembert betting system has its pros and cons that should be evaluated, but it is worth considering as an option if you are still not confident enough to make use of other riskier roulette betting methods.

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1. Introduction 2. Methodology 3. Cost Function 4. 5. Evaluating the model 6. Implementation 7. 8. Key takeaways Last Updated: Mar 27, 2024 # Logistic Regression Introduction Arun Nawani 0 upvote ## Introduction Have you ever wondered how your email account accurately segregates regular emails, important emails, and spam emails? Itâ€™s not a very complex trick and weâ€™ll learn the secret behind it. This is done with a supervised learning model called Logistic regression (However, it can be done with other machine learning algorithms also, but for the sake of this blog, weâ€™ll stick to Logistic regression). Logistic regression is employed in supervised learning tasks. More specifically, it is used for classification tasks. We know that name throws some people off. But the regression in the logistic regression is slightly misleading. It is NOT a regression model. Logistic regression is a probabilistic function. That means it makes use of probabilities of events to make its prediction. ## Methodology Suppose we are given a task, say we are given a customerâ€™s banking history and are tasked to find if the customer can be sanctioned a loan. Basically we need to find if given a loan, will the customer default on payment or not. We can use logistic regression for this purpose. It will be a binary classification between â€˜Yesâ€™ or â€˜Noâ€™. Logistic regression makes use of a sigmoid function and it is of the form - We know the straight line equation - y = w0 + w1 We know the sigmoid function has a range between 0 and 1. So letâ€™s divide the above equation by 1-y. y / (1-y) : 0 for y = 0 and âˆž for y = 1 But we require our function to be between -âˆž to +âˆž. For that, weâ€™ll take logarithm so the new equation is: Log (y / (1-y)) = w0 + w1x Upon simplifying our final equation then becomes - Here y = predicted probability belonging to the default class( default class is 1(yes)) w0 + w1x = the linear model within logistic regression. Also, the function is of the form of a sigmoid function The Sigmoid function has a range between 0 and 1. And therefore forms an S-like curve. The logistic function predicts the probability of an outcome. Hence its value lies anywhere between 0 and 1. And thatâ€™s where it gets its name from. We choose a threshold value above which the final prediction would be 1 and 0 otherwise. Letâ€™s talk about the linear equation w0 + w1x within the logistic function. Why do we need the logistic regression function in the first place if it stems from linear regression? Itâ€™s because the linear regression equation isnâ€™t confined within a range, unlike logistic regression. And it would be a very difficult task to assign a threshold value for class membership for a linear regression function. Thus we feed the predicted value to a sigmoid function which makes it Logistic regression having a range between 0 and 1. Now since the range is between 0 and 1(no outliers) it would be convenient to do a probabilistic classification. It represents a linear relationship between the input features and the final output. Here x = input feature w0 = bias term w1 = weight associated with the input variable Now suppose we take 0.5 as our threshold value. That means A predicted value >0.5 from the logistic function would have the final prediction as 1 and, A predicted value â‰¤0.5 from the logistic function would have the final prediction as 0. This is also called the decision boundary. Plotting the graph clears what makes logistic regression different from linear regression. ## Cost Function We know the cost function tells us how erroneous our model is. In the case of linear regression, our cost function was: - Cost function = 1/N âˆ‘(y(i) - y(i)predicted )2 The linear regression cost function is used to find the average of errors in all predicted values corresponding to the actual value of y. But this cost function is suitable for logistic regression since logistic regression doesnâ€™t have a continuous output variable, unlike linear regression. Suppose we were to use this cost function for a logistic regression model that does a binary classification(0 and 1). This would make the predicted value and the actual value either 0 or 1. This way we may end up at the local minima instead of the global minima. Remember, the cost function is minimum at the global minima. So we would require a new cost function for logistic regression. And itâ€™s given below. Here y(i) = the actual output for ith training example hÎ¸x(i) = Predicted output for the ith training example by our model Now we know that looks a bit overwhelming but letâ€™s break it down term by term. Suppose we train a logistic regression model and it correctly predicts the value 1. So that means the predicted value of y would be closer to 1. In this case, the second term in the logistic regression cost function becomes 0 (1-y = 1-1 = 0). The first term would be- y * log(ypredicted)= 1*log(~1) = ~0 ( low error) Since the value is close to 1, therefore the cost function would compute a low error. Refer the log and -log graph given below. Our predicted value was correct and therefore it makes sense for the error to be low. In case had our model predicted 0 instead of 1, it would have led to a higher error since the log function for an input value closer to 0 would make our error exponentially high. y * log(ypredicted)= 1*log(~0) = -âˆž (high error) Gradient descent is a technique of altering the weights of data points based on the cost function. LogLoss cost function is calculated at each input-output data point. • We take a partial derivative with respect to bias and weight to find the slope of the cost function at every point. • Gradient descent then updates the values bias and weight values based on slope. This is an iterative process. • The iterations are done until the gradient descent reaches the minimum cost. The rate at which the gradient descent reaches the minima is called the learning rate. Feature scaling is essential for gradient descent to work efficiently. We suggest you learn gradient descent in depth. You can refer to our blog on gradient descent if you wish to learn the math behind it. ## Evaluating the model Once we train the model it is necessary to evaluate how accurate our model really is. There are various evaluation metrics for this purpose. • Accuracy - It represents the number of correct predictions upon total number of predictions. • ROC AUC score it stands for Receiver Operating Characteristic Curve. The area under the ROC AUC curve describes the relationship between the true positive rate (ratio of samples that were correctly predicted belonging to the correct class) and the false positive rate (the ratio of samples for which we incorrectly predicted their class membership). More area under the curve signifies better predictions, hence a better model. ## Implementation Weâ€™ll start by importing the necessary libraries and the dataset. Here is the datasetThe dataset is about sales of a product via social media advertisement campaign. Our task at hand will be to predict if the item was purchased or not considering the given features. ``````import numpy as np import pandas as pd import matplotlib.pyplot as plt from sklearn.model_selection import train_test_split from math import exp plt.rcParams["figure.figsize"] = (10, 6) class LogisticRegression: def PltFun(self): plt.scatter(self.data['Age'], self.data['Purchased']) plt.show() # Split the training dataset and test dataset def Training(self): self.X_train, self.X_test, self.y_train, self.y_test = train_test_split(self.data['Age'], self.data['Purchased'], test_size=0.20,random_state=10) def normalize(self): self.X_train -= self.X_train.mean() # Method to make predictions def Sigmoid(self, b0, b1,x=None): if x is None: self.X_test-=self.X_test.mean() return np.array([1 / (1 + exp(-1*b0 + -1*b1*i)) for i in self.X_test]) else: return np.array([1 / (1 + exp(-1*b0 + -1*b1*i)) for i in x]) # Method to train the model def logistic_regression(self): # Initializing variables self.normalize() b0 = 0 b1 = 0 L = 0.001 epochs = 300 for epoch in range(epochs): y_pred = self.Sigmoid(b0, b1,self.X_train) D_b0 = -2 * sum((self.y_train - y_pred) * y_pred * (1 - y_pred)) # Derivative of loss wrt b0 D_b1 = -2 * sum(self.X_train * (self.y_train - y_pred) * y_pred * (1 - y_pred)) # Derivative of loss wrt b1 # Update b0 and b1 b0 = b0 - L * D_b0 b1 = b1 - L * D_b1 return b0, b1 def Accuracy(self , y_pred): accuracy=0 for i in range(len(y_pred)): if y_pred[i] == self.y_test.iloc[i]: accuracy += 1 print(f"Accuracy = {accuracy / len(y_pred)}") def Plot_result(self,y_pred): plt.clf() plt.scatter(self.X_test, self.y_test) plt.scatter(self.X_test, y_pred, c="red") plt.show() def predictor(self,x): x = [1 if p >= 0.5 else 0 for p in x] return x My_model = LogisticRegression() My_model.PltFun() My_model.Training() b0,b1=My_model.logistic_regression() y_pred = My_model.Sigmoid(b0,b1) y_pred=My_model.predictor(y_pred) My_model.Plot_result(y_pred) My_model.Accuracy(y_pred)`````` The accuracy of the model is >80%. 1. What is a sigmoid function? Ans. The sigmoid function is given by 1 / ( 1 + e-z ) and its range is between 0 and 1. 2. Why is logistic regression called a probabilistic function? Ans. Logistic regression generates the probability of output given the input features. We then set a threshold value to classify the probability prediction. 3. Mention some real-life use cases of logistic regression. Ans. a. A bank can use a logistic regression model to predict if a customer would default on a loan b. A logistic regression model can be trained to tell if a tumour is benign or not. ## Key takeaways In this blog, we have thoroughly covered logistic regression. Its mathematical significance, its cost function, and its step-by-step python implementation. However, consider this a starting point. There is always room for improvement. If you want to ace your next data science interview, check out our machine learning courses curated by our faculty from Stanford University and industry experts. Happy Learning!! Live masterclass

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You are Here: Home >< Maths Stats 2 (S2) OCR (not MEI) 22nd June 2012 Watch 1. (Original post by studentofskhs) Yes . Btw how did you do 8iii)? I done 1-(p(x=0)+p(x=1)+p(x=2)) using s1 binominal rules. Not sure if i'm right though. I got a small value like 0.01 ish I think i did that it was a binomial of (20, 0.05) and then i did greater than or equal to 4 which is 1 - less than or equal to 3. Don't know if that's right though. 1) Random number generator 000 to 999 2 Var(V)=173 3) P(X>=7) = 0.0664 larger than 0.025 ACCEPT H0 4) i) independent ii) 0.114 iii) 0.422 iv) 0.268 5) 1.1153 < 1.645 ACCEPT H0 6) i) 0.0449 ii) 0.0494 7) ii) a=6 (k=1/72) iii) Var(X) = 1.35 8) i) CR >33.1 ii) Type I error iii) 0.0159 , mean not 30 because probability so small iv) mean is 33.58 Any thoughts guys? 3. (Original post by jamib0y) 1) Random number generator 000 to 999 2 Var(Vbar)=173 I think you mean Var(V), not V bar. 3) P(X>=7) = 0.0664 larger than 0.025 ACCEPT H0 4) i) independent ii) 0.114 iii) 0.422 iv) 0.268 5) 1.1153 < 1.645 ACCEPT H0 6) i) 0.0449 ii) 0.0494 7) ii) a=6 (k=1/72) iii) Var(X) = 1.35 8) i) CR >33.1 ii) Type I error iii) 0.0159 iv) mean is 33.58 Any thoughts guys? I've forgotten many answers - those that I remember, I agree with. 4. (Original post by CraigKirk) I've forgotten many answers - those that I remember, I agree with. Yeah, that's what I mean 5. (Original post by jamib0y) 1) Random number generator 000 to 999 2 Var(Vbar)=173 3) P(X>=7) = 0.0664 larger than 0.025 ACCEPT H0 4) i) independent ii) 0.114 iii) 0.422 iv) 0.268 5) 1.1153 < 1.645 ACCEPT H0 6) i) 0.0449 ii) 0.0494 7) ii) a=6 (k=1/72) iii) Var(X) = 1.35 8) i) CR >33.1 ii) Type I error iii) 0.0159 iv) mean is 33.58 Any thoughts guys? From vaguely remembering my answers, i think it all looks good. 6. for 8)iii), Does anyone know for sure whether you're supposed to put that the mean is 30 or not 30? 7. (Original post by Bwob) for 8)iii), Does anyone know for sure whether you're supposed to put that the mean is 30 or not 30? Yeah, I put down it's not 30 because the probability is so small 8. What did people say for 7 i)? I don't think I expressed myself very well. It's generally the questions with words involved I didn't like - again 8iii) I got 0.0155 or something, but I said since the probability is small, it probably is 30, which goes against what most people have been saying :/ 9. (Original post by Bb King) What did people say for 7 i)? I don't think I expressed myself very well. It's generally the questions with words involved I didn't like - again 8iii) I got 0.0155 or something, but I said since the probability is small, it probably is 30, which goes against what most people have been saying :/ Graph of , starting from 0 and ending at a. I commented that as X takes ranges closer to a, the probability of X increases. If ranges of X fall below 0 or above a, the probability of X is zero. 10. (Original post by JongKey) From vaguely remembering my answers, i think it all looks good. im so sillyy!! i tried to find the critcial region for question 3 and found x-1 =7 and being me said therfore x=7 and rejected h0 :/ . 3 markss gonee. sigh. think i got 68 or 67 from the looks of thingss.. but ive probs made some very silly mistakes :/ 11. Can you lose marks for giving a greater degree of accuracy than 3 significant figures? 12. (Original post by Matthew692692) Can you lose marks for giving a greater degree of accuracy than 3 significant figures? In general, no, as long as your answer rounds to the answer given in the mark scheme, or falls in a range given in the mark scheme. 13. (Original post by CraigKirk) In general, no, as long as your answer rounds to the answer given in the mark scheme, or falls in a range given in the mark scheme. Thank god for that, I usually give them to more :P 14. (Original post by Matthew692692) Thank god for that, I usually give them to more :P Yeah, I gave many of mine today to 4sfs, as in statistics mark schemes there often tend to be four decimal places in the answer. 15. (Original post by jamib0y) 1) Random number generator 000 to 999 2 Var(Vbar)=173 3) P(X>=7) = 0.0664 larger than 0.025 ACCEPT H0 4) i) independent ii) 0.114 iii) 0.422 iv) 0.268 5) 1.1153 < 1.645 ACCEPT H0 6) i) 0.0449 ii) 0.0494 7) ii) a=6 (k=1/72) iii) Var(X) = 1.35 8) i) CR >33.1 ii) Type I error iii) 0.0159 , mean not 30 because probability so small iv) mean is 33.58 Any thoughts guys? Everything correct, well done. Bits you've missed out: 1. Need to mention 15 unique numbers. 2. (i) should be Var(V), but you wouldn't lose a mark (ii) "We don't know the distribution of V"; "n>30 so CLT applies" 6. (i) Normal approximation is suitable as np and nq both exceed 5. 7. (i) positive quadratic graph between x=0 and x=a going through (0,0) and not extending below x=0 or above x=a. Then something along the lines of "X can take any value between 0 & a (but not including 0). X is more likely to take values closer to a than to 0." 16. (Original post by Mr M (jr)) Everything correct, well done. Bits you've missed out: 1. Need to mention 15 unique numbers. 2. (i) should be Var(V), but you wouldn't lose a mark (ii) "We don't know the distribution of V"; "n>30 so CLT applies" 6. (i) Normal approximation is suitable as np and nq both exceed 5. 7. (i) positive quadratic graph between x=0 and x=a going through (0,0) and not extending below x=0 or above x=a. Then something along the lines of "X can take any value between 0 & a (but not including 0). X is more likely to take values closer to a than to 0." Surely question 3 - it was a 1 tail test as it said 'Test at the 5% sig level that the study is correct'. I know it doesn't affect the overall result (I checked this) but you lose 2 marks for doing it as a 2 tail rather than 1 tail? 17. (Original post by Mr M (jr)) Everything correct, well done. Bits you've missed out: 1. Need to mention 15 unique numbers. 2. (i) should be Var(V), but you wouldn't lose a mark (ii) "We don't know the distribution of V"; "n>30 so CLT applies" 6. (i) Normal approximation is suitable as np and nq both exceed 5. 7. (i) positive quadratic graph between x=0 and x=a going through (0,0) and not extending below x=0 or above x=a. Then something along the lines of "X can take any value between 0 & a (but not including 0). X is more likely to take values closer to a than to 0." Yeah, thanks, I just didn't have the time to write all the above but got it all! Thanks! 18. (Original post by CraigKirk) Yeah, I gave many of mine today to 4sfs, as in statistics mark schemes there often tend to be four decimal places in the answer. Annoyingly for some reason I gave the final one to 3sfs though :/ I NEVER do that in stats!! 19. FFS, couldn't have gotten an easier paper and I still managed to muck it up; lost a good 10 marks at the least. Well, at least it'll contribute towards bringing the grade boundaries down (by whatever negligible value) for all of you concerned. 20. (Original post by Matthew692692) Annoyingly for some reason I gave the final one to 3sfs though :/ I NEVER do that in stats!! I nearly did that too, managed to see that right at the end when I was checking :P Updated: August 14, 2012 TSR Support Team We have a brilliant team of more than 60 Support Team members looking after discussions on The Student Room, helping to make it a fun, safe and useful place to hang out. This forum is supported by: Today on TSR Stuck for things to do this summer? Come and get some inspiration.

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View more editions Introduction to Programming with C plus plus # TEXTBOOK SOLUTIONS FOR Introduction to Programming with C plus plus 2nd Edition • 660 step-by-step solutions • Solved by publishers, professors & experts • iOS, Android, & web Over 90% of students who use Chegg Study report better grades. 2013 Chegg Homework Help Survey SAMPLE SOLUTION Chapter: Problem: • Step 1 of 3 Count and compute the sum and average of numbers Program plan: • Check whether the integer numbers are positive or negative. • If the integer number is positive, increment the positive number count; otherwise, increment the negative number count. • Calculate the sum of numbers. • Calculate the average of numbers. • Display the positive number count, the negative number count, the sum of numbers, and the average of numbers. • Step 2 of 3 Program: /********************************************************** *This program demonstrates how to determine the number of * *positive and negative numbers entered, and also to * *compute the sum and average of numbers. * **********************************************************/ #include using namespace std; //Function main int main() { //Declare the variables int number; int sum=0; int count=0; int positive=0,negative=0; double ave=0; cout<<"Counting the positive and negative numbers and computing the average of numbers:"<<endl<<endl; Read the input value for ‘number’. cout<<"Enter the integer value, the program exits if the input is 0: "; cin>>number; This part of the coding is used to count the positive and negative numbers, and to calculate the sum and average of numbers. //Loop executes until the condition leads to false while(number!=0) { Check whether the number is positive or negative. /*The if condition checks whether the number is greater than 0. If yes, then it increments the positive number count; otherwise, it increments the negative number count.*/ if(number>0) { positive++; } else { negative++; } Calculate the sum of numbers. sum+=number; Count the total number of inputs. //Increment the count variable count++; cin>>number; } Calculate the average of numbers. //Calculate the average ave=(double)sum/count; Display the results. //Print the result cout<<"Total numbers of inputs: "<<count<<endl; cout<<"The number of positives is: "<<positive<<endl; cout<<"The number of negatives is: "<<negative<<endl; cout<<"The sum is: "<<sum<<endl; cout<<"The average is: "<<ave<<endl; //Pause the system system("pause"); //Return the value return 0; } • Step 3 of 3 Output: Counting the positive and negative numbers, and computing the average of numbers: Enter the integer value, the program exits if the input is 0: 1 –5 8 –2 4 6 3 0 Total numbers of inputs: 7 The number of positives is: 5 The number of negatives is: 2 The sum is: 15 The average is: 2.14286 Corresponding Textbook Introduction to Programming with C plus plus | 2nd Edition 9780136097204ISBN-13: 0136097200ISBN: Y Daniel LiangAuthors:

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orthocenter synonyms, orthocenter pronunciation, orthocenter translation, English dictionary definition of orthocenter. * The three heights (altitudes) of a triangle intersect at one point (are concurrent at a point), called the orthocentre of the triangle. Altitude of a Triangle, Definition & Example, Finding The Orthocenter, Acute Right & Obtuse Triangle - Duration: 11:15. It only takes a minute to sign up. translation and definition "orthocenter", English-Japanese Dictionary online. Here are three important theorems involving centroid, orthocenter, and circumcenter of a triangle. This is part of the series of posts on theorems in secondary school geometry.Proofs of the theorems and application problems will be provided in the next few posts. The line segment needs to intersect point C and form a right angle (90 degrees) with the "suporting line" of the side AB.Definition of "supporting line: The supporting line of a certain segment is the line The orthocenter is one of the triangle's points of concurrency formed by the intersection of the triangle's 3 altitudes.. Orthocenter Calculator is a free online tool that displays the intersection of the three altitudes of a triangle. This geometry video tutorial explains how to identify the location of the incenter, circumcenter, orthocenter and centroid of a triangle. This line containing the opposite side is called the extended base of the altitude. An altitude is the line perpendicular from a base that passes through the opposite vertex. The center of the incircle is called the incenter, and the radius of the circle is called the inradius.. The triangle is one of the most basic geometric shapes. Orthocenter of a Triangle In geometry, we learn about different shapes and figures. Definition of the Orthocenter of a Triangle. Orthocenter of a Triangle. Find the coordinates ofthe orthocenter of this triangle. Word of the day. This is a matter of real wonderment that the fact of the concurrency of altitudes is not mentioned in either Euclid's Elements or subsequent writings of the Greek scholars. Origin. Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. To download free study materials like NCERT Solutions, Revision Notes, Sample Papers and Board Papers to help you to score more marks in your exams. Step 1 : Incircle. In geometry, an altitude of a triangle is a line segment through a vertex and perpendicular to (i.e., forming a right angle with) a line containing the base (the side opposite the vertex). The orthocenter of a triangle is the point where the perpendicular drawn from the vertices to the opposite sides of the triangle intersect each other. Segment from a vertex that is perpendicular to the opposite side or line containing the opp. Orthocenter of a Triangle (Definition, How to Find, Video, & Examples) The orthocenter of a triangle, or the intersection of the triangle's altitudes, is not something that comes up in casual conversation. Mid 19th century. Define orthocenter. The timing of the first proof is still an open question; it is believed, though, that even the great Gauss saw it necessary to prove the fact. Here $$\text{OA = OB = OC}$$, these are the radii of the circle. Dealing with orthocenters, be on high alert, since we're dealing with coordinate graphing, algebra, and geometry, all … 2. Concurrent Math with the definitions A. I.e., the three heights of a triangle are cut in the orthocenter. It symbolizes from the capital letter H letter. Where all three lines intersect is the "orthocenter": Let's learn these one by one. forming a right angle with) a line containing the base (the opposite side of the triangle). The Organic Chemistry Tutor 4,974 Chemistry. Orthocenter doesn’t need to lie inside the triangle only, in case of an obtuse triangle, it lies outside of the triangle. Ruler. ... Deriving the barycentric coordinates of a triangle's orthocenter, using the areal definition of such coordinates. In right triangle, orthocenter is located on the triangle. Note: The orthocenter's existence is a trivial consequence of the trigonometric version Ceva's Theorem; however, the following proof, due to Leonhard Euler, is much more clever, illuminating and insightful.. Orthocenter Formula - Learn how to calculate the orthocenter of a triangle by using orthocenter formula prepared by expert teachers at Vedantu.com. See definitions & examples. The circumcenter of a triangle is the center of a circle which circumscribes the triangle.. (US orthocenter) Pronunciation ... sɛntə/ noun Geometry . are A (0, 0), N (6, 0), and D (–2, 8). Constructing Orthocenter of a Triangle - Steps. These three altitudes are always concurrent.In other, the three altitudes all must intersect at a single point , and we call this point the orthocenter of the triangle. It has several important properties and relations with other parts of the triangle, including its circumcenter, incenter, area, and more. In geometry, an altitude of a triangle is a straight line through a vertex and perpendicular to (i.e. Altitude in a triangle is a bisector lines for 3 angles in a triangle. An incircle is an inscribed circle of a polygon, i.e., a circle that is tangent to each of the polygon's sides. The Centroid is the point of concurrency of the medians of a triangle. Circumcenter. Three altitudes intersecting at the orthocenter. An altitude is the perpendicular segment from a vertex to its opposite side. Geometry Dictionary. Orthocenter definition is - the common intersection of the three altitudes of a triangle or their extensions or of the several altitudes of a polyhedron provided these latter exist and meet in a point. A geometrical figure is a predefined shape with certain properties specifically defined for that particular shape. A key part of learning is adding to your vocabulary. Orthocenter Draw a line segment (called the "altitude") at right angles to a side that goes to the opposite corner. razoo / rɑːˈzuː / noun. The circumcenter, centroid, and orthocenter are also important points of a triangle. Orthocenter Definition #In the diagram, O=Orthocenter(A,B,C) # The intersection of the three altitudes of the vertices # of a triangle whose vertices are Points A, B, C # hence the intersection of any two of them, its existence is proved by the OrthocenterExists Theorem. Compass. In obtuse triangle, orthocenter is … Concurrency is an excellent word to learn in geometry. The orthocenter of an acute triangle. Circumcenter definition, the center of a circumscribed circle; that point where any two perpendicular bisectors of the sides of a polygon inscribed in the circle intersect. BYJU’S online orthocenter calculator tool makes the calculation faster and it displays the orthocenter of a triangle in a fraction of seconds. Let's build the orthocenter of the ABC triangle in the next app. From ortho- + centre. To construct orthocenter of a triangle, we must need the following instruments. 1. The orthocenter is a term that is used exclusively within the scope of the geometry and refers to that point of intersection where converge the three altitudes of a triangle. orthocenter ( plural orthocenters ) ( geometry ) The intersection of the three lines that can be drawn flowing from the three corners of a triangle to a point along the opposite side where each line intersects that side at a 90 degree angle; in an acute triangle , it is inside the triangle; in an obtuse triangle , it is outside the triangle. Incenter: Where a triangle’s three angle bisectors intersect (an angle bisector is a ray that cuts an angle in half); the incenter is the center of a circle inscribed in (drawn inside) the triangle. Now, let us see how to construct the orthocenter of a triangle. In mathematics, it means a point shared by three or more lines. Circumcenter 3. The Orthocenter is the point of concurrency of the altitudes, or heights, as they are commonly called. The intersection of the extended base and the altitude is called the foot of the altitude. The orthocenter of a triangle is the intersection of the triangle's three altitudes. The common point of the perpendicular bisectors of a triangle B. Orthocenter It is the point where the three "altitudes" of a triangle meet and the orthocenter can be inside or outside of the triangle. Perpendicular Bisector of a Triangle 2. Proof of Existence. 1. This lesson focuses on points of concurrency in triangles. The point of intersection of the three perpendiculars drawn from the vertices of a triangle to the opposite sides. Orthocenter : Orthocenter is an intersection point of 3 altitudes of a triangle. Video Definition Centroid Incenter Circumcenter Orthocenter Facts. If four points form an orthocentric system, then each of the four points is the orthocenter of the other three. Orthocenter 4. See more. Ask … Geometry dictionary is the place where we can find meaning, definition and explanation etc for the geometric terms which are being often used by the students.Most of the students find it difficult to understand some geometric terms when they do theorems and problems on Geometry. Answers and explanations (–8, –6) The orthocenter of a triangle is the point where the three altitudes of the triangle intersect. An altitude of a triangle is perpendicular to the opposite side. In geometry, an orthocentric system is a set of four points on a plane, one of which is the orthocenter of the triangle formed by the other three.. n. The point of intersection of the three altitudes of a triangle. An "altitude" is nothing but a line that goes through a vertex (corner point) and is at right angles to the opposite side. In acute triangle, orthocenter is located inside the triangle. Construct triangle ABC whose sides are AB = 6 cm, BC = 4 cm and AC = 5.5 cm and locate its orthocenter. 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Sets & Cardinalities Mathematical Logic Relations & Functions Proof Techniques Results & Conjectures ### 100 What is the empty (null/void) set? The only set with zero cardinality. ### 100 What is a statement? A declarative sentence that is either true or false. ### 100 What is bijective? A relation and its inverse are functions if and only if they satisfy this property. ### 100 What is a counterexample? To disprove a statement, it suffices to give this. ### 100 What is a lemma? A mathematical result that is used as an auxiliary result, often to prove a theorem. ### 200 What is the set of rational numbers? Every element in this set can be written as the quotient of an integer and a natural number. ### 200 What it is a disjunction? The combination of two statements using a logical OR. ### 200 What is an equivalence relation? A relation that is reflexive, symmetric, and transitive. ### 200 What is QED (quod erat demonstrandum)? The English translation of this Latin abbreviation is "what had to be shown". ### 200 What is the Fundamental Theorem of Arithmetic? Every integer greater than 1 can be written as a unique product of prime numbers. ### 300 What are pairwise disjoint sets? A collection of nonempty subsets of a set A is called a partition of A if its elements contain all elements of A and satisfy this property. ### 300 What is the universal quantifier (∀)? This quantifier means FOR ALL. ### 300 What is surjective (onto)? If a function's range equals its codomain, then it has this property. ### 300 What is a proof by contradiction? To show that P ⇒ Q, suppose that P is true and Q is false. ### 300 What is Goldbach's Conjecture? Every even integer greater than 2 can be written as the sum of two prime numbers. ### 400 What are the integers modulo n? Given a positive integer n, this set contains all equivalence classes of integers that have the same remainder upon division by n. ### 400 What is a necessary condition? A requirement without which a conclusion is always false. ### 400 What is the identity? The composition of a bijective function with its own inverse. ### 400 What is without loss of generality (wlog)? A commmon justification to shorten a proof if different cases are identical or very similar. ### 400 What is Euclid's Lemma? Let m and n be integers, and p be a prime number: p | mn ⇒ p | m ∨ p | n ### 500 What is aleph null? The cardinal number of each denumerable set. ### 500 What is the Modus Ponens? P ∧ (P ⇒ Q) ⇒ Q ### 500 What is a dictionary (alphabetical/lexicographic) order ? eight five four nine one seven six three two When sorting the numbers from 1 to 9, 1 < 2 but 2 > 3 > 4 > 5, 5 < 6 but 6 > 7 > 8, 8 < 9 but 9 < 1. ### 500 What is the Principle of Mathematical Induction? ( ∀ n ≥ n0: P(n) ) ≡ ( P(n0) ∧ ∀ k ≥ n0: P(k) ⇒ P(k+1) ) ### 500 What is the Schröder-Bernstein Theorem? If the cardinal number of a set A is both less-than-or-equal and greater-than-or-equal than the cardinal number of a set B, then A and B are numerically equivalent. # Math 3000 Abstract Math Jeopardy Press F11 for full-screen mode

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https://brilliant.org/discussions/thread/prove-displaystyle-int_01-fraclogxlog1/

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× # Prove this closed form of $$\displaystyle \int_0^1\frac{(\ln(1+x))^2\ln(x)\ln(1-x)}{1-x} \, dx$$ $\int_0^1\frac{(\ln(1+x))^2\ln(x)\ln(1-x)}{1-x} \, dx=\dfrac{7}{2}\zeta(3){(\ln 2)^2}-\dfrac{\pi^2}{6}{(\ln 2)^3}-\dfrac{\pi^2}{2}\zeta(3)+{6}\zeta(5)-\dfrac{\pi^4}{48}\ln2$ Prove that the equation above is true. Notation: $$\zeta(\cdot)$$ denotes the Riemann Zeta function. This was found in another mathematics form and it was unanswered there. This is a part of the set Formidable Series and Integrals. 1 year, 9 months ago MarkdownAppears as *italics* or _italics_ italics **bold** or __bold__ bold - bulleted- list • bulleted • list 1. numbered2. list 1. numbered 2. list Note: you must add a full line of space before and after lists for them to show up correctly paragraph 1paragraph 2 paragraph 1 paragraph 2 [example link](https://brilliant.org)example link > This is a quote This is a quote # I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world" # I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world" MathAppears as Remember to wrap math in $$...$$ or $...$ to ensure proper formatting. 2 \times 3 $$2 \times 3$$ 2^{34} $$2^{34}$$ a_{i-1} $$a_{i-1}$$ \frac{2}{3} $$\frac{2}{3}$$ \sqrt{2} $$\sqrt{2}$$ \sum_{i=1}^3 $$\sum_{i=1}^3$$ \sin \theta $$\sin \theta$$ \boxed{123} $$\boxed{123}$$ Sort by: Use this expansion, $\large \ln ^2 (1+x) = \sum_{r=0}^{\infty} \dfrac{H_{r} (-1)^r x^{r+1}}{r+1}$ Then it will be left to evaluate the integral $\large \int_{0}^{1} x^{r+1} \dfrac{\ln(x) \ln(1-x)}{1-x} dx$ This is derivative of beta function. - 1 year, 9 months ago How will you evaluate the resulting summation? - 1 year, 9 months ago wow nice one! Do you know how to prove it? - 1 year, 9 months ago Comment deleted Feb 29, 2016 I'll not give up on that problem. I have to solve it first. - 1 year, 9 months ago Integrate the generating function of harmonic number. - 1 year, 9 months ago I made it on my own and i dont know whether it is already there or not - 1 year, 9 months ago - 1 year, 9 months ago Now @Ishan Singh can solve this.. Integral can be written as: $\displaystyle \sum_{r,s,t\geq 0} \frac{H_r(-1)^r}{(r+1)(t+1)(r+s+t+3)^2}$ - 1 year, 8 months ago @Pi Han Goh add it to you set - 1 year, 9 months ago - 1 year, 9 months ago Thanks. Do try it. - 1 year, 9 months ago Hint : Convert into derivative of Beta function. - 1 year, 9 months ago Yup I did that but at some point, I have to take natural logarithm of (-1). Which is imaginary, but the closed form ain't contain any imaginary term. - 1 year, 9 months ago No. Take limit. For example this - 1 year, 9 months ago can u clearly explain how u applied the limit there? - 1 year, 9 months ago That will take $$2 -3$$ pages. I may post it when I'm free. - 1 year, 8 months ago You have to do something with the $$\ln^2 (1+x)$$ term and afterwards it gets converted into the limit. Another method is to use generating function of Harmonic numbers. - 1 year, 9 months ago I got imaginary term while using that Harmonic relation. You are referring this $$\sum_{n=1}^{\infty} H_{n} x^{n} = \dfrac{-ln(1-x)}{1-x}$$ no? - 1 year, 9 months ago Yes. I'm referring to that generating function. - 1 year, 9 months ago Kk I'll give another shot at Feynman's way. - 1 year, 9 months ago @Mark Hennings could you help us out? - 1 year, 9 months ago I got it. Give me one hour of time. - 1 year, 9 months ago

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# Cubic and cube roots It was said that he radical sign was first used by Rene Descartes, but long before he used this, it has been developed in Germany in the 1500s. The radical ( sign was invented to find the square root or the cube root, 4th root, 5th root etc., depending on the index of the radical sign. Now if you remember, in previous chapter, we learned all about the different number systems and we have mentioned about the rational and irrational numbers. Radicals can belong to either of which depending of course if the radicals would be able to fit into the definition of whichever group of number. In this lesson we will get to be able to identify whether or not a certain radical is rational. We will be specifically going to study about a square root or a cube root of a particular number. We will first look into what a square and a cube are since these would be the two fundamental concepts. In order to solve a square root, or a cube root, one must know how to square or cube a number. These will be thoroughly discussed in 4.1 and 4.2. For 4.3 to 4.8 we will be looking closer into how to simplify and evaluate these radicals through applying what we have learned from chapters on square and square roots, estimating square roots, and rational numbers from earlier grades. We would also be learning how to convert them into either entire radicals to mixed radicals or vice versa, by applying our knowledge of factoring out the perfect squares or the perfect cubes in the equation. Apart from converting the radicals from one form to another we are also going to learn more about how to combine different radicals and simplify them depending on the operation used and learn to rationalize a certain radical equation. ### Cubic and cube roots Whenever we see “roots”, let it be cubic roots or square roots, we know for sure that we will need to do prime factorization to find out the prime factors of the numbers. In this section, we use factors and multiples to find perfect cube whole numbers and cubic roots.

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28th International Conference on Technology in Collegiate Mathematics 2016 ¤ March 12, 2016 Mapping Diagrams for Complex Variable Functions Visualized Dynamically with GeoGebra 1:30 p.m. - 2:00 p.m. Part I Mapping Diagrams for Real Functions Complex Arithmetic Part II Complex Functions Part III Calculus for Complex Functions Martin Flashman Professor of Mathematics Humboldt State University http://users.humboldt.edu/flashman/Presentations/MD.ICTCM.CV.3_12_16.html Abstract: Mapping diagrams provide a valuable tool for visualizing complex variable functions. Using GeoGebra, demonstrated dynamic diagrams will allow users to see important properties of complex functions. Examples include complex arithmetic, key functions, and calculus concepts. Background and References to other work on Mapping Diagrams for real variables ¤ 1.1.Background: Mapping Diagrams for Real Functions 1.1.1 Real Functions. ¤ Understanding Real Functions: Tables, Mapping Diagrams, and Graphs 1.2. Real Linear Functions. ¤ Real Linear functions are the key to understanding calculus. Linear functions are traditionally expressed by an equation like : $f(x)= mx + b$. Mapping diagrams for real linear functions have one simple unifying feature- the focus point, determined by the numbers $m$ and $b$, denoted here by $[m,b]$ . Mapping Diagrams and Graphs of Real Linear Functions Visualizing real linear functions using mapping diagrams and graphs. This is a Java Applet created using GeoGebra from www.geogebra.org - it looks like you don't have Java installed, please go to www.java.com Notice how points on the graph pair with arrows and points on the mapping diagram. 2.1 A Review of Complex Numbers and Arithmetic: $\mathbb{C}$¤ Definition: The "number" $i$: To solve the equation $x^2 +1 =0$ we create a symbol $i$ and declare that $i^2 = -1$ so $i$ solves the equation and we write $$i = \sqrt{ -1 }.$$ Definitions: A complex number $z$ is a number that can be expressed in the form $z =a +bi$ where $a$ and $b$ are real numbers. If $z =a +bi$ then $a$ is called the real part of $z$ and $b$ (and sometime $bi$) is called the imaginary part of $z$. Examples: $3 + 2i$ and $-2 - i$. 2.2 Complex Geometry- The Complex Plane Complex numbers are identified with points in a cartesian plane by having $a +bi$ identified with the point with coordinates $(a,b)$ or with position vectors by identifying $a+bi$ with the vector $$. With this identification i is identified with the point (0,1) and -i is identified with (0,-1). 2.2.1 Complex Number Norm (Magnitude): The norm of z is defined by |z| = |a+bi| = \sqrt{ a^2 +b^2} 2.2.2 Polar Representation of z: ¤ Using trigonometry we have the identification:$$z = |z| \cos( \theta) + |z| \sin(\theta) i = |z| [\cos( \theta) + \sin(\theta) i] = |z| cis( \theta) $$where a = \cos( \theta ), b = \sin(\theta). The angle \theta determined by z can be measured in degrees or radians and restricted to be in a specific interval. For example \theta \in [0,2 \pi) or \theta \in (-\pi, \pi]. Thus the angle can be considered a function of z, called the argument of z: Arg(z) =\theta. 2.2.3 Exponential Representation of z: ¤ Consider the Taylor Series for the functions:$$e^x = 1 + x + \frac{x^2}2 +\frac{x^3}{3!} + \frac{x^4}{4!} + \frac{x^5}{5!} + \frac{x^6}{6!} + \frac{x^7}{7!} ...\cos(x) = 1 - \frac{x^2}2 + \frac{x^4}{4!} - \frac{x^6}{6!} + ...\sin(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} ...$$Then using \theta in radian measure:$$e^{i\theta}= \cos(\theta) + \sin(\theta)i = cis (\theta)$$and$$z =|z|e^{i \theta}$$. Note: When \theta =\pi this equation demonstrates that e^{\pi i} = \cos(\pi) + sin(\pi)i = -1. 2.3 Complex Arithmetic 2.3.1 Complex Addition: z_1+z_2 ¤ If z_1 = a_1 + b_1i and z_2 = a_2 + b_2i then$$z_1+z_2 =a_1 + a_2 + (b_1+ b_2) i Example: If $z_1 = 2+3i; z_2= 1 - i$ then $z_1+z_2= (2+1) + (3-1) i = 3 + 2i$. The addition can be thought of as vector addition - with separate addition of the real and imaginary parts. Addition can be visualized in the complex plane by using parallelograms. 2.3.2 Complex Multiplication: $z_1 \cdot z_2$ ¤ Algebraically:If $z_1 = 2+3i; z_2= 1 - i$ then $z_1 \cdot z_2= (2+3i) \cdot (1-i) = 2 \cdot 1 - 3\cdot i^2 + 3i \cdot 1 +2 \cdot (-i) = (2+3) +(3- 2)i = 5+i$. This is easier to understand complex multiplication geometrically using the polar or the exponential representation (in radian measure): $z_1=|z_1| [\cos( \theta_1) + \sin(\theta_1) i] = |z_1|e^{i\theta_1} ;\ z_2=\cdot |z_1| [\cos( \theta_2) + \sin(\theta_2) i] = |z_2|e^{i\theta_2}$ $z_1 \cdot z_2= |z_1| [\cos( \theta_1) + \sin(\theta_1) i] \cdot |z_1| [\cos( \theta_2) + \sin(\theta_2) i]$ $= |z_1|\cdot |z_1| [\cos( \theta_1) + \sin(\theta_1) i] \cdot [\cos( \theta_2) + \sin(\theta_2) i]$ $= |z_1|\cdot |z_1| [\cos( \theta_1)\cos( \theta_2) - \sin(\theta_1)\sin(\theta_2) + (\sin(\theta_1)\cos( \theta_2) + \sin(\theta_2) \cos(\theta_1) )i]$ $=|z_1|\cdot |z_1| [\cos( \theta_1 + \theta_2) + \sin(\theta_1 +\theta_2) i]$ $=|z_1|\cdot |z_1| cis (\theta_1 +\theta_2)$ or more simply using $\theta$ in radian measure: $z_1 \cdot z_2= |z_1|e^{i\theta_1}\cdot |z_2|e^{i\theta_2}$ $=|z_1|\cdot |z_1| e^{(\theta_1 +\theta_2)i}$ Example: If $z_1 = \sqrt 2\ cis(45°); z_2= 2\ cis(30°)$ then $z_1\cdot z_2= 2\sqrt 2\ cis (75 °)$.¤ 2.3.3 Complex Inverses: $\frac1z, z \ne 0$ ¤ Example: Express $\frac 1{2+3i}$ in the standard form of $a + bi$. Solution: $\frac 1{2+i} =\frac 1{2+i} \cdot \frac {2-i}{2-i} = \frac{2-i}{4+1} = \frac 2{5} - \frac 3{5}i$. Fact: If $a^2+b^2 \ne 0$ then $\frac1{a+bi} = \frac a{a^2 + b^2} - \frac b{a^2 + b^2}i$ 2.3.4 Geometry for $\frac1z, z \ne 0$ Geometric understanding of $\frac1z, z \ne 0$ comes best by using the polar or exponential representation. That is, if $z=|z|\ cis(\theta) = |z|e^{i\theta}$ then $\frac1z = \frac1{|z|} \ cis(-\theta) = \frac1{|z|} e^{-i\theta}$ On to Part II Complex Functions References:¤ Mapping Diagrams from A(lgebra) B(asics) to C(alculus) and D(ifferential) E(quation)s. A Reference and Resource Book on Function Visualizations Using Mapping Diagrams The Sensible Calculus Program

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# 157.8 lbs to kg – How to Convert Pounds to Kilograms Source: bing.com Have you ever encountered the measurement of 157.8 pounds (lbs) and wanted to know its equivalent in kilograms (kg)? As different countries and industries use varying units of measurement, it is common to need to convert between units. In this article, we will discuss the conversion of 157.8 lbs to kg and provide insight into the usage and history of these units of measurement. ## Understanding Pounds and Kilograms Source: bing.com Pounds and kilograms are both units of mass that are used to measure the weight of an object. The pound is a unit of mass commonly used in the United States and a few other countries, whereas the kilogram is the standard measurement of mass in most other countries around the world. A pound is defined as 0.45359237 kilograms, whereas a kilogram is defined as the mass of a particular cylinder of platinum-iridium alloy kept at the International Bureau of Weights and Measures. ## Conversion of 157.8 lbs to kg Source: bing.com To convert 157.8 lbs to kg, we need to multiply the amount of pounds by the conversion factor of 0.45359237. Therefore, 157.8 lbs is equivalent to: 157.8 x 0.45359237 = 71.578 kg This means that 157.8 lbs is equal to 71.578 kg. To convert between pounds and kilograms, simply multiply the amount of pounds by 0.45359237 or divide the amount of kilograms by 2.20462. ## History of Pounds and Kilograms Source: bing.com The pound has a long history and was first used in ancient Rome. Originally, the pound was defined as 12 ounces, with each ounce being the weight of 437.5 grains of barley. This definition has since changed, with the pound now defined as 0.45359237 kg. The kilogram, on the other hand, was first defined in the late 18th century during the French Revolution. The French Academy of Sciences commissioned a group to create a new system of measurement that was based on natural standards. They defined the kilogram as the mass of 1 liter of water at its maximum density, which is 4°C. This definition has since been revised to be the mass of a particular cylinder of platinum-iridium alloy kept at the International Bureau of Weights and Measures. ## Usage of Pounds and Kilograms Source: bing.com As mentioned earlier, pounds are commonly used in the United States, while kilograms are the standard measurement of mass in most other countries. Industries such as agriculture, manufacturing, and automotive use these units of measurement for various purposes such as weighing products, determining shipping costs, and measuring the weight of vehicles. However, with the increasing globalization of trade and the internet, it is becoming more important to be able to convert between units of measurement to communicate effectively. Converting 157.8 lbs to kg is just one example of how the ability to convert between units of measurement is essential in modern life. ## Conclusion Converting 157.8 lbs to kg is a simple process that involves multiplying the amount of pounds by the conversion factor of 0.45359237. However, understanding the history and usage of pounds and kilograms provides insight into the reasons why these units of measurement are still used today. Whether for business, travel, or personal reasons, the ability to convert between units of measurement is a valuable skill that can be used in many different areas of life.

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https://magedkamel.com/18-effective-area-for-a-built-up-section

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# 18-The Effective Area for a Built-Up Section. ## The Effective Area for a Built-Up Section. This post will check a practice problem for a built-up section, problem no.3-19 quoted from Prof. Mccormack’s book, fifth edition. It is required to compute the effective net area of the built-up section shown in Fig 3-19. The holes are punched for 3/4 inches. The U factor is given as equal to 0.90. ### The gross area for a Built-Up Section. The built-up section consists of two channels of C10x25 and two plates that have a dimension of 1/2×11 inches. The thickness of each plate is 1/2 inch and the length is 11 inches. We will estimate these four elements, since we have two channels we can estimate the gross area for one channel and multiply by two. Similarly, we can find the area of one plate and then multiply it by 2. From the table of areas for channels Table 1-5, we can find that for section C10x25 its area is equal to 7.35 inch^2. For a plate, it is easy to estimate, the area of a rectangle of (1/2″x11″), the area of one plate is equal to the product of (1/2×11) the result is 5.50 inch^2. The areas of the two channels are equal to (2×7.35)=14.70 inch^2. While the area of the two plates equals (2*5.50)=11.0 inch^2. The sum of areas will give us a total area of 14.70 inch^2, please refer to the next image for more details. The bolts used are of diameter 3/4 inches, to get the hole diameter we add 1/8 inch to the bolt diameter so the diameter of the hole will be equal to 7/8 inches. ### The net area for a Built-Up Section. For the Effective Area for a Built-Up Section, we need to find the net areas for the two channels and the net area for the two plates. 1-for the net area for plates use section (1-1). We have 4 bolts, and the area of these four bolts is equal to the hole diameter by the number of bolts multiplied by the thickness of the plate, this value will be deducted from the gross area of the plates. The same section 1-1, we can use to get the net area of the two channels, again we have four bolts connecting the flanges of the two channels to the upper and lower plates. We estimate the area of the holes by multiplying 4x the diameter of the holes by the thickness of the flange of the C channel. The final net area of the built-up section equals 22.42 square inches. The net area for channels is 13.74 square inches. The net area for the two plates is 9.25 square inches. Please refer to the next slide image for more details. ### The Effective Area for a Built-Up Section. The final value for the effective area for a built-up section will be equal to U*An, U is given as equal to 0.90, then the effective area of a built-up section will be equal to the product of (UxAn) or 0.9x(13.174+9.25)=20.18 inch^2. For staggered connection, please refer to posts 16 and 17 for the details of a solved problem Problem 3-4-3 for MC section-staggered bolts.

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https://educationexpert.net/mathematics/1651956.html

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15 October, 08:44 # What is the answer to 7 (3+x+4) +1 1. 15 October, 09:07 0 Distribute the 7. =21+7x+28 Simplify by combining any terms that you can. = 49+7x To solve for X you need to get the variable alone. Start to do this by moving the 49. Subtract 49, what you do one side you have to also do to the other. So you have - 49=7x now to get the X alone divide both sides by seven. -7=x 2. 15 October, 10:02 0 7 (3+x+4) 21 + 7x + 28 7x + 49

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# wikiHow to Checkmate in 3 Moves in Chess You know the 2-move checkmate, or Fool's Mate, and you know the 4-move checkmate, or Scholar's Mate, but do you know the 3-move checkmate? Grab a friend, play white, and your next game of chess will take longer to set up than to play. You can achieve checkmate in three moves with capturing, or without capturing. For either of these methods to work requires some pretty bad play from your opponent, but maybe you can catch her cold at the start. ### Method 1 Getting Checkmate in Three Moves while Capturing 1. 1 Move your King Pawn forward to e4. In both of these methods the key piece for you is your Queen. The Queen is the piece that you are going to use to achieve the checkmate, so your first move should be to open up space for the Queen to move diagonally. Moving the King Pawn forward two spaces to square e4 achieves this (e4). • If you're unfamiliar with algebraic chess notation, check out the wikiHow article to brush up. • As well as freeing your queen, you need your opponent to expose their king. If black then moves their bishop pawn 2 spaces to f5 to tempt white, the checkmate in three moves is on! 2. 2 Capture your opponent's Pawn at f5. Now use your Pawn to capture your opponent's advanced Pawn by attacking on the diagonal. Notated, that's e4xf5. Here you are trying to encourage your opponent to move their Knight Pawn forward two spaces to g5, so it is alongside your Pawn. • This isn't a smart move from your opponent, but maybe you can lull her into it. • The idea of this move is to make sure nothing can block off your route to your opponent's King after you make your next move. 3. 3 Move your White Queen to h5 (Qh5). Checkmate! Now you can move your Queen on the diagonal to h5 and you have your opponents King pinned. That's game over! You'll notice that if your opponent hadn't moved their Pawn forward two in their last turn they could have blocked off your Queen by putting a pawn in her way by g6. • You really need your opponent to play into your hands to pull off this three-move checkmate. 4. 4 Call out checkmate! Now you can take the King with your Queen on the diagonal and celebrate a very swift victory. If your opponent has fallen into the trap they will likely be a bit annoyed, so don't gloat too much! ### Method 2 Getting Checkmate in Three Moves Without Capturing 1. 1 Move your King Pawn to d3. This is a very similar method to the previous one. You are basically aiming to get your opponent's Bishop and Knight Pawns forward one and two squares respectively, while freeing your Queen to enable it to move onto h5. The end result is the same as the previous method. • You are trying to tempt your opponent to move her Bishop and Knight Pawns. • You need you opponent to respond by bringing out her Bishop Pawn one square to f6. • It can also work if she moves her Knight Pawn forward two squares on this turn, as long as she moves the Bishop Pawn on her next move. 2. 2 Move your Queen Pawn forward to e4. The next move for you to make has to free up your Queen so it can get into a checkmate position on the next move. To do this, move the White King Pawn ahead two squares to e4. Now you have opened up an avenue for your Queen to reach h5. • In order to clear the way to your opponent's King you need her to move her Knight Pawn ahead two spaces to g5. 3. 3 Move the White Queen to h5 (Qh5). Checkmate! And that's it, you have trapped your opponent's King in the same position as the previous method, but this time you did it without even capturing a single piece. Game. Set. Match. Over.[1] • Again, this looks simple and it is. So don't expect it to work very often! • In theory, there are loads of variations on this. The key moves are getting your Queen to h5, and your opponent's Bishop and Knight Pawns out of the way of her King. ## Community Q&A Search • Can my opponent castle to get out of checkmate when my queen is on Qh5? wikiHow Contributor No, there are pieces in the way! It is also illegal to castle out of check. • What should I do if it doesn't work? wikiHow Contributor You can play another opening. It won't always work and it usually only works with players that are absolute beginners in the game. • When can my king swap with my rook? wikiHow Contributor Here are the conditions: when your bishop and knight are not in between your rook and king, your king and rook have not moved yet, there are no pieces attacking the space between your rook and king, and when doing so will not result in check. • What can I do if the opposite player doesn't move as I wish? wikiHow Contributor If the opponent doesn't make the moves that allow you to checkmate him/her in three moves, play another opening. It won't always work and it usually only works with players who are absolute beginners in the game. • How do I move the king? wikiHow Contributor You can move your king in any direction, just like the queen, but only one square at a time. Be careful where you move your king, however; the game is over if your opponent takes your king. • Can we revive pieces? wikiHow Contributor Some people adopt rules allowing players to recover captured pieces, but normally you can revive a captured piece only by promoting a pawn. • What is a checkmate? wikiHow Contributor You achieve checkmate (and win the game) by placing your opponent's king in check in such a way that the opponent cannot escape check in his/her next move. • How should I position the king? wikiHow Contributor For defensive purposes, the king is often "castled," which is explained in How to Castle in Chess. • Isn't there a mistake here? The first method has the king and queen on different squares to the second method. All the drawings above are correct. The king (with the cross on top) starts on a square of the opposite color. The queen starts on the matching-color square. • Is there such thing as a 1 move checkmate? wikiHow Contributor No, there isn't. The way these fast checkmates work is by moving through the kingside white square diagonal (comprising squares h5, g6, f7, and e8). In order to free up this diagonal, two pawn moves must occur. • What is check! N where should I move my piece to? How to check my opponent's piece? Not checkmate 200 characters left ## Warnings • For his to work you need an opponent who is either very cooperative, or perhaps not quite awake. • Be wary of trying this in a more serious game, as it not likely to come off. If they don't play right into your hand, the 3-move checkmate won't work. ## Things You'll Need • Chess board and pieces • Cooperative opponent

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back start next [start] [1] [2] [3] [4] [5] [6] [7] [8] [ 9 ] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47] [48] [49] [50] [51] [52] [53] [54] [55] [56] [57] 9 the effect of inflation which reduces the purchasing power of money over time. Inflation-adjusted calculations will be discussed in Section 9.4. As a consequence of the above principle, it is obvious that two or more amounts of money payable at different points in time cannot be compared until all the amounts are accumulated or discounted to a common date. This common date is called the comparison date, and the equation which accumulates or discounts each payment to the comparison date is called the equation of value. One device which is often helpful in the solution of equations of value is the time diagram. A time diagram is a one-dimensional diagram in which units of time are measured along the one dimension and payments are placed on the diagram at the appropriate points. Note that payments in one direction are placed on the top of the diagram and payments in the other direction are placed on the bottom of the diagram. The comparison date is denoted by an arrow. Figure 2.1 is an example of a time diagram used in the solution of Example 2.4. The time diagram is not necessary in the solution of equations of value; it IS merely an aid in visualizing the problem. With some practice the reader can usually dispense with a time diagram on simpler problems. However, time diagrams are usually helpful in the solution of more complex problems. One of the properties of compound interest is that the choice of the comparison date makes no difference in the answer obtained. Thus, there is a different equation of value for each comparison date, but they all produce the same answer. This important property of compound interest will be illustrated in the solution of Example 2.4. The reader is cautioned that under other patterns of interest, e.g. simple interest or simple discount, the choice of a comparison date does affect the answer obtained. This illustrates once again the inherent inconsistency in using simple interest or simple discount. t;. The reader should be aware that the problems involving accumulated values and present values already considered in the first two chapters are examples of equations of value. Example 2.4 illustrates a more general type of problem. Example 2.4 In return for a promise to receive \$600 at the end of 8 fears, a person agrees to pay \$100 at once, \$200 at the end of 5 years, and to make a further payment at ffie end of 10 years. Find the payment at the end of 10 years if the nominal rate of interest is 8% convertible semiannually. We shall first work the problem with a comparison date of the present. The time diagram is shown in Figure 2.1. wording of a problem rrny be different depending upon the point of view. Examples and exercises phrased from both points of view appear, and the reader should not let the different phraseology be a source of confusion. As an example, recall the discussion in Section 1.7 involving the use of the words "paid" or "credited." To some readers the word "paid" may seem more normal from the vantage point of the borrower, while "credited" may seem more normal from the vantage point of the lender. Many other such examples could be cited - Complex financial transactions often involve more than two parties. For example, a business firm analyzing its rate of return on a major investment in a new plant is involved with a multiplicity of parties. However, the basic principles developed to analyze two-party transactions can readily be extended to analyze these more complex transactions. d) In practical applications involving interest the terminology can become confusing. Many terms have ambiguous meanings (e.g. see the discussion on the use of the word "discount" in Section 1.7). Furthermore, as we shall see in succeeding chapters some terms used unfortunately do not convey an intuitive description of the transactions involved (e.g. see the discussion on the terms "annuity-immediate" and "annuity-due" in Section 3.3). Finally, many parties involved in financial transactions simply do not always use terms with the precise meanings and definitions contained in this book. The reader is admonished in real-world applications to look beyond the stated terms and be certain to understand the exact nature of the financial transactions in question. There simply is not total consistency in terminology among the large and diverse number of parties involved in financial transactions involving interest. 2.5 EQUATIONS OF VALUE It is a fundamental principle in the theory of interest that the value of an amount of money at any given point in time depends upon the time elapsed since the money was paid in the past or upon the time which will elapse in the fiiture before it is paid. We have already seen this in many of the examples and exercises considered thus far in the first two chapters. This principle is often characterized as the recognition of the time value of money. This process would be in contrast to financial calculations not involving the effect of interest, in which case it would be said that such calculations do not recognize the time value of money. The reader is cautioned that "recognition of the time value of money" reflects the effect of interest, but not 8 9 10 600 Figure 2.1 Time Diagram for Example 2.4 Since interest is convertible semiannually, we will count time periods in half-years. The equation of value is 100 + 200v "> + Xvo = 600v 1 at 4% 600v*- 100 - 200v< 600(.53391) - 100 - 200(.67556) .45639 = \$186.76. We could also have chosen a different comparison date and obtained a different equation of value. For example, if the comparison date were chosen to be the end of the 10th year, then the arrow in the time diagram would be under 10 and the equation of value would be 100(1.04)20 + 200(1.04)° + X = 600(1.04)" X = 600(1.04)" - 100(1.04)2° - 200(1.04)° = 600(1.16986)- 100(2.19112)-200(1.48024) = \$186.76. Thus, the same answer is obtained. The two equations of value are equivalent. If both sides of the first one are multiplied by (1.04)°, the second one is obtained. The reader can verify that if other comparison dates are chosen, the same answer is obtained. 2.6 UNKNOWN TIME As discussed in Section 2.4, if any three of the four basic quantifies entering into an interest problem are given, then the fourth can be determined. In this section we consider the situation in which the length of the investment period is the unknown. The best method of solving for unknown time involving a single payment is to use logarithms. This technique will be illustrated in Example 2.5. As indicated above in Section 2.4, we are assuming the availability of a pocket calculator with exponential and logarithmic functions. An alternative approach with less accuracy that can be used if such a calculator is unavailable is Unear inte olation in the interest tables. This technique is also illustrated in Example 2.5. A situation sometimes arises in which several payments made at various points in time are to be replaced by one payment numerically equal to the sum This approximation is denoted by t and is often called using the method of equated time. It is possible to prove that the value of t is always greater than the true value of t, or, alternatively, that the present value using the method of equated time is smaller than the true present value. Consider quantities each equal to v 2 quantities each equal to v, and so forth until there are s„ quantities each equal to v". The arithmetic mean of these quantities is Sjv + S2V + 5j + 2 + The geometric mean of these quantities is However, we know that the arithmetic mean of n positive numbers, not all of „ which are equal, is greater than the geometric mean, and thus we have ijv> + S2v + • • • + s/" 5j + 2 + + 5„ of the other payments. The problem is to find the point in time at which the one payment should be made such that it is equivalent in value to the payments made separately. Let amounts s,S2, . . ,s„ be paid at times fj, fj. • • • . respectively. The problem is to find time t, such tiiat + \$2 + • • • + paid at time t is equivalent to tiie payments of , • • . *n made separately. The flindamental equation of value is (S, + :?2 + • • • + n> = + 2 + • • • + V" - which is one equation in one unknown t. As an exercise, the reader will be Risked to find an exact expression for t. As a first approximation, t is often calculated as a weighted average of the various times of payment, where the weights are the various amounts paid, i.e. llflfL = k=X (2.9) The left-hand side is the true present value which exceeds the present value given by the method of equated time on the right-hand side. Thus, the value of t from formula (2.9) is always greater than the true value of t from formula (2.8). The method of equated time is useful in analyzing the average length of financial transactions. This application will be discussed further in Section 9.8. Another interesting question often asked is how long it takes money to double at a given rate of interest. We can analyze this problem as follows: giving (1 + 0" = 2 n loggd + 0 = logg2 It is possible to derive an approximation to the exact result given in formula (2.10) as follows: log, 2 loged + 0 .6931 log,(l + 0 The second factor evaluated for i = 8% is 1.0395. Thus we have (2.11) This is frequently called the rule of 72, since it can be applied immediately by dividing 72 by the rate of interest expressed as a percentage (i.e. as 100/). The rule of 72 produces surprisingly accurate results over a wide range of interest rates. Illustrative values are provided in Table 2.1. Table 2.1 Length of Time It Takes Money to Double Rate of interest Rule of 72 Exact value 17.67 11.90 9.01 7.27 6.12 4.19 Example 2.5 Find the length of time necessary for \$1000 to accumulate to \$1500 if invested at 6% per annum compounded semiannually: (1) by use of logarithms, and (2) by interpolating in the interest tables. Let n be the number of half-years. The equation of value is 1000(1.03)" = 1500 (1.03)" =1.5. 1. Using logarithms n log J.03 = logl.5 log e 1-5 .405465 log, 1.03 Thus, the number of years is 6.859. .029559 = 13.717. 2. From the interest tables, (1.03)" = 1.46853 and (1.03)" = 1.51259, so that 14 13 < n < 14. Performing a linear interpolation 1.50000 - 1.46853 n = 13 -I- 1.51259 - 1.46853 = 13.714. Thus, the number of years is 6.857. This method is equivalent to the assumption of simple interest during the final fraction of an interest conversion period. This point was discussed in more detail in Section 2.2. Example 2.6 Payments of \$100, \$200, and \$500 are due at the ends of years 2, 3, and 8, respectively. Assuming an effective rate of interest of 5% per annum, find the point in time at which a payment of \$800 would be equivalent: (1) by the method of equated time, and (2) by an exact method. 1. By the method of equated time using formula (2.9) 100 • 2 200 • 3 -I- 500 100 -h 200 -I- 500 2. The exact equation of value is 800v = lOOv + 200v + 500v = 6 years. [start] [1] [2] [3] [4] [5] [6] [7] [8] [ 9 ] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47] [48] [49] [50] [51] [52] [53] [54] [55] [56] [57]

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# Math PLZ HELP posted by . 1. Which of the following expressions is written in scientific notation? 1. 73.4 x 105 2. 0.09 × 107 3. 80 x 103 4. 4.22 x 10–3 2. Which of the following is 0.0000000708 written in scientific notation? 1. 7.08 x 10–8 2. 7.8 x 10–8 3. 708 x 10–10 4. 70.08 x 10–9 3. Which expression represents the largest number? 1. 40.1 x 10–6 2. 4.1 x 10–7 3. 0.411 x 10–7 4. 0.04001 x 10–5 4. Which expression is equal to 1/8000 written in scientific notation? 1. 8.0 x 103 2. 1.25 x 10–4 3. 125 x 10–5 4. 1.25 x 104 • Math PLZ HELP - if you know anything about scientific notation, #1,2 should be easy. For #3, convert all to same power of 10, then it is easy to compare. #4: 1/8000 = .000125 • Math PLZ HELP - 1.)A 2.)C 3.)B 4.)B ## Similar Questions 1. ### Math Use scientific notation to divide the following two numbers. Express the answer using scientific notation; retain at least three decimal places. 6.2 • 107 / 4.15 • 103 11. 17, 200 written in scientific notation is 1.7 x 10^5 21. 0.00105 written in scientific notation is 1.05 x 10^-3 are these correct? 3. ### Math Which of the following expressions is written in scientific notation? 4. ### Math Which of the following is 0.0000000708 written in scientific notation? 5. ### math 1. Which of the following expressions is written in scientific notation? 6. ### Math! Please Check My Awnser! Hello! I need some help with math! I have tried to do this but I am unsure about my answers. My I will identify my answers with a "<--". Also, if I have missed one, please tell me the right answer! Also, Thanks for your help in … 7. ### Homework help plz( Steve or Reiny) Determine if the number is written in scientific notation. If not, explain 32 * 10^4. (1 point) No; it is not written as a power of 10. No; the first factor is not a number between 1 and 10. Yes; the number is written in scientific … 8. ### Math Hi everyone! I am extremely struggling with this questions and was hoping someone would go over them with me. Thank you! 1. Which of the following expressions is written in scientific notation? 9. ### Algebra Which of the following expressions is written in scientific notation? 10. ### Algebra Which of the following is 0.0000000708 written in scientific notation? More Similar Questions

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Actuarial Outpost Fall 2017 MLC Progress Thread Register Blogs Wiki FAQ Calendar Search Today's Posts Mark Forums Read FlashChat Actuarial Discussion Preliminary Exams CAS/SOA Exams Cyberchat Around the World Suggestions #451 Yesterday, 07:06 PM Jim Daniel Member SOA Join Date: Jan 2002 Location: Davis, CA College: Wabash College B.A. 1962, Stanford Ph.D. 1965 Posts: 2,558 Quote: Originally Posted by PerpetualMotion Can someone ELI5 the differences in these variances from Fall 2014 (ii) Drug A will be given to 1000 subjects comprising Cohort A. • With probability 80%, each subject in Cohort A will have q = 0.20 . • With probability 20%, each subject in Cohort A will have q = 0.05 . • Conditional on q, the lives have independent future lifetimes. ** I understand that everyone in the group will experience whichever q is chosen so we use (un?)conditional variance formula. I don't really know why this is though. (iii) Drug B will be given to 1000 subjects with independent future lifetimes, comprising Cohort B. The drug will affect different subjects differently and independently. • With probability 80%, any given subject in Cohort B will have q = 0.20 . • With probability 20%, any given subject in Cohort B will have q = 0.05 . ** This one you end up finding E(Q) and conditional variance of Q because any given subject can experience different Qs. I really just don't understand this topic, so if anyone can break it down I'd really appreciate it. Or if someone can point me to TIA videos or other online resources. This is a major flaw in my understanding of variance and I'm really trying to understand. ii) says there is an 80% probability that you have 1000 trials with q = 0.2 [so a Binomial RV with m=1000 and q = 0.2] and a 20% probability that you have 1000 trials with q = 0.05 [so a Binomial RV with m=1000 and q = 0.05]. Thus the distribution is an 80/20 mixture of these two Binomial RV's. Get the mean and second moment by mixing and then get the variance. [My website has a free study note on mixture distributions.] iii) says that each individual has an independent 80% probability of its q being 0.2 and a 20% probability of its q being 0.05. Thus, by mixing, each individual has a probability 0.8 (0.2) + (0.2) (0.05) =0.17 of death, and there are 1000 such individuals, so the number of deaths is Binomial with parameters 1000 and 0.17. So the mean and variance just come from this Binomial. __________________ Jim Daniel Jim Daniel's Actuarial Seminars www.actuarialseminars.com [email protected] #452 Yesterday, 07:15 PM PerpetualMotion Member SOA Join Date: Jan 2013 Location: Kentucky(Louisville) Studying for Exam MLC College: University of Louisville Alumni Favorite beer: Zombie Dust Posts: 325 Wow, that actually makes perfect sense. I'm going to check out that study note. I really appreciate it, I just know there's going to be a question like this on the exam __________________ P,FM MFE , C, MLC Modules 1-8 IA FA And NUH is the letter I use to spell Nutches Who live in small caves, known as Nitches, for hutches, These Nutches have troubles, the biggest of which is the fact there are many more Nutches than Nitches. #453 Yesterday, 07:37 PM Steveo1794 Member SOA Join Date: May 2016 Location: Ohio Studying for MLC College: Miami University (OH) Favorite beer: Anything Posts: 32 Quote: Originally Posted by PerpetualMotion Wow, that actually makes perfect sense. I'm going to check out that study note. I really appreciate it, I just know there's going to be a question like this on the exam Did this question last night... Confused the hell out of me but once I really thought about it, it made sense. I think about it like this: The scenario in b(ii) has five total groups of 1000 each. In four of those five groups, everyone has q=.20. In the other one, everyone has q=.05. The scenario in b(iii) is one group of 1000. 80% of the 1000 have q=.20 and the other 20% has q=.05. Hope this helps. --- Side note - Are we allowed to use pens on the written answer? Obviously not on the MC, but the WA? Last edited by Steveo1794; Yesterday at 07:40 PM.. #454 Yesterday, 07:42 PM Jim Daniel Member SOA Join Date: Jan 2002 Location: Davis, CA College: Wabash College B.A. 1962, Stanford Ph.D. 1965 Posts: 2,558 Both of Steveo1794's interpretations are incorrect. __________________ Jim Daniel Jim Daniel's Actuarial Seminars www.actuarialseminars.com [email protected] #455 Yesterday, 08:21 PM Steveo1794 Member SOA Join Date: May 2016 Location: Ohio Studying for MLC College: Miami University (OH) Favorite beer: Anything Posts: 32 I wasn't referring to the literal scenarios presented in the problem, I was referring to how to distinguish between the "each subject will have" and "any given subject". That phrasing confused me, and thinking about it like that helps me. Does the below explanation make sense? If not, please correct it. Scenario b(ii) is 80% chance that 100% of the cohort will have q=.20 and a 20% chance that 100% of the cohort will have q=.05. Scenario b(iii) is simply an 80% chance of a member of the cohort having q=.20 and a 20% chance of someone in the same cohort having q=.05. __________________ P FM MFE C MLC VEE FAP APC #456 Yesterday, 08:45 PM Jim Daniel Member SOA Join Date: Jan 2002 Location: Davis, CA College: Wabash College B.A. 1962, Stanford Ph.D. 1965 Posts: 2,558 Quote: Originally Posted by Steveo1794 I wasn't referring to the literal scenarios presented in the problem, I was referring to how to distinguish between the "each subject will have" and "any given subject". That phrasing confused me, and thinking about it like that helps me. Does the below explanation make sense? If not, please correct it. Scenario b(ii) is 80% chance that 100% of the cohort will have q=.20 and a 20% chance that 100% of the cohort will have q=.05. Scenario b(iii) is simply an 80% chance of a member of the cohort having q=.20 and a 20% chance of someone in the same cohort having q=.05. My problem with how you described it is the following: correct solutions based on the statements you gave yield incorrect answers for both cases. However, you may have interpreted your statements in an incorrect way that produced correct answers in boith cases. It's good to be lucky. Your explanation of b(ii) is correct. Your explanation of b(iii) may be correct, but the wording strikes me as a little confusing. It would be clearer to say: For each member of the cohort, there is an 80% chance that that member's q is 0.2 and a 20% chance that that member's q is 0.05, and the results are independent from member to member. By the way, I agree that the exam problem as stated is poorly worded, with the "any given subject" and "each subject" difficult to distinguish---and imagine what it's like for non-native English speakers! Of course, exam questions are often poorly worded, and it's not all that rare for them to use textbook terms to mean something different from their meaning in the textbook, or to be missing data, or for the wrong answer to be considered correct. Quality control truly sucks. Candidates need to be robust problem solvers in that the first part of the problem is to figure out what the problem really is. __________________ Jim Daniel Jim Daniel's Actuarial Seminars www.actuarialseminars.com [email protected] #457 Yesterday, 09:44 PM MySpaceTom SOA Join Date: Oct 2016 Posts: 7 The Written Answer instructions at the beginning of the exam state " Do not write answers to more than one question per sheet". Does this mean every part (a, b, and c) to question 1, must be on an individual sheet? Or, is it okay to answer parts a, b, and c all on the same sheet? #458 Yesterday, 10:38 PM PerpetualMotion Member SOA Join Date: Jan 2013 Location: Kentucky(Louisville) Studying for Exam MLC College: University of Louisville Alumni Favorite beer: Zombie Dust Posts: 325 Quote: Originally Posted by Jim Daniel By the way, I agree that the exam problem as stated is poorly worded, with the "any given subject" and "each subject" difficult to distinguish---and imagine what it's like for non-native English speakers! Of course, exam questions are often poorly worded, and it's not all that rare for them to use textbook terms to mean something different from their meaning in the textbook, or to be missing data, or for the wrong answer to be considered correct. Quality control truly sucks. Candidates need to be robust problem solvers in that the first part of the problem is to figure out what the problem really is. I'm the first to admit that life isn't fair but that really seems like bs from the soa standpoint. We put in hundreds of hours worth of work for each of these exams, they should be held to a higher standard. Some problems are unnecessarily complex and it just doesn't feel right. The further I progress, the more it all seems like a cash grab. I hope when I'm an FSA I get to write problems that actually tests relevant knowledge. Until then. __________________ P,FM MFE , C, MLC Modules 1-8 IA FA And NUH is the letter I use to spell Nutches Who live in small caves, known as Nitches, for hutches, These Nutches have troubles, the biggest of which is the fact there are many more Nutches than Nitches. #459 Today, 12:10 AM NoFunnyStuff Member SOA Join Date: Jul 2009 Location: Green Bay, WI Studying for MLC College: Georgia State University Alumni Posts: 164 Quote: Originally Posted by noonelikesmodules Doing a profit/gain question. Specifies surrender and mortality RATES. Doesn't specify about timing of surrenders. Do we always assume EoY surrenders? Most of the time yes, because if you have already paid for your insurance in BOY you would not surrender before the end of the year for something you have already paid for. Hope that this help. __________________ Prelim:P FM C MFE MLC VEE: FINANCE ECONOMICS STATISTICS Thread Tools Display Modes Linear Mode Posting Rules You may not post new threads You may not post replies You may not post attachments You may not edit your posts BB code is On Smilies are On [IMG] code is On HTML code is Off All times are GMT -4. The time now is 08:06 AM. -- Default Style - Fluid Width ---- Default Style - Fixed Width ---- Old Default Style ---- Easy on the eyes ---- Smooth Darkness ---- Chestnut ---- Apple-ish Style ---- If Apples were blue ---- If Apples were green ---- If Apples were purple ---- Halloween 2007 ---- B&W ---- Halloween ---- AO Christmas Theme ---- Turkey Day Theme ---- AO 2007 beta ---- 4th Of July Contact Us - Actuarial Outpost - Archive - Privacy Statement - Top

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# How To Measure For A Chandelier ### Allow 2 to 3 inches of chandelier length per foot of wall height. How to measure for a chandelier. A general rule of thumb is to choose a chandelier that is 10 12 inches narrower than your table. So if you have a 10 ft ceiling multiply that by 3 and you get 30 or 30 diameter. So if your room measures 10 x 14 the diameter of the fixture should be about 24. For example if your room is 10 x 16 the sum of those equals 26. One standard method for determining the right size for a chandelier is to add the length and width dimensions of the room together measured in feet. Measure your room s length and width in feet and add those two numbers together. How to measure a room for a chandelier. Then use that number as the width in inches for your chandelier. To find this number you need to take a couple of measurements. So for each foot it translates to two and a half to three inches for your chandelier height. To determine height of chandelier. The sparkle and dazzle of a well placed chandelier defines and enhances the mood of any room. Measure the width and length of the room. The chandelier should be 26 wide. How to measure a proper sized chandelier should be related to the size of the room and the size of the table over which it is to hang. Measure from the installation point in the ceiling all the way to the floor. Multiply the height by approximately 3 1 foot wall height to 3 chandelier height per foot measurement. A chandelier that is too large can easily overpower a room and. Knowing the height of a chandelier will help you choose a chandelier that s proportional to the height of your space. This is the diameter of the chandelier that will best suit the room. I got eight feet. For example a foyer 12 feet x 12 feet in size calls for a 24 inch diameter chandelier fixture. A simple way to determine a chandelier size is to add the dimensions of the room together in feet and then convert the answer to inches. Measure the ceiling height from the point of installation point where you plan to hang the fixture to the floor. Here are a few more quick guidelines to help.

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## Which expression is equivalent to 3(4m – 2) – 2(m + 5) A. 10m – 16 B. 14m + 4 C. -16m + 10 D Question Which expression is equivalent to 3(4m – 2) – 2(m + 5) A. 10m – 16 B. 14m + 4 C. -16m + 10 D. 12m – 6 in progress 0 1 year 2021-07-27T16:04:03+00:00 2 Answers 9 views 0 A. 10m – 16 Step-by-step explanation: First, distribute 3 to all terms within the corresponding parenthesis, and -2 likewise: 3(4m – 2) = 3 * 4m = 12m 3 * -2 = -6 3(4m – 2) = 12m – 6 -2(m + 5) = -2 * m = -2m -2 * 5 = -10 -2(m + 5) = -2m -10 Combine like terms: 12m – 6 -2m – 10 12m – 2m – 6 – 10 (12m – 2m) + (-6 – 10) 10m – 16 ~ 10 m – 16 (Answer A) Step-by-step explanation: We use distributive property to eliminate parenthesis: 12 m – 6 – 2 m -10 and now combine like terms: [12 m – 2 m] – 6 – 10 10 m – 16 Therefore, answer option A. is the correct one

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# Find length of segment in triangle In triangle $\bigtriangleup ABC$, the known sides are: $AB=5$, $BC=6$ and $AC=7$. A circle passes through points $A$ and $C$, crosses straight lines $BA$ and $BC$ at points $K$ and $L$, which is non-vertex angle, respectively. Segment KL As to the incircle of the triangle ABC. Please find the length of the segment $KL$, and show me how to. - A picture would help. – Gigili Jun 7 '12 at 6:59 Proofreading would help even more. Is $S$ the same as $C$? Crosses what straight line? What does it mean to say that $L$ is a "non-vertex angle"? What does "respectively" mean is this context? Sorry, this question makes no sense at all. Please think about what you really mwean to ask, and edit the question accordingly. – Gerry Myerson Jun 7 '12 at 7:15 Much better. Almost there. What does "Segment KL As to the incircle..." mean? – Gerry Myerson Jun 7 '12 at 11:01 1. For any $\triangle ABC$ in which A circle passes through points $A$ and $C$, crosses straight lines $BA$ and $BC$ at points $K$ and $L$, which is non-vertex angle, respectively we have that $\triangle ABC \sim \triangle BKL$. Indeed, $AKLC$ is inscribed quadrilateral $\Rightarrow \angle AKL+\angle ACL=\pi$, but also we have $\angle AKL+\angle BKL=\pi \Rightarrow \angle ACL=\angle BKL$; $\angle KBL = \angle ABC$. As $\triangle ABC \sim \triangle BKL \Rightarrow |BK|=6x, \ |BL|=5x, |KL|=7x$. So we should find $x$ from additional condition of this problem. 2. I guess, that by Segment KL As to the incircle of the triangle ABC TS means following: For inscribe circle of $\triangle ABC$ - $KL$ is tangent line. In this case we can write, that $|KL|+|AC|=|AK|+|LC| \ \mathbf{^{*)}} \Rightarrow$ $\Rightarrow 7x+7=(5-6x) + (6 - 5x)=11-11x \Rightarrow$ $\Rightarrow 18x=4 \Rightarrow x=\frac{2}{9}$. $|KL|=7x=\frac{14}{9}$ 3. $\mathbf{^{*)}}$ As $AKLC$ is tangent quadrilateral thats why |KL|+|AC|=|AK|+|LC|. (See picture). -

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# Areas and perimeters: math exercises in 5th grade corrected in PDF. Figure areas with corrected exercises in fifth grade are always essential to the student’s understanding. Thus, the latter will have to be able to convert quantities but also to calculate the area usual figures such as the square, the rectangle, the rhombus, the parallelogram, the trapezoid or the area of a disk of radius R. You will have many exercises and problems to solve on the areas of geometric figures to develop solid skills in calculus and geometry. These work materials are similar to those in your textbook. In addition, the correction allows students to identify errors made in calculating the area of a figure in order to fill in gaps in mathematics and progress throughout the school year in fifth grade. ## Series of exercises on areas and perimeters ### Exercise #1: 1. the intruder: On the grid below, six figures have been drawn. Knowing that the unit of area is the square, calculate the area of each of the 6 figures and find the intruder. 2. Calculate the area of the following figures: ### Exercise 2 areas and perimeters : Using the square as a unit of area, give the area of rectangle ABCD and then the area of each of the parallelograms. ### Exercise #3: The figure below is a parallelogram. 1° Calculate its area. 2° Calculate its perimeter. ### Exercise #4: Consider the parallelogram below. ( a and d denote the heights ) . Circle the products that express thearea of this parallelogram? a $\times$ d c $\times$ d b $\times$ d a $\times$ b a $\times$ c b $\times$ c ### Exercise 5 areas and perimeters : ABCD is a square of side 5 cm. The two half-discs have diameters [AB] and [AD]. Calculate the area, in cm², of the blue surface to the nearest hundredth. ### Exercise #6: Calculate the area of the orange surface. ### Exercise 7 areas and perimeters : A disc has a diameter of 10 cm. Using the key of the calculator, give an approximate value of its area, in cm² : a. to the nearest unit; b. to the nearest hundredth. ### Exercise #8: Calculate the perimeter of the polygon ABCDE below. ### Exercise #9: It took 73.20 m of rope to install the three ropes of this boxing ring. How long is the side of this square ring? ### Exercise #10: Calculate the areas of these triangles, then arrange these areas in ascending order. ### Exercise #11: The orange and green surfaces are delimited by semicircles with centers D, B and E. Also, AD = 1.5 cm. Calculate the area, in cm², of the green surface to the nearest hundredth. ### Exercise #12: In handball, the goal area consists of two quarter discs and a rectangle. Calculate the area, in m², of this goal area to the nearest hundredth. Cette publication est également disponible en : Français (French) العربية (Arabic) Télécharger puis imprimer cette fiche en PDF Télécharger ou imprimer cette fiche «areas and perimeters: math exercises in 5th grade corrected in PDF.» au format PDF afin de pouvoir travailler en totale autonomie. ## D'autres fiches dans la section 5th grade math exercises Télécharger nos applications gratuites Mathématiques Web avec tous les cours,exercices corrigés. ## D'autres articles analogues à areas and perimeters: math exercises in 5th grade corrected in PDF. • 97 Fractions with corrected 5th grade math exercises will be of great help. The student must know the definition and be able to graph the fraction of a figure or quantity and also know how to represent it on a graduated line. The goal is also to develop computational skills… • 97 This proctored assignment is intended for teachers or students who want to review the chapter on square roots in 10th grade. In order to do well, you will have to start by reading the statement in its entirety. This will allow you to have a good overview of the subject… • 97 The corrected math exercises on geometry in space in 5th grade will help you understand the chapter. The student should be able to perform volume conversions but also know by heart the formula for the volume of a cube, a rectangle, a right block and a cylinder. In addition, many… Les dernières fiches mises à jour Voici la liste des derniers cours et exercices ajoutés au site ou mis à jour et similaire à areas and perimeters: math exercises in 5th grade corrected in PDF. . Mathématiques Web c'est 2 145 968 fiches de cours et d'exercices téléchargées. Copyright © 2008 - 2023 Mathématiques Web Tous droits réservés | Mentions légales | Signaler une Erreur | Contact . Scroll to Top Mathématiques Web FREE VIEW

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https://brainly.com/question/358736

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2015-03-17T18:27:56-04:00 Because 3 times 1/6 is 3/6 which can be simplified to 1/2 2015-03-17T18:41:46-04:00 ### This Is a Certified Answer Certified answers contain reliable, trustworthy information vouched for by a hand-picked team of experts. Brainly has millions of high quality answers, all of them carefully moderated by our most trusted community members, but certified answers are the finest of the finest. Because 1/2 ≠ 1/6. We know that 1/6 < 1/2, so we can set up an equation to see how many copies are needed for them to be equal. (1/6)x = 1/2 [(1/6)x] × 6 = [1/2] × 6 x = 6/2 = 3 This equation shows that 1/6 × 3 = 1/2, therefore we need 3 copies of 1/6 to equal 1 copy of 1/2.

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https://math.answers.com/Q/How_do_you_work_the_prime_factor_problem

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0 # How do you work the prime factor problem? Updated: 8/18/2019 Wiki User 14y ago 1. Find out all the factors for the given number 2. Select the prime numbers out of them 3. Try to divide the number by those prime numbers and see if any combination is possible. Ex: 8 as a prime factor multiple = 2 * 2 * 2 Wiki User 14y ago Earn +20 pts Q: How do you work the prime factor problem? Submit Still have questions? Related questions ### What is the LCM with prime factorization using a factor tree using numbers 3 and 4? That's a lot of extra work for this problem but here goes. 3 is already prime so it doesn't really have a factor tree or prime factorization. The prime factorization of 4 is 2 x 2 which looks like this in a factor tree.42,23 and 4 have no common prime factors, so the LCM is their product, 12 ### Is a factor a prime number? a factor is a factor , a prime number is a prime number, but a prime factor is a factor which is a prime number. it has to depend on what number number. exp 13 is a prime number ### How do you show work for finding prime numbers? Use a factor tree. ### Is three a prime factor? Yes. Any time three is a factor, it is a prime factor. ### Is 6 a prime factor of 132? It is a factor of 132, not a prime factor. ### What is a common prime factor? All numbers have factors. Some factors are prime numbers. A prime factor is a factor that is a prime number. A common prime factor is a prime factor that appears on the list of factors of two or more given numbers. ### Is 19 factor or prime? 19 is a prime factor. ### Is 35 a prime or a factor? 35 can be a factor, but it is not prime. ### Do factor trees always work for determining the greatest common factors? If you construct them correctly, factor trees always work to determine the prime factorization of a number. Once you compare the prime factorizations of two or more numbers, it is relatively easy to find the greatest common factor of them from there. ### What is the prime factor of 37? There is no factorization. 37 is a prime number.37 is already prime. Its only prime factor is itself. ### Is 83 a prime factor or composite factor? 81 is not prime. Its prime factorization is 3x3x3x3. ### Is 160 a prime factor or composite factor? Is 19 a composite or prime factor

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Q: # What is the answer to a multiplication problem called? A: The solution to a multiplication problem is called the "product." For example, the product of 2 and 3 is 6. When the word "product" appears in a mathematical word problem, it is a sign that multiplication is necessary. ## Keep Learning Credit: Mayr CC-BY 2.0 Each of the four basic arithmetical operations has a special term indicating the result. For addition, the word is "sum," for subtraction it is "difference," and for division it is "quotient." The word product comes from the Latin word "producere," meaning to "bring forth" or "draw out." The word quotient comes from the Latin word "quotiens," meaning "how many times." The word sum is also from Latin and is short for "summus," meaning "highest." Sources: ## Related Questions • A: You can find printable multiplication charts online on MathWorksheets4kids.com. On the left-hand side of the homepage, click on "Multiplication" under the heading "Basic Topics." Then click on "Multiplication Tables and Charts." Filed Under: • A: A partial product multiplication algorithm is the process in which each part of one number is multiplied by each part of another number, after which the products are added together. An understanding of place value is necessary to use this algorithm, and the largest numbers are multiplied first. Filed Under: • A: To perform partial product multiplication, you use the distributive property of numbers, multiplying each digit of a number by each digit of the other number and adding the results while taking the place value of each digit into account. One of the most common types of multiplication problems involves a pair of two-digit numbers. When working with these problems, add a total of four partial products.

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# Center of Mass and Moment of Inertia ## Homework Statement Find the center of mass of the collection of mass points in Figure P.12 and then find the moment of inertia of the system about an axis through the center of mass and parallel to the y-axis. ## Homework Equations Center of Mass Moment of Inertia ## The Attempt at a Solution M1 = 1kg, r1 = (1i + 0j) m M2 = 2kg, r2 = (4i + 0j) m M3 = 3kg, r3 = (6i + 0j) m Mass total = 6kg X of CM = 1/6 (0 + 8 + 18) = 13/3 m Y of CM = 1/6 (0 + 0 + 0) = 0 m I = mR$$^{2}$$ I = 6 (4.333)$$^{2}$$ I = 112.6667 kgm$$^{2}$$ The answer was wrong. Thanks for the help #### Attachments • p10-12.gif 3 KB · Views: 369

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× Get Full Access to Discrete Mathematics And Its Applications - 7 Edition - Chapter 2.3 - Problem 79e Get Full Access to Discrete Mathematics And Its Applications - 7 Edition - Chapter 2.3 - Problem 79e × # a) Show that if a set S has cardinality m, where m is a ISBN: 9780073383095 37 ## Solution for problem 79E Chapter 2.3 Discrete Mathematics and Its Applications | 7th Edition • Textbook Solutions • 2901 Step-by-step solutions solved by professors and subject experts • Get 24/7 help from StudySoup virtual teaching assistants Discrete Mathematics and Its Applications | 7th Edition 4 5 1 339 Reviews 24 3 Problem 79E Problem 79E a) Show that if a set S has cardinality m, where m is a positive integer, then there is a one-to-one correspondence between S and the set {1. 2,...,m). b) Show that if S and T are two sets each with m elements. where m is a positive integer, then there is a one-to-one correspondence between S and T. Step-by-Step Solution: Step 1 of 3 J N'= d ot70 d2, d'rt ojto.q'd h_ q'q...q ol'- d *o fu yxv-tr^'' \-Jn larngE d bqSe ,' n Upo^mt /l- p,4,d., Yn*Lt =d \- _4_h dn-Y!^ 4n4 : &.r.d Y : /' vn -/ -Yni --]il ct'01 h-h d' d:d ,7- - I -l* dvo Q^=o , A. h.'o Uo - nof rdef iro{ , th b) = ,r*'orh...* , o{^'^=6)* ) nYJ'i*':l }J l^ G l-c )-e 11 Yx &*,U # d rohn3I b*1e** b {:;u*a.*{-wy \$e{s*];,c|*e&*rilfi',,*kc * =*nv i^1un*q b =n,amkrJ bo*rj*#{r*s {. b ^&\ t I^ n[r= 21 n{pi n&5= ,,nL'=' rf.) nr+j 'h[o) n lv10&ig d Ltnianv,wr+* d,*,rh tlrtdx hasw 4*-5 ou\ *,1d ,lo,.l 1 e"L,1lg%,,, T*\* n*wbw I Q: u'{dv\ ,\ (r,n^ 'lh+ b=+A) t "+-'&\ x ax I-X p* -5 t'I *"; +*' w*4r.utr1,tlo oW ) Step 2 of 3 Step 3 of 3 ##### ISBN: 9780073383095 Discrete Mathematics and Its Applications was written by and is associated to the ISBN: 9780073383095. Since the solution to 79E from 2.3 chapter was answered, more than 390 students have viewed the full step-by-step answer. This textbook survival guide was created for the textbook: Discrete Mathematics and Its Applications, edition: 7. The answer to “a) Show that if a set S has cardinality m, where m is a positive integer, then there is a one-to-one correspondence between S and the set {1. 2,...,m).________________b) Show that if S and T are two sets each with m elements. where m is a positive integer, then there is a one-to-one correspondence between S and T.” is broken down into a number of easy to follow steps, and 58 words. The full step-by-step solution to problem: 79E from chapter: 2.3 was answered by , our top Math solution expert on 06/21/17, 07:45AM. This full solution covers the following key subjects: Where, correspondence, show, Integer, set. This expansive textbook survival guide covers 101 chapters, and 4221 solutions. ## Discover and learn what students are asking Calculus: Early Transcendental Functions : The Natural Logarithmic Function: Integration ?In Exercises 1-26, find the indefinite integral. $$\int \frac{4 x^{3}+3}{x^{4}+3 x} d x$$ Statistics: Informed Decisions Using Data : Applications of the Normal Distribution ?If X is a normal random variable with mean 40 and standard deviation 10 and P(X < 38) = 0.4207, then P(X ? 38) = _________ . Statistics: Informed Decisions Using Data : The Normal Approximation to the Binomial Probability Distribution ?In a binomial experiment with n trials and probability of success p, if ________, the binomial random variable X is approximately normal with ?x= and Statistics: Informed Decisions Using Data : Inference about Measures of Central Tendency ?Write a paragraph that describes the logic of the test statistic in a right-tailed sign test. Statistics: Informed Decisions Using Data : Inference about the Difference between Two Medians: Dependent Samples ?Effects of Exercise To determine the effectiveness of an exercise regimen, a physical therapist randomly selects 10 women to participate in a study. S #### Related chapters Unlock Textbook Solution

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https://jeopardylabs.com/play/2nd-grade-math-review-495

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Addition Subtraction Word Problems What is the missing number? 100 42 + 67 = What is 109? 100 98 - 32 = What is 66? 100 Jake has 95 baseball cards. He gives 24 cards to his little sister. How many cards does Jake have left? What is 71 baseball cards? 100 8 - 2 = 9 - ___ What is 6 ? 200 58 + 32 = What is 90? 200 81 - 65 = What is 16? 200 Heidi is having a party. She needs to make 75 cookies. So far she has only made 56 cookies. How many more cookies does Heidi need to make? What is 19 cookies? 200 12 - ___ = 9 - 4 What is 5? 300 39 + 52 = What is 91? 300 75 - 49 = What is 26? 300 Mrs. Dallman has 37 books in her library. Miss Stern has 26 more books in her library. How many books does Miss Stern have in her library? What is 63 books? 300 19 - 8 = ___ - 10 What is 21? 400 45 + 38 = What is 83? 400 60 - 29 = What is 31? 400 1st grade has 46 students, 2nd grade has 30 more students. How many students does 2nd grade have? What is 76 students? 400 15 - ____ = 20 - 10 What is 5? 500 65 + 25 = What is 90? 500 82 - 47 = What is 35? 500 Jenna read some pages in her book on Sunday. She read 27 pages on Monday. Now she is on page 50 in her book. How many pages did she read on Sunday? What is 23 pages? 500 20 - 5 = ___ - 5 What is 15? Click to zoom

435

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## Conversion formula The conversion factor from years to seconds is 31556952, which means that 1 year is equal to 31556952 seconds: 1 yr = 31556952 s To convert 2936 years into seconds we have to multiply 2936 by the conversion factor in order to get the time amount from years to seconds. We can also form a simple proportion to calculate the result: 1 yr → 31556952 s 2936 yr → T(s) Solve the above proportion to obtain the time T in seconds: T(s) = 2936 yr × 31556952 s T(s) = 92651211072 s The final result is: 2936 yr → 92651211072 s We conclude that 2936 years is equivalent to 92651211072 seconds: 2936 years = 92651211072 seconds ## Alternative conversion We can also convert by utilizing the inverse value of the conversion factor. In this case 1 second is equal to 1.0793167066353E-11 × 2936 years. Another way is saying that 2936 years is equal to 1 ÷ 1.0793167066353E-11 seconds. ## Approximate result For practical purposes we can round our final result to an approximate numerical value. We can say that two thousand nine hundred thirty-six years is approximately ninety-two billion six hundred fifty-one million two hundred eleven thousand seventy-two seconds: 2936 yr ≅ 92651211072 s An alternative is also that one second is approximately zero times two thousand nine hundred thirty-six years. ## Conversion table ### years to seconds chart For quick reference purposes, below is the conversion table you can use to convert from years to seconds years (yr) seconds (s) 2937 years 92682768024 seconds 2938 years 92714324976 seconds 2939 years 92745881928 seconds 2940 years 92777438880 seconds 2941 years 92808995832 seconds 2942 years 92840552784 seconds 2943 years 92872109736 seconds 2944 years 92903666688 seconds 2945 years 92935223640 seconds 2946 years 92966780592 seconds

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# Shapes ## Areas of 2D shapes These are some common two-dimensional shapes and the formulae for calculating their areas: ### Triangle The area of a triangle is half its height times its base: \$A = ½h⋅b\$ This works for all triangles. If you made a copy of a triangle, and cut up the copy so that you made a rectangle with the original triangle, you would see that the rectangle has the dimensions of the height and base of the triangle. The area is therefore half of this rectangle! ### Heron's formula The area of a triangle can also be calculated using Heron's formula: \$\$A = √{s(s-a)(s-b)(s-c)}\$\$ Where \$a\$, \$b\$ and \$c\$ are le lengths of the sides of the triangle, and \$s = {a+b+c}/2\$ Square: \$A = L^2\$, where L is the length of one side. Rectangle: \$A = L ⋅ W\$, where L is the length, and W is the width. Rhomboid (elongated diamond): \${pq}/2\$, where p and q are the lengths of the two diagonals. Rhombus ('pushed over rectangle'): \$L ⋅ H\$, where L is the length and H is the height (perpendicular to L). ## Triangles In two dimensions (such as on a flat piece of paper), the angles of a triangle all add up to 180°. The equilateral triangle has all three side and all three angles equal. The angles are 60° each. The isoceles triangle has two sides the same length. This requires two angles to be same as well. Trapezium: one set of two parallel sides Isoceles trapezium: two sides equal length, the other two sides parallel Parallelogram: two sets of parallel and equal length sides Rhombus: two sets of two parallel sides, 4 equal lengths, two sets of identical angles ## Lines of Symmetry A useful way to describe a shape is to state its number of reflection lines of symmetry. These are the number of straight lines that may be drawn through a shape, across which a reflection of the shape would result in an identical shape. ### Triangle lines of symmetry Triangles may have zero, one or three lines of symmetry, depending on their type. An equilateral triangle has 3 lines of symmetry. Equilateral triangles have 3 lines of symmetry Line of symmetry for an isoceles triangle An isoceles has only one. Other types of triangles have no lines of symmetry, so their orientation is unique. A square reflects across a horizontal line through its centre, and a vertical line through its centre. It also has lines of symmetry across its diagonals. It therefore has four lines of reflection symmetry. A rhombus, like a rectangle, has 2 lines of symmetry A rhombus and a rectangle have only 2 lines of symmetry. A parallelogram has no lines of symmetry. ## Rotational Symmetry A shape which has rotational or radial symmetry is one which, when rotated 360° reassumes an identical form to the starting position one or more times. Order of rotational symmetry: the number of times an object takes an identical shape while being rotated through 360°. e.g. a square has rotational symmetry of 4, a rhombus 2, a triangle 3. ## Site Index ### Latest Item on Science Library: The most recent article is: Trigonometry View this item in the topic: Vectors and Trigonometry and many more articles in the subject: ### Mathematics Mathematics is the most important tool of science. The quest to understand the world and the universe using mathematics is as old as civilisation, and has led to the science and technology of today. Learn about the techniques and history of mathematics on ScienceLibrary.info. ### Great Scientists #### Dmitri Mendeleev 1834 - 1907 Dmitri Mendeleev, 1834 - 1907, was a Russian chemist who developed the modern Periodic Table of Elements. ### Quote of the day... We've long since won the argument about climate change; we've just lost the fight. And that's because it's not about data, it's about power.

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# Show that the matrix $A = \begin{bmatrix} 1 & -1 & 5 \\ -1 & 2 & 1 \\ 5 & 1 & 3 \end{bmatrix}$ is a symmetric matrix. ## 1 Answer Toolbox: • If A_{i,j} be a matrix m*n matrix , then the matrix obtained by interchanging the rows and column of A is called as transpose of A. • A square matrix A=[a$_{ij}$] is said to be symmetric if A'=A that is $[a_{ij}]=[a_{ji}]$ for all possible value of i and j. Given $A = \begin{bmatrix} 1 & -1 & 5 \\ -1 & 2 & 1 \\ 5 & 1 & 3 \end{bmatrix}$ Transpose of a matrix can be obtained by interchanging the rows and columns. $A'=\begin{bmatrix}1 & -1 & 5\\-1 & 2 & 1\\5 & 1 & 3\end{bmatrix}$ A square matrix is said to be symmetric if A'=A. $\Rightarrow [a_{ji}]=[a_{ij}]$ for all values of i & j. $a_{21}=-1=a_{12}$ $a_{31}=5=a_{13}$ $a_{32}=1=a_{23}$ $a_{11}=1=a_{11}$ $a_{22}=2=a_{22}$ $a_{33}=3=a_{33}$ Hence $a_{ji}=a_{ij}$ Therefore A is symmetric matrix. answered Apr 12, 2013 1 answer 1 answer 1 answer 1 answer 1 answer 1 answer 1 answer

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BROWSE ALPHABETICALLY LEVEL: Elementary Advanced Both INCLUDE TOPICS: Basic Math Algebra Analysis Biography Calculus Comp Sci Discrete Economics Foundations Geometry Graph Thry History Number Thry Phys Sci Statistics Topology Trigonometry space – Tarski Truth Theorem space Any abstract set with a structure defined on it, such as an order relation, metric, etc.Cf. Euclidean space, Hilbert space, metric space, topological space. sphere A closed surface, all points of which are equidistant from a given point, called the center. In 3-dimensional Euclidean space, the equation of a sphere of radius r and center (h, j, k) isThe term sphere may also refer to the solid bounded by this surface, and the interior is then called the open sphere of radius r.More generally, a sphere may be defined as the set of points in n-dimensional space (or any metric space) equidistant from a given point. The unit sphere in n-dimensional space is typically denoted S n - 1. Thus, the unit sphere in ordinary 3-space is denoted S2, and the unit circle in the plane is denoted S1. square A regular polygon having four equal sides and four right angles. stationary set If a is an ordinal, a set S in a is called stationary if S has non-empty intersection with every closed unbounded subset of a. story problem ARTICLE A mathematical problem presented as a real-world situation. See the article for problem solving techniques. subset A set A is a subset of a set B if every element of A is also an element of B. If in addition B is a subset of A, then A = B, but if not then A may be said to be a proper subset of B.Cf. superset. subtract To subtract a number m from a number n is to calculate the difference of m and n. If m is less than n we take the positive difference, otherwise we take the negative of the difference. This is tantamount to adding the negative of m to n. successor In a structure with an order relation defined upon it, the successor of an element a is the least element greater than a, if such exists.Cf. predecessor. sumset Given a set A, the sumset of A, denoted byis the set containing all of the elements of the elements of A, that is, it is the union of the elements of A. sumset axiom An axiom of set theory which states that if A is any set, then the sumset of A is also a set. sup Abbreviation of supremum. superset A set A is a superset of a set B if every element of B is an element of A.Cf. subset. supplemental angles Two angles are supplemental if they add up to 180 degrees (p radians).Cf. complementary angles. supremum The supremum of any subset of a linearly ordered set is the least upper bound of the subset. In particular, the supremum of any set of numbers is the smallest number in the set which is greater than or equal to every number in the set. In a complete linear order the supremum of any bounded set always exists.Cf. infimum, least upper bound axiom. surd (rare) An irrational root of a number, e.g., the square root of two. Related article: Irrationality of the Square Root of 2 surjection A surjective function, i.e., a function that maps at least one element of its domain to each element of its range.Cf. injection, bijection. Suslin tree Set Theory: For a an infinite cardinal, an a-Suslin tree is a tree T such that |T| = a, and every chain and every antichain of T has cardinality less than a.Cf. Aronszajn tree. symbolic logic Logic reduced to syntax, i.e., which works only with uninterpreted symbols. The two most often used kinds of symbolic logic are the propositional calculus and the predicate calculus. symmetric difference The symmetric difference of two sets A and B is the set of those elements that are in either A or B but not both.Cf. intersection, union. symmetric relation A relation “ ~ ” on a set X is symmetric if for any two elements x and y in X we have x ~ y if and only if y ~ x. The relation “ ~ ” is called asymmetric if for any two elements x and y we have that x ~ y implies it is not true that y ~ x, and it is called antisymmetric if whenever x ~ y and y ~ x then x = y. Note that a relation may be neither symmetric, asymmetric, nor antisymmetric.Cf. reflexive relation, transitive relation. Tarski Truth Theorem Let T be a mathematical theory. Denote formulas of T by j, y, etc., and let Tj be the statement that “the sentence j is true in T ” Choose a canonical numbering of the formulas of T (e.g., Gödel numbering) that assigns to each sentence of T a unique positive integer, and denote the positive integer associated with a sentence j by ‘j’. Then there is no formula y in T such that y(‘j’)Tj. In other words, “truth” in a theory T is not definable in T. space – Tarski Truth Theorem HOME | ABOUT | CONTACT | AD INFO | PRIVACYCopyright © 1997-2013, Math Academy Online™ / Platonic Realms™. Except where otherwise prohibited, material on this site may be printed for personal classroom use without permission by students and instructors for non-profit, educational purposes only. All other reproduction in whole or in part, including electronic reproduction or redistribution, for any purpose, except by express written agreement is strictly prohibited. Please send comments, corrections, and enquiries using our contact page.

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# Difference between revisions of "2012 AIME I Problems/Problem 6" ## Problem 6 The complex numbers $z$ and $w$ satisfy $z^{13} = w,$ $w^{11} = z,$ and the imaginary part of $z$ is $\sin{\frac{m\pi}{n}}$, for relatively prime positive integers $m$ and $n$ with $m Find $n.$ ## Solutions ### Solution 1 Substituting the first equation into the second, we find that $(z^{13})^{11} = z$ and thus $z^{142} = 1.$ So $z$ must be a $142$nd root of unity, and thus the imaginary part of $z$ will be $\sin{\frac{2m\pi}{142}} = \sin{\frac{m\pi}{71}}$ for some $m$ with $0 \le m < 142.$ But note that $71$ is prime and $m<71$ by the conditions of the problem, so the denominator in the argument of this value will always be $71$ and thus $n = \boxed{071.}$ ### Solution 2 Note that $w^{143}=w$ and similar for $z$, and they are not equal to $0$ because the question implies the imaginary part is positive. Thus $w^{142}=z^{142}=1$, so each is of the form $sin(2 \pi k/142)$ where $k$ is a positive integer between $0$ and $141$ inclusive. This simplifies to $sin(pi*k/71)$, and $071$ is prime, so it is the only possible denominator, and thus is the answer.

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# HSPT Math : Arithmetic ## Example Questions ### Example Question #51 : Concepts Fill in the blank with the proper sign. __________ Explanation: In order to solve this problem, we will use a number line. Numbers to the left of zero on the line are less than numbers to the right. Therefore: ### Example Question #11 : Understand An Inequality On A Number Line: Ccss.Math.Content.6.Ns.C.7a Fill in the blank with the proper sign. __________ Explanation: In order to solve this problem, we will use a number line. Numbers to the left of zero on the line are less than numbers to the right. Therefore: ### Example Question #51 : Concepts Fill in the blank with the proper sign. __________ Explanation: In order to solve this problem, we will use a number line. Numbers to the left of zero on the line are less than numbers to the right. Therefore: ### Example Question #51 : Negative Numbers Fill in the blank with the proper sign. __________ Explanation: In order to solve this problem, we will use a number line. Numbers to the left of zero on the line are less than numbers to the right. Therefore: ### Example Question #52 : Concepts Fill in the blank with the proper sign. __________ Explanation: In order to solve this problem, we will use a number line. Numbers to the left of zero on the line are less than numbers to the right. Therefore: ### Example Question #53 : Concepts Fill in the blank with the proper sign. __________ Explanation: In order to solve this problem, we will use a number line. Numbers to the left of zero on the line are less than numbers to the right. Therefore: ### Example Question #11 : Understand An Inequality On A Number Line: Ccss.Math.Content.6.Ns.C.7a Fill in the blank with the proper sign. __________ Explanation: In order to solve this problem, we will use a number line. Numbers to the left of zero on the line are less than numbers to the right. Therefore: ### Example Question #51 : Concepts Fill in the blank with the proper sign. __________ Explanation: In order to solve this problem, we will use a number line. Numbers to the left of zero on the line are less than numbers to the right. Therefore: ### Example Question #471 : The Number System Fill in the blank with the proper sign. __________ Explanation: In order to solve this problem, we will use a number line. Numbers to the left of zero on the line are less than numbers to the right. Therefore: ### Example Question #11 : Understand An Inequality On A Number Line: Ccss.Math.Content.6.Ns.C.7a Fill in the blank with the proper sign. __________

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In the verge of coronavirus pandemic, we are providing FREE access to our entire Online Curriculum to ensure Learning Doesn't STOP! # Pre-number Math Go back to 'Maths' ## Introduction to Pre-number Math Building pre-number math skills is a prerequisite to understanding numbers. So even before young children (aged 3 to 5 years) start with numbers, a fair amount of time should be invested in building these pre-number skills. Pre-number skills like matching, sorting, classifying, ordering and comparing will set the stage to build a strong number sense. Numbers should be introduced only after this. It’s after the numbers that the number names can be introduced. Because once the child is familiar with the numbers, associating it with the spelling is easier. ## The Big Idea At Cuemath pre-number math skills are built-in preschool years. It begins by identifying and working with objects like colour counters, pattern blocks and number blocks. Then moving on to working with dots. Counting the dots and associating it with the right numbers. After the child has gained decent proficiency over this they move to counting and identifying only numbers. Thus, laying a strong foundation for number sense. ## How do I understand? A preferable way to introduce pre-number skills is by using physical objects like pattern blocks. Children can be asked to match, sort, classify pattern blocks. From the concrete (pattern blocks) one can move to matching and sorting images. Children can now be introduced to different quantities. An association between the number and the quantity needs to be formed. This association will help the children to understand the largeness or smallness of a number. Avoid introducing numbers in its abstract form. Rote memorization of numbers without visualizing its magnitude won’t build a good number sense. ## How to Visualise numbers using counters? This video explains how numbers can be logically introduced. Instead of rote memorizing them here is an interesting way to understand numbers. Using counters will strengthen the number sense. ## Sub Topics Here are a few links that will take you through the journey that every Cuemath student undertakes in the pursuit of understanding Pre-Number Math along with practice worksheets:

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# Tag: Bernoulli • ## A brief Introduction to Probability Distribution for Machine Learning Probability Distributions are prevalent in many fields, namely, computer science, stock market, astronomy and economics. In this blog we are going to see different probability distribution for Machine Learning and their properties. Also, we will discuss key statistical points for the simple models. Note, all the discussion in this blog is based on the assumption that all data points are independent and identically distributed. “A probability distribution for Machine Learning is a statistical method that describes all the possible values and likelihoods that a random variable can take within a given interval.” Always remember the issue of choosing an appropriate distribution relates to the problem of model selection. ###### Discrete Probability Distributions It is distribution of all possible values of a discrete random variable together with an indication of their probabilities. Some examples of well known discrete probability distribution for Machine Learning are: • Bernoulli distributions. • Poisson distribution. • Uniform distribution. • Multinomial distributions. ###### Bernoulli Distribution Bernoulli Distribution is widely use for distribution of categorical outcomes. Concept behind logistic regression is best example of Bernoulli distribution. Let’s consider you have toss a coin ‘c’ , c ∈ {0, 1}. such that c = 1 representing heads and c = 0 representing tails. Let the probability of head is denoted by ‘μ’ ,such that: P(x=1 | μ) = μ ; 0 =< μ =< 1 So, P(x=0 | μ) = 1 – μ The probability distribution for flipping a coin is therefore can be written as: Bern (c | μ) = μc (1 -μ)1-c This is known as Bernoulli Distribution. From this equation we can easily said that this distribution is normalize. Now, suppose we have a data set X = {c1, c2, c3,….cn} of observed value of c. So, for this we can construct maximum likelihood function, which is function of μ. Such that P( X | μ) is given as: P(X | μ) = ∏ P(c | μ) = ∏ μc (1 -μ)1-c where; c = {c1, c2,…….cn} Now, we can estimate a value for μ by maximizing the logarithm of the likelihood and it is given as: lnP(X | μ) = ΣP(cn | μ) = Σ{cn ln μ + (1 + cn) ln(1 – μ)} Now if you remember, this is a same equation we have used in logistic regression for the error calculation. ###### Poisson Distribution A Poisson distribution is a measure of how many times an event is likely to occur within given period of time. It’s super handy because it’s pretty simple to use and is applicable for tons of things, there are a lot of interesting processes that boil down to “events that happen in time or space.” So, the probability for total ‘E’ events in ‘X’ period of time is given as: P(E events in X period of time) = e(-( E / T ) * X) * { (E / T) * X)k} / K! Let λ = (E / T) * X) Then, P(E events in X period of time) = eλ * {λ}k/ K! Where T is time for each event, X is total time interval and λ is a rate parameter which is the expected number of events in the interval, i.e. (E / T) * X). ###### Continuous Probability Distributions It is distribution of all possible values of a continuous random variable together with an indication of their probabilities. Some examples of well-known continuous probability distribution for Machine Learning are: • Normal or Gaussian distribution. • Power-law distribution. • Pareto distribution. ###### Gaussian Distribution The Gaussian, also known as normal distribution, is a widely used model for the distribution of continuous variables. When you work on large data it is observed that most of the data is closer to mean and the very less frequent data is observed towards the extremes, which is nothing but a gaussian distribution, and this is what central limit theorem tells us. In the case of a single variable ‘x’, gaussian distribution is given as: Where μ is the mean and σ² is the variance. For the single real variable, the distribution that maximizes the entropy is the Gaussian. Also, the Gaussian Distribution arises is when we consider the sum of multiple random variables. They may used to find non-linear regression as well as to reduce dimensionality by identifying which dimensions of a dataset have larger variance. Lots of phenomenon in nature follows the Gaussian Distribution. Like, Our height, weight, blood pressure etc. Hence it is a widely used distribution and favorite of many data scientists. 😉 I hope this article helped you in your data science journey. Was it explanatory? If you have any doubts and want to see more articles on distributions, please do write in the comment section. ###### Written by – Sarang Deshmukh Co – Founder and Developer @capablemachine.com | Performance Engineer @AMDOCS

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# Kmap Calculator ## Kmap Calculator: A Helpful Tool for Simplifying Boolean Functions When working with Boolean functions in digital electronics, simplifying expressions can be a tedious and time-consuming task. This is where a Kmap calculator comes in handy. A Karnaugh map, or Kmap for short, is a graphical method used to simplify Boolean algebra expressions. By using a Kmap calculator, you can easily reduce complex Boolean functions to their simplest form, making circuit designing and analysis more efficient and error-free. ## How Does a Kmap Calculator Work? A Kmap calculator takes as input the truth table of a Boolean function and creates a Karnaugh map representation of the function. The Kmap is then used to identify patterns and groupings of ones in the truth table, which can be used to simplify the function. The calculator applies the rules of Boolean algebra to combine adjacent ones in the map, resulting in a simplified expression that is equivalent to the original function. Also Check This Residual Calculator ## Benefits of Using a Kmap Calculator There are several advantages to using a Kmap calculator when working with Boolean functions: • Efficiency: Simplifying Boolean expressions manually can be time-consuming and prone to errors. A Kmap calculator automates the simplification process, saving time and reducing the risk of mistakes. • Accuracy: By applying the rules of Boolean algebra systematically, a Kmap calculator ensures that the simplified expression is equivalent to the original function. • Visual representation: Kmaps provide a visual representation of the Boolean function, making it easier to identify patterns and groupings for simplification. • Ease of use: Kmap calculators are user-friendly and require minimal input from the user, making them accessible to beginners and experts alike. Also Check This Reflection Calculator ## How to Use a Kmap Calculator Using a Kmap calculator is a straightforward process that involves the following steps: 1. Enter the truth table of the Boolean function into the calculator. 2. Generate the Karnaugh map representation of the function. 3. Identify patterns and groupings of ones in the map. 4. Combine adjacent ones to simplify the expression. 5. Verify the simplified expression with the original function to ensure correctness. ## Applications of Kmap Calculators Kmap calculators are commonly used in digital electronics and computer science for simplifying Boolean functions in circuit design, logic gates, and programming. They are also utilized in telecommunications, control systems, and signal processing for optimizing algorithms and reducing computational complexity. Also Check This Double Angle Calculator ## Conclusion In conclusion, a Kmap calculator is a valuable tool for simplifying Boolean functions in digital electronics. By automating the simplification process and providing a visual representation of the function, Kmaps make it easier to analyze and design circuits accurately. Whether you are a student learning Boolean algebra or a professional working in the field of digital electronics, a Kmap calculator can help streamline your work and improve efficiency. Try using a Kmap calculator for your next Boolean function simplification task and experience the benefits for yourself!

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0 # What is 2000000 divided by 1000000? Updated: 4/28/2022 Wiki User 12y ago 2 Mara Denesik Lvl 10 3y ago Wiki User 12y ago 2000000/1000000 = 2 Earn +20 pts Q: What is 2000000 divided by 1000000? Submit Still have questions? Related questions ### What is 1000000 plus 2000000? 1000000 + 2000000 = 3,000,000 2000000 ### What is 2 percent in 1000000? 2 percent of 2000000 = 40000 2% of 2000000 = 2% * 2000000 = 2%/100% * 2000000 = 0.02 * 2000000 = 40000 ### How many kilograms are there in two million milligrams? There are 1000000 milligrams in one kilogram. Therefore, 2000000 milligrams is equal to 2000000/1000000 = 2 kilograms. 1/2 80000 20000 ### What is x divided by 2000000 if x equals 00002? 00002/2000000 = 2/2000000 = 0.000001 14072.6147 ### 2 kilometers are to how many milimeters? 1 kilometre = 1000000 millimetres so 2 km = 2*1000000 = 2000000 millimetres. Simple! ### How many micrograms are in 2 grams? There are 1000000 micrograms in one gram. Therefore, 2 grams is equal to 2 x 1000000 = 2000000 micrograms.

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# Density (polytope) Updated on Edit Like Comment In geometry, the density of a polytope represents the number of windings of a polytope, particularly a uniform or regular polytope, around its center. It can be visually determined by counting the minimum number of facet or face crossings of a ray from the center to infinity. The density is constant across any continuous interior region of a polytope that crosses no facets. For a non-self-intersecting (acoptic) polytope, the density is 1. ## Contents Tessellations with overlapping faces can similarly define density as the number of coverings of faces over any given point. ## Polygons The density of a star polygon is the number of times that the polygonal boundary winds around its center; it is the winding number of the boundary around the central point. For a regular star polygon {p/q}, the density is q. It can be visually determined by counting the minimum number of edge crossings of a ray from the center to infinity. ## Polyhedra Arthur Cayley used density as a way to modify Euler's polyhedron formula (VE + F = 2) to work for the regular star polyhedra, where dv is the density of a vertex figure, df of a face and D of the polyhedron as a whole: dv VE + df F = 2D For example, the great icosahedron, {3, 5/2}, has 20 triangular faces (df = 1), 30 edges and 12 pentagrammic vertex figures (dv = 2), giving 2·12 − 30 + 1·20 = 14 = 2D. This implies a density of 7. The unmodified Euler's polyhedron formula fails for the small stellated dodecahedron {5/2, 5} and its dual great dodecahedron {5, 5/2}, for which VE + F = −6. The regular star polyhedra exist in two dual pairs, with each figure having the same density as its dual: one pair (small stellated dodecahedron—great dodecahedron) has a density of 3, while the other (great stellated dodecahedron–great icosahedron) has a density of 7. Hess further generalised the formula for star polyhedra with different kinds of face, some of which may fold backwards over others. The resulting value for density corresponds to the number of times the associated spherical polyhedron covers the sphere. This allowed Coxeter et al. to determine the densities of the majority of the uniform polyhedra. For hemipolyhedra, some of whose faces pass through the center, the density cannot be defined. Non-orientable polyhedra also do not have well-defined densities. ## Polychora There are 10 regular star polychora or 4-polytopes (called the Schläfli–Hess polychora), which have densities between 4, 6, 20, 66, 76, and 191. They come in dual pairs, with the exception of the self-dual density-6 and density-66 figures. ## References Density (polytope) Wikipedia

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Email Print # Voltage systems, Ohms Law, and DC-DC converters Oct 1, 2013 Voltage systems: Boats up to about 60 feet in length generally have a 12 Volt DC electrical system. This is largely due to the fact that many pieces of boating electrical gear are based on automotive/industrial electrical devices and so it is cheaper to use these mass produced items. Larger boats tend to have 24-Volt DC electrical systems for the same reason of keeping costs down and increasing savings. This may sound a bit confusing, but after we bone up on Ohms Law this seeming paradox will become clear. Ohms Law deals with the relationship of Voltage (E), Current (I), Resistance (R) and Power (P) of an electrical circuit and if we know any two quantities we can plug them into the formula and come up with the third. The various forms of Ohms Law follow: E=IR, I=E/R, R=E/I and P=IE, I=P/E, E=P/I. For our purposes the most important formula is the one dealing with current (I) because it will determine what size wire we need to operate safely. Notice that current in the Power formula I=P/E is a function of the power in watts divided by the system voltage, so if we double the voltage our current will be cut in half. Since electrical wire gauge is determined by its current carrying capacity, increasing voltage allows us to decrease the size of electrical cabling and save money. Example #1: P=240 Watts, E=12 VDC so I=240/12 which equals 20 amps current, but if we go to a 24-volt system then it works out as in Example #2: P=240 Watts, E=24 VDC so I=240/24 which equals 10 Amps current. Since a longer boat will need longer cable lengths it is doubly important to keep the cable gauge to a safe minimum or the costs will skyrocket and that is the big advantage of using a 24-VDC system versus 12 VDC. But what if your boat has a 24-VDC electrical system and you need to install a new electronic device that is built for 12 VDC? Again the current is the problem because I=E/R and given the same resistance if you tap into 24 VDC with a device designed for 12 VDC then you will double the current and most likely smoke your electronics. This situation can be resolved in one of three ways: 1) using a device that is rated for 10 to 35 VDC, 2) tapping off 12 volts from a 24 volt system, or 3) by using a 24 Volt to 12 Volt DC-DC converter. The simplest solution is to check the data plate on your new electronic device and hope that it is rated for 10-35 VDC in which case you could just connect it into your 24 Volt system. If this is not the case then you can tap off 12 volts from the middle of the 24-volt stack, but I suggest that this only be done on an emergency basis for the following reasons. Tapping will cause disproportionate charging of the batteries, overcharging one and undercharging the other which leads to shortened battery life. Also, you need to use the proper size wiring and fuse or circuit breaker it to protect the wiring from overheating during a short, etc. If you still want to tap off the 24-volt stack, then hook the positive conductor to where the two batteries are connected together (+ to -) and connect the ground lead to the output black lead. Again this method is not recommended. The optimum solution to our problem is to procure and install a device known as a 24 Volt to 12 Volt DC converter. These devices are rated for peak output and continuous output current. Just make sure that the continuous output rating matches your constant load requirements. In order to cut down on any potential RF interference from a DC-DC converter make sure to buy one that is listed as FCC “Class B” which should keep interference to a minimum. A comparison price for these taken from the latest West Marine catalog is \$79.99 for the 6 Amp model, \$119.99 for the 12 Amp model, and \$ 969.99 for the 50 Amp model. As you can see the price varies considerably depending on the continuous output current rating. If your boat has a NMEA 2000 network it may be as easy as tapping into the 12 Volt trunk line using a tee and running a drop cable to your 12-volt device. Most newer marine electronics have provision for hooking into an NMEA network so it should be an easy installation. You can buy NMEA 2000 cables and accessories from West Marine and they even have a NMEA 2000 starter kit for only \$87.99. Well, fellow sailors, enough for his time, so until we talk again: “Clear Skies and Following Seas!” Edit Module Old to new | New to old Oct 2, 2013 02:58 pm Posted by Arnold Fine article, thank you for writing it! I've always remember Ohm's Law as W (watts) = V (volts) x A (amps). My silly memory hook for this is War=Vetrans x Adminstration. Oct 2, 2013 03:45 pm Posted by Tim Thanks, Arnold. Interesting memory hook!

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# Partial Correlation & Semi-Partial: Definition & Example Contents: ## What is Partial Correlation? Partial correlation measures the strength of a relationship between two variables, while controlling for the effect of one or more other variables. For example, you might want to see if there is a correlation between amount of food eaten and blood pressure, while controlling for weight or amount of exercise. It’s possible to control for multiple variables (called control variables or covariates). However, more than one or two is usually not recommended because the more control variables, the less reliable your test. Partial correlation has one continuous independent variable (the x-value) and one continuous dependent variable (the y-value); This is the same as in regular correlation analysis. In the blood pressure example above, the independent variable is “amount of food eaten” and the dependent variable is “blood pressure”. The control variables — weight and amount of exercise — should also be continuous. ## Notation A period in the subscript separates the correlated variables and the controlled for variables. For example, correlating caloric intake (X1) against blood pressure (X2), while controlling for weight (X3), is written as: r12.3 Alternatively, a bar is used instead of a period and subscript: r(1,2|3). ## Running the Test The correlation coefficient, r, is also used to show the results from partial correlation. Like the regular correlation coefficient, rpartial returns a value from -1 to 1. Graphs showing a correlation of -1, 0 and +1 Partial correlation is usually carried out by running multiple regression analysis. Some software programs include partial correlation. For example, in SPSS choose Analyze > Correlations > Partial. ## How to Interpret the Result If the partial correlation, r12.3, is smaller than the simple (two-variable) correlation r12, but greater than 0, then variable 3 partly explains the correlation between X and Y. ## Semi-Partial Correlation Semi-partial correlation is almost the same as partial. In fact, many authors use the two terms to mean the same thing. However, others do make the following subtle distinction: With semi-partial correlation, the third variable holds constant for either X or Y but not both; with partial, the third variable holds constant for both X and Y. For example, the semi partial correlation statistic can tell us the particular part of variance, that a particular independent variable explains. It explains how one specific independent variable affects the dependent variable, while other variables are controlled for to prevent them getting in the way. To find it, calculate the correlation between the dependent variable and the residual of the prediction of one independent variable by the others. ## Example Suppose we use a set of data (from a 2002 paper from Abdi et al.) which lists three variables over six children. Each child was tested for memory span (Y) and speech rate (X2), and their age was also noted. A correlation statistic was desired which predicts Y (memory span) from X1 and X2 (age and speech rate). Normally, in a situation where X1 and X2 were independent random variables, we’d find out how important each variable was by computing a squared coefficient of correlation between X1 and X2 and the dependent variable Y. We would know that these squared coefficients of correlation were equal to the square multiple coefficient of correlation. But in a case like ours, X1 and X2 are anything but independent. Speech rate is highly dependent on age, and so using the squared coefficient will count the contributions of each variable several times over. ## References 1. Abdi, Herve. Part (Semi Partial) and Partial Regression Coefficients. Retrieved from https://www.utdallas.edu/~herve/Abdi-PartialRegressionCoefficient2007-pretty.pdf on May 8, 2018 2. Abdi, H., Dowling, W.J., Valentin, D., Edelman, B., & Posamentier M. (2002). Experimental Design and research methods. Unpublished manuscript. Richardson: The University of Texas at Dallas, Program in Cognition. 3. Brannick, M. Partial and Semipartial Correlation. Retrieved from http://faculty.cas.usf.edu/mbrannick/regression/Partial.html on May 8, 2018 4. Sharpa, J. (2007). Business Stats. Pearson Education India. 5. STATISTICA Help. Semi-Partial (or Part) Correlation. Retrieved from http://documentation.statsoft.com/STATISTICAHelp.aspx?path=glossary/GlossaryTwo/S/SemiPartialorPartCorrelation on May 8, 2018 6. Weatherburn, C. (1949). A First Course Mathematical Stats. CUP Archive. ------------------------------------------------------------------------------ Need help with a homework or test question? With Chegg Study, you can get step-by-step solutions to your questions from an expert in the field. If you'd rather get 1:1 study help, Chegg Tutors offers 30 minutes of free tutoring to new users, so you can try them out before committing to a subscription. If you prefer an online interactive environment to learn R and statistics, this free R Tutorial by Datacamp is a great way to get started. If you're are somewhat comfortable with R and are interested in going deeper into Statistics, try this Statistics with R track.

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Upcoming SlideShare × # Pembahasan soal ipa 241 views 191 views Published on 0 Likes Statistics Notes • Full Name Comment goes here. Are you sure you want to Yes No • Be the first to comment • Be the first to like this Views Total views 241 On SlideShare 0 From Embeds 0 Number of Embeds 0 Actions Shares 0 6 0 Likes 0 Embeds 0 No embeds No notes for slide ### Pembahasan soal ipa 1. 1. PEMBAHASAN SOAL IPA 1. Besaran fisika/ Besaran Pokok (BP)/Besaran Turunan (BT) Satuan / (SI) / (Non SI) Keterangan Panjang (BP) meter (SI) Benar Suhu (BP) celsius (non SI) Salah Kecepatan (BT) km/jam (non SI) Salah Berat (BP) gram (non SI) Salah Kunci Jawaban : A 2. Data dari gambar gelas ukur : Vair + benda = 85 ml V air = 60 ml V benda = 25 ml Kunci Jawaban : C 3. Data dari gambar neraca tiga lengan : mbenda = 125 gram ρ = 7,9 g/cm³ → V = g/cm³7,9 gram125 = 15,95 cm³ Kunci Jawaban = A 4. Dari gambar : Hal ini dilakukan agar kawat tidak mudah putus di malam hari ketika suhu rendah . Berarti : A. → salah B. → salah C. → salah D. → benar Kunci Jawaban : D 5. - Proses A – B = Proses Kenaikan suhu - ∆t = 100°C - 60°C = 40°C - Q = m . c. ∆t = 2 . 4200 . 40 = 336.000 joule Kunci Jawaban : A V m =ρ ρ m V = 2. 2. 6. Contoh gerak Jenis gerak 1 GLBB diperlambat 2 GLBB dipercepat 3 GLBB diperlambat 4 GLBB diperlambat 5 GLBB dipercepat Kunci Jawaban : D (1), (3) dan (5) 7. ρ = 1 g/cm³ = 1.000 kg/m³ g = 10 m/s² h = 60 cm – 35 cm = 25 cm = 0,25 m Ph = 1.000 kg/m³ . 10 m/s² . 0,25 m = 2.500 N/m² Kunci Jawaban : D 8. Dari Gambar ; Baterai : energi kimia → energi listrik Voltmeter : energi listrik → energi gerak Jadi : Energi kimia → energi listrik → energi gerak Kunci Jawaban : B 9. 3 detik menghasilkan 18 getaran Hz satuan dari frekuensi Frekuensi : jumlah getaran setiap detik Jadi : detik3 getaran18 =f = 6 get/det = 6 Hz Kunci Jawaban : B 10. Data dari gambar : s = 20 meter w = 600 joule W = F . S = F = S W = meter20 joule600 = 30 m Kunci Jawaban : B 11. Resultan gaya = 25 N – 5 N = 20 N hgPh ××= ρ 3. 3. Perpindahan (s) = 10 m W = F x S = 20 N x 10 m = 200 N.m = 200 joule Kunci Jawaban : A 12. A. Prinsip kerjanya : bidang miring B. Prinsip kerjanya : gerak dipermudah oleh roda C. Prinsip kerjanya : tuas D. Prinsip kerjanya : bidang miring Kunci Jawaban : C 13. T = 0,4 sekon f = T 1 = 4,0 1 = 4 10 Hz = 2,5 Hz Kunci Jawaban : D 14. Bunyi dapat merambat pada : zat padat, zat cair, dan gas Bunyi merambat pada zat padat lebih cepat dari zat cair dan lebih cepat dari gas Kunci Jawaban : D 15. Data dari gambar : So = 12 cm f = 10 cm SofSifSiSo 111111 −=→=+ = 12 1 10 1 − = 60 56 − Si 1 = 60 1 Si = 60 Kunci Jawaban : D 16. f = 12 cm So = 15 cm 4. 4. SofSifSiSo 111111 −=→=+ = 15 1 12 1 − = 60 45 − = 60 1 Si = 60 cm Kunci Jawaban : D 17. Dari data pada gambar : Gambar I : Titik bayangan jatuh di depan retina → cacat matanya : MIOPI Gambar II : setelah pakai kacamata titik bayangan jatuh tepat di retina Berarti lensa kacamata yang dipakai berlensa : cekung (-) Kunci Jawaban : D 18. Agar sisir plastik bermuatan listrik maka harus digosok dengan kain wol Dari peristiwa penggosokan terjadi : Elektron kain wol berpindah ke sisir plastik sehingga sisir plastik kelebihan elektron. Jadi sisir plastik bermuatan listrik negatif (-) Kunci Jawaban : B 19. Rtotal = 2 Ω + 3 Ω = 5 Ω I yang mengalir : Ω5 V02 = 4 A Volt menunjuk angka : V = I x R2 = 4 x 3 = 12 V Kunci Jawaban : C 20. Rtotal : Ω= × = ++ ×+ 15 54 15 69 663 6)63( I = A R V total 7,1 9 15 54 15 6 15 54 6 ==×== Kunci Jawaban : C 21. Biaya = daya x waktu x tarif = 000.1 800.3051004 Rpjamwatt ×××× = Rp. 48.000,00 Kunci Jawaban : C 22. Berdasarkan arah gosokan dan jenis kutub yang digosokkan : Besi PQ : Ujung P → kutub utara Ujung Q → kutub selatan Besi RT : Ujung R → kutub utara Ujung T → kutub selatan Kunci Jawaban : C 23. Berdasarkan arah arus pada lilitan dan kaidah tangan kanan, maka : 5. 5. Baja KL : Ujung K → kutub utara Ujung L → kutub selatan Sifat magnet : Baja → permanen Kunci Jawaban : A 24. Nama nama planet berdasarkan gambar : 1. Merkurius 5. Yupiter 2. Venus 6. Saturnus 3. Bumi 7. Uranus 4. Mars 8. Neptunus Saturnus berada diantara Yupiter (5) dan Uranus (7) Kunci Jawaban : C 25. Posisi kedua gambar : Gambar I : pasang maksimum Gambar II : pasang maksimum atau pasang purnama Kunci Jawaban : D 26. Kunci Jawaban : D Ciri mahluk hidup pada gambar adalah seekor ayam sedang mengerami telurnya adalah berkembangbiak 27. Kunci Jawaban : B Ciri tumbuhan dikotil dan monokotil No Organ Monokotil Dikotil 1 Akar Serabut Tumggang 2 Batang Tidak berkambium, tidak bercabang Letak jaringan pembuluh tersebar Berkambium, dan bercabang Letak jaringan pembuluh teratur 3 Daun Tulang daun lurus berbentuk pita Tulang daun menyirip atau menjari 4 Bunga Jumlah mahkota/kelopak berkelipatan 3 Jumlah mahkota/kelopak berkelipatan 2, 4, atau 5 5 Biji Lembaga berkeping satu Lembaga berkeping 2 28. Kunci Jawaban : C

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## Monday, February 19, 2024 ### January 2024 Algebra 2, Part II This exam was adminstered in January 2024. More Regents problems. ### Algebra 2 January 2024 Part II: Each correct answer will receive 2 credits. Partial credit can be earned. One mistake (computational or conceptual) will lose 1 point. A second mistake will lose the other point. It is sometimes possible to get 1 point for a correct answer with no correct work shown. 25. Factor the expression x3 + 4x2 - 9x - 36 completely. Factor by grouping, and then factor the quadratic you get after the first step. There are two ways to group, and either should work in any question of this kind. x3 + 4x2 - 9x - 36 (x3 + 4x2) - (9x + 36) x2(x + 4) - 9(x + 4) (x2 - 9)(x + 4) (x + 3)(x - 3)(x + 4) You can also switch the two middle terms around. This is just the way I learned it, so I usually do it, especially if it helps me avoid factoring out a minus sign. x3 - 9x + 4x2 - 36 (x3 - 9x) + (4x2 - 36) x(x2 - 9) + 4(x2 - 9) (x + 4)(x2 - 9) (x + 4)(x + 3)(x - 3) Note: This was Very Similar to Question 25 on the Auguest 2023 Regents. Right down to the (x + 3)(x - 3). 26. Determine if x + 4 is a factor of 2x3 + 10x2 + 4x - 16. Explain your answer. If (x + 4) is a factor of the polynomial, then the value of the polynomial must be 0 when x = -4. 2(-4)3 + 10(-4)2 + 4(-4) - 16 = 0 Since the expression is equal to zero when x = -4, then (x + 4) must be a factor. You could also solve this using polynomial division. (x + 4) divides evenly, with no remainder, so it is a factor. 27. An initial investment of \$1000 reaches a value, V(t), according to the model V(t) = 1000(1.01)4t, where t is the time in years. Determine the average rate of change, to the nearest dollar per year, of this investment from year 2 to year 7. Calculate V(7) and V(2). Subtract them and divide by 7 - 2, which is 5. You are looking for the rate of change (or slope, if you prefer). V(7) = 1000(1.01)4(7) = 1321.29 V(2) = 1000(1.01)4(2) = 1082.86 Rate of change = (1321.29 - 1082.86) / 5 = 47.686, which is \$48 to the nearest dollar. 28. When ( 1 / ∛(y2) ) y4 is written in the form yn, what is the value of n? Justify your answer. Use the laws of exponents to change the radical into a fraction. The combine the terms. ( 1 / ∛(y2) ) y4 ( 1 / (y2/3) y4 (y-2/3) y4 y10/3 n = 10/3. 29. The heights of the members of a ski club are normally distributed. The average height is 64.7 inches with a standard deviation of 4.3 inches. Determine the percentage of club members, to the nearest percent, who are between 67 inches and 72 inches tall. They don't use the chart with the normal distribution and all the standard deviations marked off any more. They just assume that you have and will use a calculator for this. You need to use the normalcdf function. Enter the command normalcdf(67,72,64.7,4.3) and you will get .2515... or 25%. All of the numbers that go into the command are in the question. Lower bound, upper bound, median, standard deviation. 30. The explicit formula an = 6 + 6n represents the number of seats in each row in a movie theater, where n represents the row number. Rewrite this formula in recursive form. A recursive function needs an initial value (a1) and an equation for an is terms of an-1. The inition value a1 = 12. Then an = an-1 + 6, because the common difference (rate of change) is 6. 31.Write (2xi3 - 3y)2) in simplest form. Square the binomial, substitute the powers of i, and Combine Like Terms. (2xi3 - 3y)2) (2xi3 - 3y)(2xi3 - 3y) 4x2i6 - 6xyi3 - 6xyi3 + 9y2 -4x2 - 12xyi3 + 9y2 -4x2 + 12xyi + 9y2 32. A survey was given to 1250 randomly selected high school students at the end of their junior year. The survey offered four post-graduation options: two-year college, four-year college, military, or work. Of the 1250 responses, 475 chose a four-year college. State one possible conclusion that can be made about the population of high school juniors, based on this survey This seems almost too simple a problem. If you divide 475/1250, you get .38 or 38%. One conclusion you can draw is that the population of high school juniors that would chose a four-year college would probably be about 38% and 62% would choose a different option. End of Part II How did you do? More to come. Comments and questions welcome. More Regents problems. ### I also write Fiction! You can now order my newest book Burke's Lore, Briefs: A Heavenly Date / My Damned Best Friend, written by Christopher J. Burke, which contains the aforementioned story and a bonus story. Order the softcover or ebook at Amazon. Also, check out In A Flash 2020, by Christopher J. Burke for 20 great flash fiction stories, perfectly sized for your train rides. Available in softcover or ebook at Amazon. If you enjoy it, please consider leaving a rating or review on Amazon or on Good Reads.

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It is currently 22 Nov 2017, 00:48 GMAT Club Daily Prep Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email. Customized for You we will pick new questions that match your level based on your Timer History Track every week, we’ll send you an estimated GMAT score based on your performance Practice Pays we will pick new questions that match your level based on your Timer History Events & Promotions Events & Promotions in June Open Detailed Calendar if |m+4|=2, what is the value of m? (a) m <0 (a) m2 +8m + Author Message Manager Joined: 27 Jul 2003 Posts: 122 Kudos [?]: 58 [0], given: 0 Location: Singapore if |m+4|=2, what is the value of m? (a) m <0 (a) m2 +8m + [#permalink] Show Tags 05 Oct 2003, 23:34 1 This post was BOOKMARKED 00:00 Difficulty: (N/A) Question Stats: 100% (00:03) correct 0% (00:00) wrong based on 11 sessions HideShow timer Statistics This topic is locked. If you want to discuss this question please re-post it in the respective forum. if |m+4|=2, what is the value of m? (a) m <0 (a) m2 +8m + 12=0 Kudos [?]: 58 [0], given: 0 SVP Joined: 03 Feb 2003 Posts: 1603 Kudos [?]: 308 [0], given: 0 Show Tags 05 Oct 2003, 23:44 m=-2 or -6 (1) not suff (2) m2 +8m + 12=0 or m=-2 or -6 not suff E Kudos [?]: 308 [0], given: 0 Non-Human User Joined: 09 Sep 2013 Posts: 15562 Kudos [?]: 283 [0], given: 0 Re: if |m+4|=2, what is the value of m? (a) m <0 (a) m2 +8m + [#permalink] Show Tags 21 Oct 2016, 22:14 Hello from the GMAT Club BumpBot! Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos). Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email. _________________ Kudos [?]: 283 [0], given: 0 Re: if |m+4|=2, what is the value of m? (a) m <0 (a) m2 +8m + [#permalink] 21 Oct 2016, 22:14 Display posts from previous: Sort by

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## A dealer deals in two products X & Y. He has ₹1,00,000/- to invest & space to store 80 pieces. Product X costs ₹ 2500/- &a Question A dealer deals in two products X & Y. He has ₹1,00,000/- to invest & space to store 80 pieces. Product X costs ₹ 2500/- & product Y costs ₹ 1000/- per unit. Construct the LPP and find the number of units of each product to be purchased. in progress 0 1 month 2021-07-31T04:49:37+00:00 2 Answers 0 views 0 8jwhsbdjr8jd sides susme sidn 2. Given : A dealer deals in two products X & Y. He has ₹1,00,000/- to invest & space to store 80 pieces. Product X costs ₹ 2500/- & product Y costs ₹ 1000/- per unit. To Find : Construct the LPP and the number of units of each product to be purchased. Solution: Let say unit to be bought for product X are x and product Y are y space to store 80 pieces x + y ≤ 80 He has ₹1,00,000/- to invest => 2500x + 1000y ≤ 100000 => 25x + 10y ≤ 1000 => 5x + 2y ≤ 200 x + y ≤ 80 5x + 2y ≤ 200 x = 40 , y = 0 x = 40/3 , y = 200/3 x = 0 , y = 80 Now further conditions are not provided that what is best to buy

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1 / 15 # Physics 2102 Physics 2102 Gabriela Gonz á lez. b. a. Physics 2102 . Circuits. DC circuits: resistances in series. Two resistors are “in series” if they are connected such that the same current flows in both. ## Physics 2102 E N D ### Presentation Transcript 1. Physics 2102 Gabriela González b a Physics 2102 Circuits 2. DC circuits: resistances in series Two resistors are “in series” if they are connected such that the same current flows in both. The “equivalent resistance” is a single imaginary resistor that can replace the resistances in series. “Walking the loop” results in :E –iR1-iR2-iR3=0  i=E/(R1+R2+R3) In the circuit with the equivalent resistance, E –iReq=0  i=E/Req Thus, 3. Multiloop circuits: resistors in parallel Two resistors are “in parallel” if they are connected such that there is the same potential drop through both. The “equivalent resistance” is a single imaginary resistor that can replace the resistances in parallel. “Walking the loops” results in :E –i1R1=0, E –i2R2=0, E –i3R3=0 The total current delivered by the battery is i = i1+i2+i3 = E/R1+ E/R2+ E/R3. In the circuit with the equivalent resistor, i=E/Req. Thus, 4. Resistors and Capacitors ResistorsCapacitors Key formula: V=iR Q=CV In series: same current same charge Req=∑Rj 1/Ceq=∑1/Cj In parallel: same voltage same voltage 1/Req=∑1/Rj Ceq=∑Cj 5. DC circuits • Loop rule: when walking along a loop, add potential differences across each element, and make the total equal to zero when you come back to the original point. • Junction rule: at every junction, total current is conserved. • Problem strategy: • Replace resistors in series and in parallel with their equivalent resistors • Draw currents in every wire, and label them • Write the loop rule for each loop (or for the loop that involves your question) • Write the junction rules for the currents. • Solve the equations for the currents. • Answer the question that was asked. 6. Bottom loop: (all else is irrelevant) 12V 8W Example Which resistor gets hotter? 7. Example • Which circuit has the largest equivalent resistance? • Assuming that all resistors are the same, which one dissipates more power? • Which resistor has the smallest potential difference across it? 8. Assume the batteries are ideal, and have emf E1=12V, E2=9V, and R=3W. Which way will the current flow? Which battery us doing positive work? If the potential at A is 0V, what is the potential at B? How much power is dissipated by the resistor? How much power is dissipated (or absorbed) by the batteries? Example 9. Example Find the equivalent resistance between points (a) F and H and (b) F and G. (Hint: For each pair of points, imagine that a battery is connected across the pair.) 10. Non-ideal batteries • You have two ideal identical batteries, and a resistor. Do you connect the batteries in series or in parallel to get maximum current through R? • Does the answer change if you have non-ideal (but still identical) batteries? 11. Light bulbs • If all batteries are ideal, and all batteries and lightbulbs are identical, in which arrangements will the lightbulbs as bright as the one in circuit X? • Does the answer change if batteries are not ideal? 12. i(t) E/R RC circuits: charging a capacitor In these circuits, current will change for a while, and then stay constant. We want to solve for current as a function of time i(t). The charge on the capacitor will also be a function of time: q(t). The voltage across the resistor and the capacitor also change with time. To charge the capacitor, close the switch on a. E + VR(t)+VC(t) =0 E -i(t)R -q(t)/C = 0 E - (dq(t)/dt) R - q(t)/C =0 A differential equation for q(t)! The solution is: q(t) = CE(1-e-t/RC) And then i(t) = dq/dt= (E/R) e-t/RC Time constant=RC 13. i(t) + C E/R - RC circuits: discharging a capacitor Assume the switch has been closed on a for a long time: the capacitor will be charged with Q=CE. +++ --- Then, close the switch on b: charges find their way across the circuit, establishing a current. VR+VC=0 -i(t)R+q(t)/C=0 => (dq/dt)R+q(t)/C=0 Solution: q(t)=q0e-t/RC=CEe-t/RC i(t) = dq/dt = (q0/RC) e-t/RC = (E/R) e-t/RC 14. Example The three circuits below are connected to the same ideal battery with emf E. All resistors have resistance R, and all capacitors have capacitance C. • Which capacitor takes the longest in getting charged? • Which capacitor ends up with the largest charge? • What’s the final current delivered by each battery? • What happens when we disconnect the battery? 15. Example In the figure, E= 1 kV, C = 10 µF, R1 = R2 = R3 = 1 MW. With C completely uncharged, switch S is suddenly closed (at t = 0). • What’s the current through each resistor at t=0? • What’s the current through each resistor after a long time? • How long is a long time? More Related

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# Second Term Examination Mathematics for Primary 1 (Basic 1) Maths Exam Questions MATHEMATICS SECOND TERM EXAMINATION PRIMARY 1 (BASIC 1) ### MATHS EXAM QUESTIONS SECTION A – Choose the correct answer from the options. 1. ( 24 + 10 ) + 10 [a] 54 [b] 64 [c] 44 2. Fatimah has 465 bags and dashes raphael 142 bag. How many bags are left. [a] 323 [b] 332 [c] 233 3. 1,5,7,9,11, 13, 15, 17 can be referred to as an __________ number. [a] high [b] odd [c] even 4. Two hundred and fourteen can be written as __________. [a] 214 [b] 241 [c] 421 5. How many 3’s are in 12? [a] 10 [b] 2 [c] 4 6. Which of this is an even number? [a] 2, 4, 6, 8 [b] 4, 3, 2, 1 [c] 8, 1, 5, 3, 2 7. 6 8 9 stands for [a] hundreds [b] tens [c] units 8. Write 234 in expanded form. [a] 200 + 300 + 4 [b] 200 + 30 + 4 [c] 200 + 3 + 40 9. Find the product of 33 and 2. [a] 77 [b] 88 [c] 66 10. A dozen is __________. [a] 12 [b] 14 [c] 24 11. 3 2 1 + 1 4 2 [a] 346 [b] 463 [c] 643 12. 9 4 6 – 1 2 4 [a] 212 [b] 248 [c] 822 13. 23 + __________ = 53 [a] 40 [b] 50 [c] 30 14. 67 – 48 = [a] 49 [b] 115 [c] 19 15. 32 x 2 [a] 54 [b] 74 [c] 64 SECTION B – Attempt all questions 1. Add up these with remaining a. 20 + 4 b. 10 + 8 c. 440 + 3 d. 220 + 6 2. Subtraction of three digit number. a. H T U b. H T U 6 9 8 8 4 7 – 5 3 3 – 2 4 6 c. H T U d. H T U 4 4 6 5 4 9 – 2 1 6 – 2 2 2 3a. Write out odd number from 1 to 39 b. Write out even number from 2 to 40 4a. Multiply these a. 1 0 x 2 b. 3 0 x 3 c. 4 3 x 2 d. 1 6 x 2 5. Write the value of each underlined digit. a. 4 9 6 b. 2 4 2 c. 9 6 9 6. How many 4s in 16? c. How many 2s in 20? d. How many 5s in 15?

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Associated Topics || Dr. Math Home || Search Dr. Math ### Truth Tables: And, Or, Implies, Not ``` Date: 06/10/2001 at 22:08:15 From: Lisa Subject: Truth Tables I have looked in the textbook about how to do truth tables, and it really doesn't explain or show how to do it. Can you please email me an explanation and some examples? Sincerely, Lisa ``` ``` Date: 06/11/2001 at 17:39:13 From: Doctor Tony Subject: Re: Truth Tables Hi Lisa, Thanks for writing to Ask Dr. Math. Essentially, truth tables allow you to evaluate every possibility for a given logic statement. Each of the basic operations (and, or, implies, not) has a truth table associated with it. The easiest way to explain is through examples. Let's form the truth tables for the set of operations I listed above, and then build a truth table for an example logic statement. Example 1: ~p (not p) We only have one logic variable, p, which can either be true or false. The first column of the truth table will contain all possible values of p. The second column will give the appropriate value for ~p. p | ~p ----------- T | F F | T That is the truth table. This was a pretty simple one, since it only involved one logic variable. Example 2: p^q (p and q) Now there are two logic variables, p and q; therefore, our truth table will have 4 rows, since there are four possible arrangements of p and q. In general, if there are n logic statements involved, there will be 2^n rows in the truth table. p q | p^q -------------- T T | T T F | F F T | F F F | F So we see that p^q is only T if both p and q are T - it makes sense. Example 3: p\/q (p or q) p q | p\/q --------------- T T | T T F | T F T | T F F | F We see that p\/q is only F if both p and q are F. Example 4: p->q (p implies q) p q | p->q --------------- T T | T T F | F F T | T F F | T Now let's do a more complicated example. Let's make a truth table for the logic statement ~p^(q\/r): p q r | ~p | q\/r | ~p^(q\/r) --------------------------------- T T T | F | T | F T T F | F | T | F T F T | F | T | F T F F | F | F | F F T T | T | T | T F T F | T | T | T F F T | T | T | T F F F | T | F | F As the last column shows, this statement is true for three out of the eight possible truth states of the logic variables. I hope this helps. If you're still stuck, please feel free to write back. - Doctor Tony, The Math Forum http://mathforum.org/dr.math/ ``` Associated Topics: High School Logic Search the Dr. Math Library: Find items containing (put spaces between keywords): Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words Submit your own question to Dr. Math Math Forum Home || Math Library || Quick Reference || Math Forum Search

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Expectation Custom Search EXPECTATION Expectation is the average of the values you would get in conducting an experiment or trial exactly the same way many times. In this discussion of expectation, we will consider two types. One is a numerical expectation and the other is a mathematical expectation. Numerical Expectation If you tossed a coin 50 times, you would expect the coin to fall heads (on the average) about 25 times. Your assumption is explained by the following definition of numerical expectation: If the probability of success in one trial is p, and k is the total number of trials, then kp is the expected number of successes in the k trials. In the above example of tossing the coin 50 times, the probability of heads (successes) is where Substituting values in the equation, we find that = 25 EXAMPLE: A die is rolled by a player. What is the expectation of rolling a 6 in 30 trials? SOLUTION: The probability of rolling a 6 in 1 trial is and the number of rolls is k=30 therefore, In words, the player would expect (on the average) to roll a 6 five times in 30 rolls. Mathematical Expectation We will define mathematical expectation as follows: If, in the event of a successful result, amount a, is to be received and the probability of success of that event is p, then ap is the mathematical expectation. If you were to buy 1 of 500 raffle tickets for a video recorder worth \$325.00, what would be your mathematical expectation? In this case, the product of the amount you stand to win and the probability of winning is where a = amount you stand to win p = probability of success and Then, by substitution Thus, you would not want to pay more than 65 cents for the ticket, unless, of course the raffle were for a worthy cause. EXAMPLE: To entice the public to invest in their development, Sunshine Condominiums has offered a prize of \$2,000 to 1 randomly selected family out of the first 1,000 families that participate in the condominium's tour. 1. What would be each family's mathematical expectation? 2. Would it be worthwhile for the Jones family to spend \$3.00 in gasoline to drive to Sunshine Condominiums to take the tour? SOLUTION: 1. 2. No; since \$3.00 is \$1.00 over their expectation of \$2.00, it would not be worthwhile for the Jones family to take the tour. PRACTICE PROBLEMS: 1. A box contains 7 slips of paper, each numbered differently. A girl makes a total of 50 draws, returning the drawn slip after each draw. a. What is the probability of drawing a selected numbered slip in 1 drawing? b. How many times would the girl expect to draw the single selected numbered slip in the 50 draws? 2. In a winner-take-all tournament among four professional tennis players, the prize money is \$500,000. Joe Conners, one of the tennis players, figures his probability of winning is 0.20. a. What is his mathematical expectation? b. Would he be better off if he made a secret agreement with the other tennis players to divide the prize money evenly regardless of who wins? ANSWERS: 2. a. \$100,000 b. Yes; he would be better off, since he would make \$125,000, which is greater than his expectation of \$100,000.

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https://ncatlab.org/nlab/show/Stirling%27s+approximation

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Contents Idea Stirling’s approximation, or Stirling’s formula, gives an asymptotic approximation to the factorial function in terms of elementary functions. Statement Define $n!$ for $n \in \mathbb{N}$ as usual by recursion: $0! = 1$ and $(n+1)! = (n+1) \cdot n!$. Stirling’s approximation says $n! \sim \frac{n^n \sqrt{2\pi n}}{e^n}$ or in other words that $\underset{n \to \infty}{\lim}\; \frac{n^n \sqrt{2\pi n}}{e^n \cdot n!} = 1.$ Proof Many proofs are known. The following is meant to bring out the essential kinship between this formula and the famous Gaussian integral identity $\int_{-\infty}^\infty e^{-x^2}\; d x = \sqrt{\pi},$ or equivalently $\int_{-\infty}^\infty e^{-x^2/2}\; d x = \sqrt{2\pi}.$ First we establish an elementary fact, accessible to a student of elementary calculus. Proposition For some constant $C$, $n! \sim C\frac{n^n \sqrt{n}}{e^n}.$ Proof Consider a trapezoidal sum estimate? for the integral $I_n = \int_1^n \log x\; dx = \left. x\log x - x\right|_1^n = n \log n - n + 1 = 1 + \log \frac{n^n}{e^n}.$ Partitioning the interval $[1, n]$ into subintervals of unit length, the corresponding trapezoidal sum is $T_n = \frac{\log 1 + \log 2}{2} + \frac{\log 2 + \log 3}{2} + \cdots + \frac{\log (n-1) + \log n}{2} = \log (n!) - \frac1{2} \log n = \log \frac{n!}{\sqrt{n}}.$ $T_n$ underestimates $I_n$ because the graph of $\log$ is “concave down” (meaning $-\log x$ is a convex function). In other words, the error terms $\int_k^{k+1} \log x\; dx - \frac{\log k + \log (k+1)}{2} \qquad (1)$ are positive, so that the sum of these error terms from $k=1$ to $n-1$, which is $I_n - T_n$, increases with $n$. However, using the error term estimate for the trapezoidal rule, the term (1) is on the order of $1/k^2$ (the maximum value of $|(\log)''(x)| = 1/x^2$ over $[k, k+1]$). Since $\sum 1/k^2$ converges, we see that the increasing sequence $I_n - T_n$ has a uniform upper bound and therefore a limit. Hence the limit $\underset{n \to \infty}{\lim} (I_n - 1) - T_n = \underset{n \to \infty}{\lim} \log \frac{n^n}{e^n} - \log \frac{n!}{\sqrt{n}} = \underset{n \to \infty}{\lim} \log \frac{n^n \sqrt{n}}{e^n \cdot n!}$ exists, and therefore $n! \sim C\frac{n^n \sqrt{n}}{e^n}$ for some constant $C$. The following lemma makes reference to the Gamma function. Lemma $\frac{\sqrt{2\pi}}{C} = \underset{n \to \infty}{\lim} \frac{\Gamma(n + \frac1{2})}{\Gamma(n) \cdot \sqrt{n}}$; in particular, this limit exists. Proof From $n! \sim C\frac{n^n \sqrt{n}}{e^n}$, we easily derive $\frac{(2n)!}{n! \cdot n!} \sim \frac{2^{2n} \sqrt{2n}}{C n}$ so that $\frac1{C} \sim \frac{(2n)!}{2^{2n} n! \cdot n!} \cdot \frac{n}{\sqrt{2n}} = \frac{(2n-1)(2n-3) \ldots 3 \cdot 1}{2 \cdot (2n-2)(2n-4) \ldots 2} \cdot \frac1{\sqrt{2n}}.$ Then, using $\Gamma\left(\frac1{2}\right) = 2\int_0^\infty e^{-x^2} dx = \sqrt{\pi}$, we have $\frac{\sqrt{2\pi}}{C} \sim \frac{(2n-1)(2n-3) \ldots 3 \cdot 1}{2 \cdot (2n-2)(2n-4) \ldots 2} \cdot \frac{\sqrt{\pi}}{\sqrt{n}} = \frac{\left(n-\frac1{2}\right)\left(n-\frac{3}{2}\right) \ldots \left(\frac1{2}\right) \Gamma\left(\frac1{2}\right)}{(n-1)! \cdot \sqrt{n}} = \frac{\Gamma\left(n + \frac1{2}\right)}{\Gamma(n) \cdot \sqrt{n}}.$ Proposition $\underset{n \to \infty}{\lim} \frac{\Gamma(n + \frac1{2})}{\Gamma(n) \cdot \sqrt{n}} = 1$. Proof From log-convexity of $\Gamma$ (see here), we may derive $\frac{\Gamma(x-\frac1{2})}{\Gamma(x - 1)} \leq \frac{\Gamma(x)}{\Gamma(x-\frac1{2})} \leq \frac{\Gamma(x+\frac1{2})}{\Gamma(x)}$ which implies that the values of $\frac{\Gamma(x+\frac1{2})}{\Gamma(x) \cdot \sqrt{x}}$, with $x$ ranging over half-integers, tend to the same limit $L$ as with $x$ ranging over whole integers. Therefore $L^2 = \underset{n \to \infty}{\lim} \frac{\Gamma(n+1)}{\sqrt{n+\frac1{2}}\cdot \Gamma(n + \frac1{2})} \cdot \frac{\Gamma(n + \frac1{2})}{\sqrt{n}\Gamma(n)} = \underset{n \to \infty}{\lim} \frac{n\Gamma(n)}{\sqrt{(n+\frac1{2})n}\cdot \Gamma(n)} = \underset{n \to \infty}{\lim} \frac{n}{\sqrt{(n+\frac1{2})n}},$ whence $L^2 = 1$, and therefore $L = 1$. References • n-Category Café post • Dan Romik, Stirling’s Approximation for n!: the Ultimate Short Proof?, The American Mathematical Monthly Volume 107 Issue 6 (2000), 556-557. doi.org/10.1080/00029890.2000.12005235 Last revised on April 12, 2024 at 22:24:35. See the history of this page for a list of all contributions to it.

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# math posted by . Integrate sinx/sinx-cosx..... Plz note sinx-cosx as a whole is in the denominator.. • math - use the good old quotient rule: if y = u/v, y' = (u'v - uv')/v^2 u = sinx v = sinx - cosx y' = [(cosx)(sinx-cosx) - (sinx)(cosx+sinx)]/(sinx-cosx)^2 you can play with that some, but I like 1/(sin(2x)-1) ## Similar Questions 1. ### Math Verify the identity . (cscX-cotX)^2=1-cosX/1+cosX _______ sorry i cant help you (cscX-cotX)=1/sinX - cosX/sinX = (1-cosX)/sinX If you square this you have (1-cosX)^2/(sinX)^2 Now use (sinX)^2 = 1 - (cosX)^2 to get (1-cosX)^2 / 1 - … 2. ### Pre-Calc Trigonometric Identities Prove: (tanx + secx -1)/(tanx - secx + 1)= tanx + secx My work so far: (sinx/cosx + 1/cosx + cosx/cosx)/(sinx/cos x - 1/cosx + cosx/cosx)= tanx + cosx (just working on the left side) ((sinx + 1 - cosx)/cosx)/((sinx … 3. ### Trigonometry. ( tanx/1-cotx )+ (cotx/1-tanx)= (1+secxcscx) Good one! Generally these are done by changing everything to sines and cosines, unless you see some obvious identities. Also generally, it is best to start with the more complicated side … 4. ### Trig........ I need to prove that the following is true. Thanks (cosx / 1-sinx ) = ( 1+sinx / cosx ) I recall this question causing all kinds of problems when I was still teaching. it requires a little "trick" L.S. =cosx/(1-sinx) multiply top and … 5. ### Mathematics - Trigonometric Identities Prove: (tanx)(sinx) / (tanx) + (sinx) = (tanx) - (sinx) / (tanx)(sinx) What I have so far: L.S. = (sinx / cosx) sinx / (sinx / cosx) + sinx = (sin^2x / cosx) / (sinx + (sinx) (cosx) / cosx) = (sin^2x / cosx) / (cosx / sinx + sinxcosx) 6. ### Math - Pre- Clac Prove that each of these equations is an identity. A) (1 + sinx + cos x)/(1 + sinx + cosx)=(1 + cosx)/sinx B) (1 + sinx + cosx)/(1 - sinx + cosx)= (1 + sin x)/cosx Please and thankyou! 7. ### maths - trigonometry I've asked about this same question before, and someone gave me the way to finish, which I understand to some extent. I need help figuring out what they did in the second step though. How they got to the third step from the second. … 8. ### Trigonometry Check Simplify #3: [cosx-sin(90-x)sinx]/[cosx-cos(180-x)tanx] = [cosx-(sin90cosx-cos90sinx)sinx]/[cosx-(cos180cosx+sinx180sinx)tanx] = [cosx-((1)cosx-(0)sinx)sinx]/[cosx-((-1)cosx+(0)sinx)tanx] = [cosx-cosxsinx]/[cosx+cosxtanx] = [cosx(1-sinx]/[cosx(1+tanx] … 9. ### Math Im really struggling with these proving identities problems can somebody please show me how to do these? 10. ### trigonometry can i use factoring to simplify this trig identity? More Similar Questions

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Last updated: # Cubic Meter Calculator What is a cubic meter?How to use the cubic meter calculatorHow to calculate the cubic meters of a box or spaceFAQs With the cubic meter calculator, you can convert any volume unit to cubic meters. On top of that, you can also calculate the cubic meters of a box of a known size, regardless of the length units you've got! If you want to convert from cubic ft to cubic meters, or convert cubic meters to cubic feet, or any other combination, this is the fastest and easiest way! So go ahead and use the calculator to calculate cubic meters and convert from and to cubic meters, cubic feet, and whatever other unit of volume you want! ## What is a cubic meter? First things first, we need to make sure we all know what a cubic meter is. A cubic meter is the SI unit for volume. It is the obvious choice for volume, given that the volume is the 3D space taken up by an object. In fact, this is the clearest when we think about a box. In that case, the formula for its volume would be: volume = length × width × height If we use meters for length, width, and height, it makes sense then that the final units for volume would be meter × meter × meter = meter³ That is why we use the cubic meter and its derivative units (cubic kilometer, cubic centimeter, etc.) as the international unit for volume. However, it is not the only one, and in places like the USA, the imperial system is more prevalent. In the imperial system, we find units such as the cubic inch, the cubic mile, and, most similar to cubic meters, the cubic foot. Just like knowing different languages, being able to convert from cubic meters to cubic feet and back from cubic feet to cubic meters is essential. In fact, the more units we can convert from and to, the better! Before we learn how to calculate cubic meters from any other unit, we should take a look at some of the common abbreviations: • Cubic meters: m³ • Cubic centimeters: cm³, cc • Cubic kilometers: km³ • Cubic feet/foot: cu ft, ft³ • Cubic inches: cu in, in³ ## How to use the cubic meter calculator The cubic meter calculator can be thought of as several calculators in one. It has two main modes: conversion and calculation, but they can also be used simultaneously with some clever operation. Assuming you do it correctly, this calculator can: • Calculate the cubic meters of a box; • Calculate the cubic feet of a box; • Calculate the cubic inches of a box; • Calculate the cubic centimeters of a box; • Convert cubic meters to cubic feet; • Convert cubic feet to cubic meters (cubic ft to cubic meters); • Convert cubic inches to liters; • Convert gallons to cubic meters; and • More! Even if it looks like the ultimate conversion tool, this cubic meter calculator is best suited for... well... calculations involving cubic meters, duh! If you are looking for the ultimate converters, check out the volume conversion, the area converter, and the distance converter, all closely related to this one but slightly more focused on pure conversion. It is now time to see how to use the cubic meters calculator by Omni Calculator to compute the volume of a box using any mix of units: 1. Select the units and input the value for the length. 2. Select the units and input the value for the width. 3. Select the units and input the value for the height. 4. At the bottom of the calculator, you will see the resulting volume in 2 different units (by default, cubic ft and cubic meters). Feel free to change if you want to. It's all effortless and very straightforward. You can mix and match units as you see fit, and the results will always be in the units you choose, from cu ft to cubic meters to gallons or liters. But we can make it even easier if you don't need to calculate the cubic meters and just want to convert volume, for example, from cubic feet to cubic meters. If you want to use the cubic meters calculator as a volume converter, you have even fewer steps to follow: 1. Collapse the top section of the calculator. 2. The fields for width, length, and height will be hidden because they are not needed. 3. Select the unit you want to input and enter the value in the top field for volume. 4. Automatically, the tool will calculate the cubic meters equivalent of the volume. It could not be easier! You don't need to remember any conversion factors, do any math or wait. Just select your unit, input the value, and enjoy the result! ## How to calculate the cubic meters of a box or space "Sure, that's fine, but how do you calculate the cubic meters of a box?" You might be saying. Maybe you don't want to use our wonderful calculator. Perhaps you want the knowledge as a backup, or maybe you're preparing for an exam. We won't judge; we only help! So, if you want to learn how to calculate the cubic meters of a box, here is how to do it: 1. Take the values for your box's length, width, and height. If possible, use meters. 2. If you haven't used meters, convert each of them individually to meters. 3. Apply the volume formula for a box: volume = length × width × height 4. Your results should now be in cubic meters. Apply the conversion factor if you want another unit. As you can see, it is not complicated, but it is needlessly time-consuming when you have a nice tool like this one, which we created at Omni Calculator. FAQs ### How many cubic centimeters in a cubic meter? There are 1,000,000 cubic centimeters in a cubic meter. To remember this, you can think about converting meters to centimeters for each of the three dimensions: length, width, and height. One meter has 100 cm, so multiplying that by itself three times, a m³ has 1,000,000 cm³. ### How many cubic feet are in a cubic meter? There are 35.315 cubic feet in a cubic meter. Converting from cubic meters to cubic feet is the same as converting from meters to feet for each of the three dimensions: length, width, and height. One meter has 3.281 feet, so multiplying that by itself three times, one m³ has 35.315 cubic ft. ### How many liters in a cubic meter? There are 1000 liters in a cubic meter. To remember this, you can think about cubic decimeters and liters since they are both the same size. One cubic meter (m³) is 1000 dm³, so since liters and dm³ are the same, there are 1000 liters in a m³. ### How many gallons in a cubic meter? There are 264.17 US gallons in 1 cubic meter, and there are 219.97 UK gallons in 1 cubic meter. You can perform any conversion from cubic meters to gallons by following this instructions: 1. Find out if you want US gallons or UK gallons. 2. Multiply the number of cubic meters by the conversion factor (264.17 for US gallons) and (219.97 for UK gallons). 3. The result will be expressed in your chosen unit. ### What is cubic meter? A cubic meter is a unit of measurement that corresponds to the volume of a square box with 1 meter-long sides. This means a little over 3 ft per side. You can calculate the volume of said box by multiplying all three dimensions together: 1 m × 1 m × 1 m = 1 m³ Another way to visualize a cubic meter is to convert it to other units. One cubic meter is the same as: • 1000 liters; • 264.17 US gallons; and • 35.32 cu ft.

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# What is Composition of Functions ? Here you will learn what is composition of functions with properties and examples. Let’s begin – ## What is Composition of Functions ? Let f : A $$\rightarrow$$ B & g : f : B $$\rightarrow$$ C be two functions. Then the function gof : f : A $$\rightarrow$$ C defined by (gof)(x) = g(f(x)) $$\forall$$ x $$\in$$ A is called the composite of the two function f & g. ## Properties : (a) In general composite of functions is not commutative i.e. gof $$\ne$$ fog. (b) The composition of functions is associative i.e. if f, g, h are three functions such that fo(goh) & (fog)oh are defined, then fo(goh) = (fog)oh. (c) The composition of two bijections is a bijection i.e. if f & g are two bijections such that gof is defined, then gof is also a bijection. Example : If f(x) = $$x^2$$ + 1, g(x) = $$1\over x-1$$, then find (fog)(x). Solution : Now (fog)(x) = f(g(x)) = f($$1\over x-1$$) = f(z), where z = $$1\over x-1$$ = $$z^2 + 1$$ [ $$\because$$ f(x) = $$x^2 + 1$$ ] = $$({1\over x-1})^2$$ + 1 = $$1\over {(x-1)^2}$$ + 1

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1. Chapter 3 Class 11 Trigonometric Functions 2. Serial order wise Transcript Ex 3.3, 23 Prove that tanβ‘4π₯ = (4 tanβ‘γπ₯ (1βtan2π₯)γ)/(1 β 6 tan2 π₯+tan4 π₯) Taking L.H.S. tan 4x = (2 tanβ‘2x)/(1 β tan2 2x) tan 4x = 2((2 tanβ‘π₯)/(1 β π‘ππ2 π₯))/(1 β ((2 tanβ‘π₯)/(1 β π‘ππ2 π₯))^2 ) = (((4 tanβ‘π₯)/(1 β π‘ππ2 π₯)))/(1 β((2 tanβ‘π₯ )^2/(1 β π‘ππ2 π₯)^2 ) ) = (((4 tanβ‘π₯)/(1 β π‘ππ2 π₯)))/(1 β((4 γπ‘ππγ^2 π₯)/(1 β π‘ππ2 π₯)^2 ) ) = (((4 tanβ‘π₯)/(1 β π‘ππ2 π₯)))/((((1 β π‘ππ2 π₯)^2 β 4 γπ‘ππγ^2 π₯)/(1 β π‘ππ2 π₯)^2 ) ) = (4 tanβ‘π₯)/(1 β tan2β‘π₯ ) Γ ((1 β tan2β‘γπ₯)2γ)/((1 βπ‘ππ2 π₯)2 β4 tan2β‘π₯ ) = (4 tanβ‘π₯)/1 Γ ((1 β tan2β‘γπ₯)γ)/((1 βπ‘ππ2 π₯)2 β4 tan2β‘π₯ ) = (4 tanβ‘γπ₯ (1 βπ‘ππ2 π₯γ))/((1 βtan2β‘π₯ )2 β4 tanβ‘2π₯ ) Using (a β b)2 = a2 +b2 β 2ab = (4 tanβ‘γπ₯ ( 1 βtan2β‘π₯)γ)/(( (1)2+(π‘ππ2 π₯)2 β2 Γ 1 Γ π‘ππ2 π₯) β4 π‘ππ2 π₯) = (4 tanβ‘γπ₯ ( 1 βtan2β‘π₯)γ)/(( (1)2+(π‘ππ2 π₯)2 β2 Γ 1 Γ π‘ππ2 π₯) β4 π‘ππ2 π₯) = (4 tanβ‘γπ₯ ( 1 βtan2β‘π₯)γ)/(1 + tanβ‘γ4 π₯ β 2 tan2β‘γπ₯ β4 tan2β‘π₯ γ γ ) = (4 tanβ‘γπ₯ ( 1 β tan2β‘γπ₯ )γ γ)/(1 + tan4β‘γπ₯ β6 π‘ππ4 π₯γ ) = R.H.S. Hence L.H.S. = R.H.S. Hence proved

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# How do you calculate payroll minutes? All you need to do is divide your minutes by 60. For example, say your employee worked 20 hours and 15 minutes during the week. Divide your total minutes by 60 to get your decimal. For this pay period, your employee worked 20.25 hours. ## What is 1/6th of an hour? Each Hour Has 60 Minutes For example, 10 minutes is 10/60 = 1/6 of an hour, and 24 minutes is 24/60 = 6/15 of an hour. ## How many hours is 7am to 5pm with a 30 minute lunch? For example, a 7:30 to 4:30 work day with a 30 minute lunch break means an 8.5 hour work day (9 hours in between, minus 30 minutes or 0.5 hours equals 8.5). If you are working shifts your break may not be a lunch break, and you may also have multiple breaks allowed. ## How do I calculate work? Work can be calculated with the equation: Work = Force × Distance. The SI unit for work is the joule (J), or Newton • meter (N • m). One joule equals the amount of work that is done when 1 N of force moves an object over a distance of 1 m. ## How do you calculate 15 minutes? To convert any fraction to decimal form, we just need to divide its numerator by the denominator. In this case, if we convert 15 minutes to hours we write it as 15/60 because 1 hour = 60 minutes. This means 15 minutes can be written as 0.25 hours in the decimal form. ## How do I calculate my hours per week? Add up the number of hours from each week to get your total. Divide by the total number of weeks. The resulting number is the average hours you would have worked during weeks when you took your previous leave. ## What is .25 on a timesheet? Conversion Chart – Minutes to Hundredths of an Hour Enter time in Oracle Self Service as hundredths of an hour. For example 15 minutes (¼ hour) equals . 25, 30 minutes (½ hour) equals . 5, etc. ## How do you turn 6 7 into a percent? Convert 6/7 to Percentage by Changing Denominator Our percent fraction is 85.714285714286/100, which means that 67 as a percentage is 85.71%. ## What is 0.3 as a percent? To write 0.3 as a percent, multiply 0.3 by 100. Append % symbol in the product obtained. So, 0.3 as a percent is 30 %. ## What is a 85? Letter grade Percentage Grade definition A+ 90-100 Excellent A 85-89 Very good A– 80-84 Very good B+ 75-79 Good B. ## Why does Europe use military time? The 24-hour clock has a long history. It is easier than specifying AM and PM, when there might be doubt. Railways were major users. The military also need to avoid ambiguity. ## How many hours is 9/5 in a day? The traditional American business hours are 9:00 a.m. to 5:00 p.m., Monday to Friday, representing a workweek of five eight-hour days comprising 40 hours in total. These are the origin of the phrase 9-to-5, used to describe a conventional and possibly tedious job. ## How are lunch timesheets calculated? Enter this formula: =SUM((C2-B2)+(E2-D2))*24 into a blank cell beside your time record cells, F2, for instance, see screenshot: Note: In the above formula: C2 is the lunch start time, B2 is the log in time, E2 indicates the log out time and D2 is the lunch end time. ## How do you calculate payroll hours for employees? You do this by dividing the minutes worked by 60. You then have the hours and minutes in numerical form, which you can multiply by the wage rate. For example, if your employee works 38 hours and 27 minutes this week, you divide 27 by 60. This gives you 0.45, for a total of 38.45 hours. ## How do you calculate payroll per hour? First, determine the total number of hours worked by multiplying the hours per week by the number of weeks in a year (52). Next, divide this number from the annual salary. For example, if an employee has a salary of \$50,000 and works 40 hours per week, the hourly rate is \$50,000/2,080 (40 x 52) = \$24.04. ## How do you calculate late minutes? How to calculate Late In: 1. Subtract the scheduled work time from the actual arrival time. Using the sample above: 10:42 – 8:00 = 2.42, or 2 hours and 42 minutes late arrival. 2. ## What percentage of one hour is 45 minutes? Therefore, 1 hour and 45 minutes is 7.291% of the total number of hours in a day. Hence the correct option is C. ## What is a 100 minute clock? A 100-minute clock is typically referred to when talking about the decimal equivalent of the minutes of an hour. In other words, 1 is 60 minutes in a 100-minute clock. ## How do you write time in hours? In general, to write time in the 24-hour system, omit the colon between hours and minutes, and follow the numerals for time with the word “hours.” The invasion began at 0823 hours . Read aloud as “oh-eight-twenty-three hours” or “zero-eight-twenty-three” (military). ## How do you calculate 30 minutes payroll? For example, if an employee works 8:30 minutes, this is 8.5 hours when converted to decimal, multiply it by their hourly wage; this results in a gross wage amount. ## How do you calculate monthly hours? A quick and easy method of calculating monthly hours is to multiply 40 hours per week by 4 weeks, yielding 160 hours for the month. The other method will provide the average number of work hours in a month. ## What is full time UK? There is no specific number of hours that makes someone full or part-time, but a full-time worker will usually work 35 hours or more a week. Part-time workers should get the same treatment for: pay rates (including sick pay, maternity, paternity and adoption leave and pay) pension opportunities and benefits.

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View more editions Solutions Introduction to Algorithms # Introduction to Algorithms (3rd Edition)Solutions for Chapter 11.P • 954 step-by-step solutions • Solved by publishers, professors & experts • iOS, Android, & web Looking for the textbook? Over 90% of students who use Chegg Study report better grades. May 2015 Survey of Chegg Study Users Chapter: Problem: SAMPLE SOLUTION Chapter: Problem: • Step 1 of 5 Longest-probe bound for hashing Open addressing is the hashing technique in which for inserting an element in the table firstly we search for the elements if it is already contained in it, the search returns either the element or the free space in the array to fill that element. In uniform hashing technique sequence of elements to be search can be any of the m! permutation inthat is this technique gives the method of searching which search in the full list for the single element in the generalized way. For the storage of n elements wherein a hash table of capacity m items it uses the open addressing technique. • Step 2 of 5 a . In the process of insertion of an element by using uniform hashing technique if the element is not found in the table then it is inserted into the first slot which do not contain any element. Ifis the load factor that is to search an element which is not found in the table then the total expected probes is maximum of. Assuming that a variable chosenrepresents the probes which do not return the required variable then . We know that then. The probability of insertion of element in a hash table that uses uniform hashing requires not less than k probes so thus, Hence Thus for insertion of element in the hash table by using uniform hashing where the probability of storage is not more thanwhen it searches more than k times. • Step 3 of 5 b . For insertion of element in the hash table by using uniform hashing where the probability of storage is not more thanwhen it searches more than k times that is Here the probes required for the insertion of element in hash table is more than . Thus, . By using formulae of logarithm: Thus Hence the probability of storage for the insertion of element is for when it probes elements in the table. • Step 4 of 5 c. The insertion of element in the table requires searches and represents the total number of searches necessary for the insertion of n elements. Consider an event A for which the number of probes X are such that, and in event the number of probes, for . Then probability of events is as: For n number of events according to the Boolâ€™s inequality: Then for the set A of n events, Hence if the insertion of element in the table requires searches and if represents the total number of searches necessary for the insertion of n elements then: • Step 5 of 5 d . The expected length of the probe sequence for k probes is given by the formulae: Breaking the sequence into two parts one from and the other from then, For the searching of element for which the number of probes less than then and for the search when number of probes are greater thanthen . Thus, Hence, the expected total lengthof the maximum length probe succession is. Corresponding Textbook Introduction to Algorithms | 3rd Edition 9780262033848ISBN-13: 0262033844ISBN:

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# Ambiguous case of the law of sines The ambiguous case of the law of sines happens when two sides and an angle opposite one of the two sides are given. We can shorten this situation with SSA. Since the length of the third side is not known, we don't know if a triangle will be formed or not. That is the reason we call this case ambiguous. In fact, this kind of situation or SSA can give the following 4 scenarios. • No triangle • 1 triangle • 1 right triangle • 2 triangles ## The first scenario of the ambiguous case of the law of sines occurs when a < h. For example, take a look at the triangle below where only two sides are given. These two given sides are a and b. An angle opposite to one side is also given. Angle A is the angle that is opposite to side a or one of the two sides. Note that SSA in this case means side a side b angle A in that order. Because a is shorter than h, a is not long enough to form a triangle. In fact, the number of possible triangles that can be formed in the SSA case depends on the length of the altitude or h. Notice that sin A = h / b Once you multiply both sides of the equation above by b, we get h = b sin A. An example showing that no triangle can be formed Method #1 Suppose A = 74°, a = 51, and b = 72. h = 72 × sin (74°) = 72 × 0.9612 = 68.20 Since 51 or a is less than h or 69.20, no triangle will be formed. Method #2 We can also show that no triangle exists by using the law of sines. a / sin A = b / sin B The ratio a / sin A is known since a / sin A = 51 / sin 74° Since we also know the length of b, the missing quantity in the law of sines is sin B. It is logical then to look for sin B and see what we end up with. 51 / sin 74° = 72 / sin B 51 sin B = 72 sin 74° sin B = (72 sin 74°) / 51 sin B = (72 × 0.9612) / 51 sin B = (69.2064) / 51 sin B = 1.3569 Since the sine of an angle cannot be bigger than 1, angle B does not exist. Therefore, no triangle can be formed with the given measurements. ## The second scenario of the ambiguous case of the law of sines occurs when a = h. When a = h, the resulting triangle will always be a right triangle. An example showing that a right triangle can be formed Method #1 Suppose A = 30°, a = 25, and b = 50. h = 50 × sin (30°) = 50 × 0.5 = 25 Since 25 or a is equal to h or 25, 1 right triangle will be formed. Method #2 Again, we can use the law of sines to show that this time sin B exists and it is equal to 90 degrees. a / sin A = b / sin B 25 / sin 30° = 50 / sin B 25 sin B = 50 sin 30° sin B = (50 sin 30°) / 25 sin B = (50 × 0.5) / 25 sin B = (25) / 25 sin B = 1 B = sin-1(1) = 90 degrees. ## The third scenario of the ambiguous case of the law of sines occurs when a > h and a > b. When a is bigger than h, again a triangle can be formed. However, since a is bigger than b, we can only have one triangle. Try to make a triangle where a is bigger than b, you will notice that there can only be 1 such triangle. An example showing that exactly 1 triangle can be formed Suppose A = 30°, a = 50, and b = 40. h = 40 × sin (30°) = 40 × 0.5 = 20 Since 50 or a is bigger than both h (or 20) and b (or 40), 1 triangle will be formed. ## Last situation: a > h and a < b When a is less than b, 2 triangles can be formed as clearly illustrated below. The two triangles are triangle ACD and triangle AED. An example showing that exactly 2 triangles can be formed Suppose A = 30°, a = 40, and b = 60 h = 60 × sin (30°) = 60 × 0.5 = 30 Since 40 or a is bigger than h and a is smaller than b or 60, 2 triangles will be formed. ## Recent Articles 1. ### How To Find The Factors Of 20: A Simple Way Sep 17, 23 09:46 AM There are many ways to find the factors of 20. A simple way is to... 2. ### The SAT Math Test: How To Be Prepared To Face It And Survive Jun 09, 23 12:04 PM The SAT Math section is known for being difficult. But it doesn’t have to be. Learn how to be prepared and complete the section with confidence here.

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https://fahrenheittocelsius.org/1471-f-to-c

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# 1471 Fahrenheit to Celsius Welcome to 1471 Fahrenheit to Celsius, or 1471 F to C in short. Here you can find what 1471 degrees Fahrenheit to Celsius is, along with a temperature converter and the formula. For 1471 (degrees) Fahrenheit we write 1471 °F, and (degrees) Celsius or centigrades are denoted with the symbol °C. So if you have been looking for 1471 °F to °C, then you are right here, too. Read on below to learn everything about the temperature conversion. You don’t have to press the blue button, unless you want to swap the conversion. Bookmark this temperature converter now. ## Formula The 1471 Fahrenheit to Celsius formula is: [°C] = ([1471] − 32) x 5 ⁄ 9. Therefore, we get: 1471 F to C = 799.444 °C 1471 °F in °C = 799.444 Celsius 1471 F in C = 799.444 degrees Celsius Right above we have given you the result in the common spelling variants of this temperature conversion. Of course, the result is also valid for 1471F to C, 1471F in C etc. Here you can change 1471 Celsius to Fahrenheit. Next, we explain the math. ## Conversion To convert the temperature start by start by deducting 32 from 1471. Then multiply 1439 by 5 over 9 to obtain 799.444 degrees Celsius. Easier, however, is using our converter above. Similar temperature conversions on our website include: Ahead is the wrap-up of our content. ## Summary How much is 1471 degrees Fahrenheit in Celsius? By reading our article so far, or by means of our converter, you already know the answer 1471 Fahrenheit in other temperature units is: • Newton: 263.817 °N • Kelvin: 1072.594 °K • Réaumur: 639.556 °Ré • Rømer: 427.208 °Ro • Delisle: -1049.167 °De • Rankine: 1930.67 °R This ends our posts about 1471 °F to °C. If you have anything to tell or in case you would like to ask something about 1471 F in C, then fill in the form below. And, if this article about 1471 F to Celsius has been helpful to you, then hit the social buttons please. Thanks for visiting fahrenheittocelsius.org. Posted in F to C

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https://stats.stackexchange.com/questions/104403/what-is-the-meaning-of-a-large-p-value

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What is the meaning of a large p-value? I understand that the $p$-value is the conditional probability of observing the test statistic or something more extreme given that the null hypothesis is true. I have read the great explanation by @user28 in this post: What is the meaning of p values and t values in statistical tests? However, do large $p$-values say anything? Does a larger $p$-value lend greater support to the null hypothesis? If I set rejection region to be $<0.05$, then does it make a difference if I get $p$-value $0.06$ or $0.99$? (After all, $0.05$ is arbitrary, and $0.06$ is so close to being rejected that if I arbitrarily set $0.05$ as $0.1$ instead, the null hypothesis would have been rejected.) Can one make any statistical use of a non-rejecting $p$-value? • One example is mentioned at the end of this answer: Fisher's re-examination of Mendel's pea experiments. – whuber Jun 23, 2014 at 14:37 • You cannot conclude that a test statistic supports the null hypothesis when the calculation of that statistic is conditional on the assumption that the null is true. That is to say, if $E$ and $H_0$ are events, you cannot say $H_0$ is true if $\Pr[E \mid H_0]$ is "large." Jun 23, 2014 at 16:04 How you should 'use' the p-value depends on how you have designed your study with regard to the analyses you will run. I discuss two different philosophies about p-values in my answer here: When to use Fisher and Neyman-Pearson framework? You may find it helpful to read that. If you have, for example, run a power analysis and intend to use the p-value to make a final decision, you should not use close to the line ('marginally significant') as a meaningful category. It is fine to use a different alpha than $0.05$ (such as $0.10$), but once you decided on it and set your study up accordingly, you should stick with it. In addition, you cannot use a large p-value as evidence for the null hypothesis. I discussed that idea in my answer here: Why do statisticians say a non-significant result means "you cannot reject the null" as opposed to accepting the null hypothesis? Reading that answer may be helpful to you as well. In my view, everything boils down to assumptions, that is, how well the models fits them. If it does agree with all of them, treat the p-value as a probability. Then you can compare p-value of 0.06 with 0.99 by concluding which of the two are more likely. Also, a lot depend on circumstances: in some cases, marginal significance shouldn't be ignored, because as you stated, rejection region can be set rather arbitrarily. But if the models satisfies the assumptions, then you should not seek to reject your hypothesis to some arbitrary level but rather investigate, how likely is the outcome that you got. • It is problematic to (mis)interpret a p-value as a probability in this sense. You seem to be applying a Bayesian-like approach but without the usual care needed to consider and defend a prior distribution. What justification can you provide to support your recommendations? – whuber Jun 23, 2014 at 14:40 • well, I was having a linear regression in mind satisfying all the conventional assumptions of the models and as a result regression coefficients having normal distributions. I suppose in this case it would be safe to treat p-value as a probability, wouldn't it? Jun 23, 2014 at 14:51 • No, it would not. There is a lot of debate about p-values, but practically everybody who has entered into it (on either side) has noted that this use of p-values is invalid. Comparing p-values does not determine whether one model is more or less likely than another. – whuber Jun 23, 2014 at 14:54 • Yes, you are right. I started googling and two source made things more clear: Hubbard, R. (2004). Alphabet soup: Blurring the distinctions between p's and α's in psychological research. Theory and Psychology, 14 (3), 295-327; Hubbard, R., & Lindsay, R.M. (2008). Why P values are not a useful measure of evidence in statistical significance testing. Theory and Psychology, 18 (1), 69-88. Obviously, I had an incorrect view of the concept! Jun 23, 2014 at 15:16 It is true that the acceptance range for the p-value of a hypothesis test is rather arbitrary, but nevertheless a lower p-value means that the test result can be accepted with more certainty, because the p-value essentially defines the confidence interval for the estimate, so a narrower confidence interval should be regarded more significant for the test.

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Print Version Page Index Solvers Puzzles Basic Strategies Tough Strategies Diabolical Strategies Extreme Strategies Deprecated Strategies Str8ts Other Order Now! Order Now! # Glossary • Almost Locked Set (ALS) – A set of N unsolved cells with N+1 candidates • Bi-location – A unit with only two occurances of a candidate X • Bi-value – A cell with only two candidates remaining • Box – Any 3x3 shaped unit in a normal sudoku, or any squiggly 9-cell shape in Jigsaw sudoku which must contain 1 to 9. There are 9 boxes in 9x9 sudoku. • Cage – A marked off group of cells for which their sum is provided in Killer Sudoku and variants. • Candidate – A possible solution for a cell. Any cell reduced to one candidate is solved. • Cell – A square on the sudoku grid for which there is one single number solution. • Chain – A sequence of links between candidates such that the two ends imply an elimination. All Loops are closed chains. • Conjugate Pair – The two remaining candidates in a bi-location are conjugate. They are used to form strong link within a unit since one must be true and the other must be false. • Contradiction – In Sudoku an illegal and therefore contradictory situation can occur if a) there are no candidates left in a cell, or b) two or more cells claim to be true. The aim of many strategies is to show a contradiction. • Continuous – In a Loop it signifies that the links between nodes strictly alternate between strong and weak links. • Cycle – Another name for a Loop. See Loop. • Discontinuous – In a Loop it signifies that there is one discontinuity where a node is beset by either two weak links or two strong links. • Grouped Node – A set of candidates of the same value (that must share a unit) that can be used to form a group with the effect of being able to treat it as a single cell. • Hinge – In a Y-Wing the middle cell which joins the two pincer cells. • Inference – Valid deductions that can be made between two linked candidates. There are inferences of the weak kind and the strong kind. • Locked Set – A group of N candidates confined to a group of N cells. All candidates that can see a locked set can be removed. Naked Pairs, Triples etc are Locked Sets. • Loop – Another name for a Cycle. See Cycle. A closed chain or sequence of candidates (which are called nodes in the loop) used to make a deduction. Loops are continuous or discontinuous. • Pincer – In a Y-Wing or Y-Wing Chain the two end nodes are pincers which attack a cell for the purposes of elimination. See also Hinge. • Strong Inference – Where two candidates X and Y are linked with strong inference then both cannot be false at the same time. Therefore if X is false Y is true and if Y is false X is true. • Strong Link – A link between two candidates (which can be a single number or a grouped node) whereby if one is deemed false the other must be true. Strong Links can be between two cells (bi-location), within a call (bi-value) or as part of an ALS or more complex structure. Article here • Unit – My preferred term for any row, column or box. Also called a house. • Variant – A variation on an original puzzle, eg Jigsaw Sudoku is a variant of Sudoku. • Weak Inference – Where two candidates X and Y are linked weak inference implies that both cannot be true at the same time. Therefore if X is true Y is false and if Y is true X is false. • Weak Link – A link between one candidate and all others in a unit or cell whereby if the candidate is deemed to be true all the others are false. A strong link can be deemed to be weak merely because "all the others" includes cases where there is only one other. Article here • XY-Wing – Another name for Y-Wing The following terms are not used on this site but may be found out in the wild • Block – Another name for a box • Chute – Three boxes in a column. • Domain – Another term for a unit. • Given – Another name for a clue. • House – Another name for a unit, sometimes also the same as a chute. • Reduction – Another name for elimination • Region – Another name for a box

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https://www.bankingcareers.in/blog/bankexam/crack-ibps-po-2017-2-recent-types-simplification-problems/

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# Crack IBPS PO 2017: 2 Recent Types Of Simplification Problems Dear Reader, IBPS PO exams are starting very soon! (In 5 months tentatively). Below tutorial will help you for sure. You already know simplification is an important topic for bank exams. In this post, you will find two types of problems asked in previous year PO exam. Example For Type 1: Find the number in place of the question mark: ? / 676.03 = 1295.6 / ? Solution: We can approximate 676.03 as 676 and 1295.6 as 1296. Therefore, the given equation can be simplified as: ?2 = 1296 x 676 Or ? = square root of (1296 x 676) = square root of (1296) x square root of (676) = 36 x 26 = 936 Tip: Remember squares of numbers up to 50 to solve such problems easily. Example For Type 2: Find the number in place of the question mark: ? = square root of (0.25 x 0.36) x 1/30 This type was surprisingly easy (in last year exams). Only one thing you have to know to solve such problems is the correct placement of decimal point in square roots. Now let us solve the above problem. square root of (0.25 x 0.36) x 1/30 = square root of (0.25) x square root of (0.36) x 1/30 = 0.5 x 0.6 x 1/30 = 0.30 x 1/30 = 0.01

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https://metanumbers.com/54194

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# 54194 (number) 54,194 (fifty-four thousand one hundred ninety-four) is an even five-digits composite number following 54193 and preceding 54195. In scientific notation, it is written as 5.4194 × 104. The sum of its digits is 23. It has a total of 5 prime factors and 16 positive divisors. There are 22,932 positive integers (up to 54194) that are relatively prime to 54194. ## Basic properties • Is Prime? No • Number parity Even • Number length 5 • Sum of Digits 23 • Digital Root 5 ## Name Short name 54 thousand 194 fifty-four thousand one hundred ninety-four ## Notation Scientific notation 5.4194 × 104 54.194 × 103 ## Prime Factorization of 54194 Prime Factorization 2 × 73 × 79 Composite number Distinct Factors Total Factors Radical ω(n) 3 Total number of distinct prime factors Ω(n) 5 Total number of prime factors rad(n) 1106 Product of the distinct prime numbers λ(n) -1 Returns the parity of Ω(n), such that λ(n) = (-1)Ω(n) μ(n) 0 Returns: 1, if n has an even number of prime factors (and is square free) −1, if n has an odd number of prime factors (and is square free) 0, if n has a squared prime factor Λ(n) 0 Returns log(p) if n is a power pk of any prime p (for any k >= 1), else returns 0 The prime factorization of 54,194 is 2 × 73 × 79. Since it has a total of 5 prime factors, 54,194 is a composite number. ## Divisors of 54194 1, 2, 7, 14, 49, 79, 98, 158, 343, 553, 686, 1106, 3871, 7742, 27097, 54194 16 divisors Even divisors 8 8 4 4 Total Divisors Sum of Divisors Aliquot Sum τ(n) 16 Total number of the positive divisors of n σ(n) 96000 Sum of all the positive divisors of n s(n) 41806 Sum of the proper positive divisors of n A(n) 6000 Returns the sum of divisors (σ(n)) divided by the total number of divisors (τ(n)) G(n) 232.796 Returns the nth root of the product of n divisors H(n) 9.03233 Returns the total number of divisors (τ(n)) divided by the sum of the reciprocal of each divisors The number 54,194 can be divided by 16 positive divisors (out of which 8 are even, and 8 are odd). The sum of these divisors (counting 54,194) is 96,000, the average is 6,000. ## Other Arithmetic Functions (n = 54194) 1 φ(n) n Euler Totient Carmichael Lambda Prime Pi φ(n) 22932 Total number of positive integers not greater than n that are coprime to n λ(n) 3822 Smallest positive number such that aλ(n) ≡ 1 (mod n) for all a coprime to n π(n) ≈ 5515 Total number of primes less than or equal to n r2(n) 0 The number of ways n can be represented as the sum of 2 squares There are 22,932 positive integers (less than 54,194) that are coprime with 54,194. And there are approximately 5,515 prime numbers less than or equal to 54,194. ## Divisibility of 54194 m n mod m 2 3 4 5 6 7 8 9 0 2 2 4 2 0 2 5 The number 54,194 is divisible by 2 and 7. ## Classification of 54194 • Arithmetic • Deficient ### Expressible via specific sums • Polite • Non-hypotenuse • Frugal ## Base conversion (54194) Base System Value 2 Binary 1101001110110010 3 Ternary 2202100012 4 Quaternary 31032302 5 Quinary 3213234 6 Senary 1054522 8 Octal 151662 10 Decimal 54194 12 Duodecimal 27442 20 Vigesimal 6f9e 36 Base36 15te ## Basic calculations (n = 54194) ### Multiplication n×y n×2 108388 162582 216776 270970 ### Division n÷y n÷2 27097 18064.7 13548.5 10838.8 ### Exponentiation ny n2 2936989636 159167216333384 8625908121971412496 467472464762118728808224 ### Nth Root y√n 2√n 232.796 37.8428 15.2577 8.84688 ## 54194 as geometric shapes ### Circle Diameter 108388 340511 9.22683e+09 ### Sphere Volume 6.66718e+14 3.69073e+10 340511 ### Square Length = n Perimeter 216776 2.93699e+09 76641.9 ### Cube Length = n Surface area 1.76219e+10 1.59167e+14 93866.8 ### Equilateral Triangle Length = n Perimeter 162582 1.27175e+09 46933.4 ### Triangular Pyramid Length = n Surface area 5.08702e+09 1.8758e+13 44249.2 ## Cryptographic Hash Functions md5 41e84168c20f59d698f99bf46ded6625 50f95adf879f0d833cdd2dace7977b37c80a0602 f3011286091aae62db60d623c71e8d7286153ce75d695a6c1d68cb8f6432f9b5 fae4bb769cc8372eb3671186735c4967dcb1763d2c2b693df8d771b65f79c177b9a0716e4cdd97d6a051358eaa199aa87d9b78060f95e5c1f5afe32c8066a858 5ee7c08e91cd1146afcd2cb5459f679c85e5ba21

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https://nrich.maths.org/public/leg.php?code=6&cl=2&cldcmpid=1130

1,568,625,303,000,000,000

text/html

crawl-data/CC-MAIN-2019-39/segments/1568514572516.46/warc/CC-MAIN-20190916080044-20190916102044-00113.warc.gz

596,288,519

9,072

# Search by Topic #### Resources tagged with Place value similar to Reach 100: Filter by: Content type: Age range: Challenge level: ### There are 75 results Broad Topics > Numbers and the Number System > Place value ### Reach 100 ##### Age 7 to 14 Challenge Level: Choose four different digits from 1-9 and put one in each box so that the resulting four two-digit numbers add to a total of 100. ### Two and Two ##### Age 11 to 14 Challenge Level: How many solutions can you find to this sum? Each of the different letters stands for a different number. ### Cayley ##### Age 11 to 14 Challenge Level: The letters in the following addition sum represent the digits 1 ... 9. If A=3 and D=2, what number is represented by "CAYLEY"? ### Football Sum ##### Age 11 to 14 Challenge Level: Find the values of the nine letters in the sum: FOOT + BALL = GAME ##### Age 11 to 14 Challenge Level: Replace the letters with numbers to make the addition work out correctly. R E A D + T H I S = P A G E ### Eleven ##### Age 11 to 14 Challenge Level: Replace each letter with a digit to make this addition correct. ### Calculator Bingo ##### Age 7 to 11 Challenge Level: A game to be played against the computer, or in groups. Pick a 7-digit number. A random digit is generated. What must you subract to remove the digit from your number? the first to zero wins. ### Nice or Nasty for Two ##### Age 7 to 14 Challenge Level: Some Games That May Be Nice or Nasty for an adult and child. Use your knowledge of place value to beat your opponent. ### All the Digits ##### Age 7 to 11 Challenge Level: This multiplication uses each of the digits 0 - 9 once and once only. Using the information given, can you replace the stars in the calculation with figures? ### Which Scripts? ##### Age 7 to 11 Challenge Level: There are six numbers written in five different scripts. Can you sort out which is which? ### The Thousands Game ##### Age 7 to 11 Challenge Level: Each child in Class 3 took four numbers out of the bag. Who had made the highest even number? ### One Million to Seven ##### Age 7 to 11 Challenge Level: Start by putting one million (1 000 000) into the display of your calculator. Can you reduce this to 7 using just the 7 key and add, subtract, multiply, divide and equals as many times as you like? ### Song Book ##### Age 7 to 11 Challenge Level: A school song book contains 700 songs. The numbers of the songs are displayed by combining special small single-digit cards. What is the minimum number of small cards that is needed? ### Tis Unique ##### Age 11 to 14 Challenge Level: This addition sum uses all ten digits 0, 1, 2...9 exactly once. Find the sum and show that the one you give is the only possibility. ##### Age 5 to 11 Challenge Level: Who said that adding couldn't be fun? ### Being Curious - Primary Number ##### Age 5 to 11 Challenge Level: Number problems for inquiring primary learners. ### Being Resourceful - Primary Number ##### Age 5 to 11 Challenge Level: Number problems at primary level that require careful consideration. ### Round the Dice Decimals 1 ##### Age 7 to 11 Challenge Level: Use two dice to generate two numbers with one decimal place. What happens when you round these numbers to the nearest whole number? ### Diagonal Sums ##### Age 7 to 11 Challenge Level: In this 100 square, look at the green square which contains the numbers 2, 3, 12 and 13. What is the sum of the numbers that are diagonally opposite each other? What do you notice? ### Coded Hundred Square ##### Age 7 to 11 Challenge Level: This 100 square jigsaw is written in code. It starts with 1 and ends with 100. Can you build it up? ### Multiply Multiples 2 ##### Age 7 to 11 Challenge Level: Can you work out some different ways to balance this equation? ### Trebling ##### Age 7 to 11 Challenge Level: Can you replace the letters with numbers? Is there only one solution in each case? ### Multiply Multiples 1 ##### Age 7 to 11 Challenge Level: Can you complete this calculation by filling in the missing numbers? In how many different ways can you do it? ### Dicey Operations in Line for Two ##### Age 7 to 11 Challenge Level: Dicey Operations for an adult and child. Can you get close to 1000 than your partner? ### Digit Sum ##### Age 11 to 14 Challenge Level: What is the sum of all the digits in all the integers from one to one million? ### Round the Dice Decimals 2 ##### Age 7 to 11 Challenge Level: What happens when you round these numbers to the nearest whole number? ### Round the Three Dice ##### Age 7 to 11 Challenge Level: What happens when you round these three-digit numbers to the nearest 100? ### Arrange the Digits ##### Age 11 to 14 Challenge Level: Can you arrange the digits 1,2,3,4,5,6,7,8,9 into three 3-digit numbers such that their total is close to 1500? ### Subtraction Surprise ##### Age 7 to 14 Challenge Level: Try out some calculations. Are you surprised by the results? ### Multiply Multiples 3 ##### Age 7 to 11 Challenge Level: Have a go at balancing this equation. Can you find different ways of doing it? ##### Age 5 to 11 Challenge Level: Try out this number trick. What happens with different starting numbers? What do you notice? ### Basically ##### Age 11 to 14 Challenge Level: The number 3723(in base 10) is written as 123 in another base. What is that base? ### Number Detective ##### Age 5 to 11 Challenge Level: Follow the clues to find the mystery number. ### Chocolate Maths ##### Age 11 to 14 Challenge Level: Pick the number of times a week that you eat chocolate. This number must be more than one but less than ten. Multiply this number by 2. Add 5 (for Sunday). Multiply by 50... Can you explain why it. . . . ### Spell by Numbers ##### Age 7 to 11 Challenge Level: Can you substitute numbers for the letters in these sums? ### Oddly ##### Age 7 to 11 Challenge Level: Find the sum of all three-digit numbers each of whose digits is odd. ### ABC ##### Age 7 to 11 Challenge Level: In the multiplication calculation, some of the digits have been replaced by letters and others by asterisks. Can you reconstruct the original multiplication? ### Six Is the Sum ##### Age 7 to 11 Challenge Level: What do the digits in the number fifteen add up to? How many other numbers have digits with the same total but no zeros? ##### Age 11 to 14 Challenge Level: Powers of numbers behave in surprising ways. Take a look at some of these and try to explain why they are true. ### Seven Up ##### Age 11 to 14 Challenge Level: The number 27 is special because it is three times the sum of its digits 27 = 3 (2 + 7). Find some two digit numbers that are SEVEN times the sum of their digits (seven-up numbers)? ### Legs Eleven ##### Age 11 to 14 Challenge Level: Take any four digit number. Move the first digit to the end and move the rest along. Now add your two numbers. Did you get a multiple of 11? ### Being Collaborative - Primary Number ##### Age 5 to 11 Challenge Level: Number problems at primary level to work on with others. ### Mini-max ##### Age 11 to 14 Challenge Level: Consider all two digit numbers (10, 11, . . . ,99). In writing down all these numbers, which digits occur least often, and which occur most often ? What about three digit numbers, four digit numbers. . . . ### Being Resilient - Primary Number ##### Age 5 to 11 Challenge Level: Number problems at primary level that may require resilience. ### Becky's Number Plumber ##### Age 7 to 11 Challenge Level: Becky created a number plumber which multiplies by 5 and subtracts 4. What do you notice about the numbers that it produces? Can you explain your findings? ### Always a Multiple? ##### Age 11 to 14 Challenge Level: Think of a two digit number, reverse the digits, and add the numbers together. Something special happens... ### The Number Jumbler ##### Age 7 to 14 Challenge Level: The Number Jumbler can always work out your chosen symbol. Can you work out how? ### Alien Counting ##### Age 7 to 11 Challenge Level: Investigate the different ways these aliens count in this challenge. You could start by thinking about how each of them would write our number 7. ### What Do You Need? ##### Age 7 to 11 Challenge Level: Four of these clues are needed to find the chosen number on this grid and four are true but do nothing to help in finding the number. Can you sort out the clues and find the number? ### Which Is Quicker? ##### Age 7 to 11 Challenge Level: Which is quicker, counting up to 30 in ones or counting up to 300 in tens? Why?

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