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https://socratic.org/questions/how-do-you-find-abs-3-2i-3
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# How do you find abs( 3-2i )?
Jun 12, 2016
$| 3 - 2 i | = \sqrt{13}$
#### Explanation:
The modulus of a complex number $a + b i$, denoted $| a + b i |$, is given by
$| a + b i | = \sqrt{{a}^{2} + {b}^{2}}$
and represents the distance of that number from the origin on the complex plane. Note that this is analogous to the absolute value of a real number.
For our given number, $3 - 2 i$, we have
$| 3 - 2 i | = \sqrt{{3}^{2} + {\left(- 2\right)}^{2}} = \sqrt{9 + 4} = \sqrt{13}$
| 181
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https://elook.org/computing/boolean-algebra.htm
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|
# Boolean algebra
<mathematics, logic> (After the logician George Boole)
1. Commonly, and especially in computer science and digital electronics, this term is used to mean two-valued logic.
2. This is in stark contrast with the definition used by pure mathematicians who in the 1960s introduced "Boolean-valued models" into logic precisely because a "Boolean-valued model" is an interpretation of a theory that allows more than two possible truth values!
Strangely, a Boolean algebra (in the mathematical sense) is not strictly an algebra, but is in fact a lattice.
A Boolean algebra is sometimes defined as a "complemented distributive lattice".
Boole's work which inspired the mathematical definition concerned algebras of sets, involving the operations of intersection, union and complement on sets.
Such algebras obey the following identities where the operators ^, V, - and constants 1 and 0 can be thought of either as set intersection, union, complement, universal, empty; or as two-valued logic AND, OR, NOT, TRUE, FALSE; or any other conforming system.
a ^ b = b ^ a
a V b
=
b V a
(commutative laws) (a ^ b) ^ c
=
a ^ (b ^ c) (a V b) V c
=
a V (b V c)
(associative laws) a ^ (b V c)
=
(a ^ b) V (a ^ c) a V (b ^ c)
=
(a V b) ^ (a V c)
(distributive laws) a ^ a
=
a
a V a
=
a
(idempotence laws) --a
=
a -(a ^ b)
=
(-a) V (-b) -(a V b)
=
(-a) ^ (-b)
(de Morgan's laws) a ^ -a
=
0
a V -a
=
1 a ^ 1
=
a
a V 0
=
a a ^ 0
=
0
a V 1
=
1 -1
=
0
-0
=
1
There are several common alternative notations for the "-" or logical complement operator.
If a and b are elements of a Boolean algebra, we define a <= b to mean that a ^ b = a, or equivalently a V b = b.
Thus, for example, if ^, V and - denote set intersection, union and complement then <= is the inclusive subset relation.
The relation <= is a partial ordering, though it is not necessarily a linear ordering since some Boolean algebras contain incomparable values.
Note that these laws only refer explicitly to the two distinguished constants 1 and 0 (sometimes written as LaTeX \top and \bot), and in two-valued logic there are no others, but according to the more general mathematical definition, in some systems variables a, b and c may take on other values as well.
< Previous Terms Terms Containing Boolean algebra Next Terms > Bookreader book titles Bookviewer bool Boolean algebraic structure binary Boolean Boolean algebra Boolean logic Boolean logic Boolean search Boole, George Booster boot
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https://vidque.com/what-is-path-compression-in-union-find/
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crawl-data/CC-MAIN-2024-18/segments/1712296818474.95/warc/CC-MAIN-20240423095619-20240423125619-00785.warc.gz
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# What is path compression in Union find?
## What is path compression in Union find?
The find() operation traverses up from x to find root. The idea of path compression is to make the found root as parent of x so that we don’t have to traverse all intermediate nodes again. If x is root of a subtree, then path (to root) from all nodes under x also compresses. The two techniques complement each other.
### What is weighting rule for union explain?
Weighted Union. A low-cost approach to reducing the height is to be smart about how two trees are joined together. One simple technique, called the weighted union rule, joins the tree with fewer nodes to the tree with more nodes by making the smaller tree’s root point to the root of the bigger tree.
#### What is path compression technique?
Path compression` is a way of flattening the structure of the tree whenever Find is used on it. Since each element visited on the way to a root is part of the same set, all of these visited elements can be reattached directly to the root.
Does Union find O 1?
The UNION operation takes O(1) time except for its two calls to FIND.
What is weighted union heuristic?
A weighted-union heuristic With this simple weighted-union heuristic, a single UNION operation can still take (m) time if both sets have (m) members. As the following theorem shows, however, a sequence of m MAKE-SET, UNION, and FIND-SET operations, n of which are MAKE-SET operations, takes O(m + n 1g n) time.
## What is Union-find algorithm with example?
A union-find algorithm is an algorithm that performs two useful operations on such a data structure: Find: Determine which subset a particular element is in. This can be used for determining if two elements are in the same subset. Union: Join two subsets into a single subset.
### Why is quick union more efficient than quick find?
Quick-union might slightly be faster in certain scenarios depending on the nature of the input. This is because with Quick-find, the union operation will always have a computational complexity greater than or equal to N. This is not the case for Quick-union, the find operation can perform computations less than N.
#### What is the union find problem?
In this lecture we describe the union-find problem. This is a problem that captures a key task one needs to solve in order to efficiently implement Kruskal’s minimum-spanning-tree algorithm. We then give two data structures for it with good amortized running time.
How an we use union-find to find the connected components of a graph?
1) As explained above, Union-Find is used to determine the connected components in a graph. We can determine whether 2 nodes are in the same connected component or not in the graph. We can also determine that by adding an edge between 2 nodes whether it leads to cycle in the graph or not.
What is the complexity of union-find?
You can do n union find (union by rank or size) operations with complexity O(n lg* n) where lg* n is the inverse Ackermann function using path compression optimization.
## What is the time complexity of union-find?
Using link-by-size, any UNION or FIND operation takes O(log n) time in the worst case, where n is the number of elements.
### What heuristics are applied to the union and find operations to improve their running times?
Heuristics to improve the running time By using two heuristics, however, we can achieve a running time that is almost linear in the total number of operations m. The first heuristic, union by rank, is similar to the weighted-union heuristic we used with the linked-list representation.
#### What is the complexity of Union find algorithm?
What is the union-find problem?
How does the weighted quick union implementation ensure that the trees in the data structure don’t get too tall?
In the weighted quick-union, we examine the size of two trees and make sure that we only link the smaller tree under the larger tree. By applying this method, we can guarantee that the tree will not grow too tall. Keep tracking the size of the tree (numbers of objects).
## What is the time complexity of Union find?
In Union by size -> When performing a union, we make the root of smaller tree point to the root of the larger. This implies O(n log n) time for performing n union find operations. Each time we follow a pointer, we are going to a subtree of size at most double the size of the previous subtree.
### Where can I use union-find?
The Union–Find algorithm is used in high-performance implementations of unification. This data structure is used by the Boost Graph Library to implement its Incremental Connected Components functionality. It is also a key component in implementing Kruskal’s algorithm to find the minimum spanning tree of a graph.
#### Does block do Union by rank or path compression?
In fact, you block does union by rank. With both union by rank and path compression, though, the expression you used can be proved easily (much more easily than the inverse Ackerman one). The proof is based on three points: On each leaf-root path, the rank of each node is increasing.
What is the difference between path compression and find ()?
With path compression, we also make 3 and 0 as the child of 9 so that when find () is called next time for 0, 1, 2 or 3, the path to root is reduced. ——–9——- / / / \\ \\ 0 4 5 6 3 / \\ / \\ 7 8 1 2 The two techniques complement each other. The time complexity of each operation becomes even smaller than O (Logn).
What does the new union algorithm look like?
Here is what the new union algorithm looks like: Now we have increased the efficiency of the union algorithm from 0 (n) to 0 (log2 (n)), because the depth of a tree will never exceed the log of n elements. This means that as the size of the data set increases, the time it takes for the union algorithm to complete will not increase by much.
## How does the size of the data set affect the Union?
This means that as the size of the data set increases, the time it takes for the union algorithm to complete will not increase by much. We can connect trees trees containing billions of elements in a matter of seconds. We can improve the algorithm even more by compressing the trees whenever we create a new union.
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https://www.jiskha.com/questions/1103039/the-wheels-of-a-scooter-have-a-diameter-of-4-5-inches-if-the-person-riding-the-scooter-is
| 1,627,082,684,000,000,000
|
text/html
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crawl-data/CC-MAIN-2021-31/segments/1627046150067.51/warc/CC-MAIN-20210723210216-20210724000216-00169.warc.gz
| 877,354,061
| 4,800
|
# Trigonometry
The wheels of a scooter have a diameter of 4.5 inches. If the person riding the scooter is traveling down hill at 15.0 mph. what is the approximate angular speed of the wheels in radians per second
1. 👍
2. 👎
3. 👁
1. Circumference = pi*D = 3.14 * 4.5 = 14.13 In. = 1.1775 Ft
V=15mi/h * 5280Ft/mi * 6.28Rad/1.1775Ft
* 1h/3600s. = 117.3 Rad/s.
1. 👍
2. 👎
2. <'ea14
1. 👍
2. 👎
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https://www.softmath.com/tutorials-3/reducing-fractions/polynomial-functions-and-their.html
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crawl-data/CC-MAIN-2024-38/segments/1725700650883.10/warc/CC-MAIN-20240907131200-20240907161200-00655.warc.gz
| 973,091,139
| 9,629
|
# Polynomial Functions and Their Graphs
Definition of a Polynomial Function
Let n be a nonnegative integer and let be
real numbers, with an ≠ 0 .
The function defined by is called
a polynomial function of x of degree n. The number an , the
coefficient of the variable to the highest power, is called the
leading coefficient.
Note: The variable is only raised to positive integer powers–no
negative or fractional exponents.
However, the coefficients may be any real numbers, including
fractions or irrational numbers like π or .
Graph Properties of Polynomial Functions
Let P be any nth degree polynomial function with real coefficients.
The graph of P has the following properties .
1. P is continuous for all real numbers, so there are no breaks,
holes, jumps in the graph.
2. The graph of P is a smooth curve with rounded corners and no
sharp corners.
3. The graph of P has at most n x-intercepts.
4. The graph of P has at most n – 1 turning points.
Example 1: Given the following polynomial functions, state the
leading term, the degree of the polynomial and the leading
coefficient.
End Behavior of a Polynomial
Odd-degree polynomials look like y = ±x3.
Even-degree polynomials look like y = ±x2 .
Power functions :
A power function is a polynomial that takes the form f(x) = axn ,
where n is a positive integer . Modifications of power functions can
be graphed using transformations.
Even-degree power functions: Odd-degree power functions:
Note: Multiplying any function by a will multiply all the y-values
by a. The general shape will stay the same. Exactly the same as it
was in section 3.4.
Zeros of a Polynomial
Example 1:
Find the zeros of the polynomial and then sketch the graph.
P(x) = x3 − 5x2 + 6x
If f is a polynomial and c is a real number for which f (c) = 0 , then c
is called a zero of f, or a root of f.
If c is a zero of f, then
• c is an x- intercept of the graph of f.
• (x − c) is a factor of f .
So if we have a polynomial in factored form, we know all of its xintercepts.
• every factor gives us an x-intercept.
• every x-intercept gives us a factor.
Example 2: Consider the function
f(x) = −3x(x − 3)4(5x − 2)(2x −1)3(4 − x)2.
Zeros (x-intercepts):
To get the degree, add the multiplicities of all the factors:
The leading term is :
Steps to graphing other polynomials:
1. Factor and find x-intercepts.
2. Mark x-intercepts on x-axis.
3. Determine the leading term.
• Degree: is it odd or even?
Sign : is the coefficient positive or negative?
4. Determine the end behavior. What does it “look like”?
Odd Degree Sign (+) Odd Degree Sign (-) Even Degree Sign (+) Even Degree Sign (-)
5. For each x-intercept, determine the behavior.
• Even multiplicity: touches x-axis, but doesn’t cross
(looks like a parabola there ).
Odd multiplicity of 1: crosses the x-axis (looks like a
line there ).
Odd multiplicity ≥ 3 : crosses the x-axis and looks like a
cubic there .
Note: It helps to make a table as shown in the examples below.
6. Draw the graph, being careful to make a nice smooth
curve with no sharp corners.
Note: without calculus or plotting lots of points, we don’t have
enough information to know how high or how low the turning
points are.
Example 3:
Find the zeros then graph the polynomial. Be sure to label the x
intercepts, y intercept if possible and have correct end behavior.
Example 4:
Find the zeros then graph the polynomial. Be sure to label the x
intercepts, y intercept if possible and have correct end behavior.
Example 5:
Find the zeros then graph the polynomial. Be sure to label the x
intercepts, y intercept if possible and have correct end behavior.
Example 6:
Find the zeros then graph the polynomial. Be sure to label the x
intercepts, y intercept if possible and have correct end behavior.
Example 7:
Given the graph of a polynomial determine what the equation of
that polynomial.
Example 8:
Given the graph of a polynomial determine what the equation of
that polynomial.
Prev Next
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# How Many Cups Are In 9 Quarts? Update New
Let’s discuss the question: how many cups are in 9 quarts. We summarize all relevant answers in section Q&A of website Myyachtguardian.com in category: Blog MMO. See more related questions in the comments below.
## Which proportion could you use to convert 9 quarts to cups?
It is simple to convert Quarts (qt) to Cups (cup). In fact, all volume conversions are easy to solve, you just need to know the ratio from one unit type to another. In this case, to convert from Quarts to Cups, all you have to do is multiply by 4. This is the ratio of one qt to one cup.
## How many cups can 9 quarts hold?
Quarts to cups conversion table
Quarts to Cups Quarts to Cups
6 quarts = 24 cups 16 quarts = 64 cups
7 quarts = 28 cups 17 quarts = 68 cups
8 quarts = 32 cups 18 quarts = 72 cups
9 quarts = 36 cups 19 quarts = 76 cups
### How to Measure Cups, Pints, Quarts, and Gallons
How to Measure Cups, Pints, Quarts, and Gallons
How to Measure Cups, Pints, Quarts, and Gallons
## How many cups are in the quart?
How many cups in a quart? There are 4 cups in 1 quart. There are 8 cups in 2 quarts.
## Does 8 cups equal 1 quart?
Table 1. Conversions: Cups to Quarts, etc.
Cups Pints Quarts
4 c 2 pt 1 qt
8 c 4 pt 2 qt
12 c 6 pt 3 qt
16 c 8 pt 4 qt
20 thg 8, 2021
## How many quarts is 6 cups 1 quart 4 cups?
Cup to Quart Conversion Table
Cups Quarts
4 c 1 qt
5 c 1.25 qt
6 c 1.5 qt
7 c 1.75 qt
## How many quarter cups are in a quart?
There are 4 cups in a quart.
## What is C to PT?
Cup to Pint Conversion Table
Cups Pints
1 c 0.5 pt
2 c 1 pt
3 c 1.5 pt
4 c 2 pt
## How do you calculate quarts?
To convert a cubic inch measurement to a quart measurement, multiply the volume by the conversion ratio. The volume in quarts is equal to the cubic inches multiplied by 0.017316.
## How many cups is 8 dry quarts?
US Cups to US Quarts (Dry) table
US Cups US Quarts (Dry)
5 cup US 1.07 US qt dry
6 cup US 1.29 US qt dry
7 cup US 1.50 US qt dry
8 cup US 1.72 US qt dry
## Does 2 pints make 1 quart?
How many pints in a quart? There are 2 pints in a quart.
## How much is a quart of food?
There are 4 cups in a quart.
### ✅ How Many Cups In A Quart
✅ How Many Cups In A Quart
✅ How Many Cups In A Quart
## How much is a 1 quart?
A quart (qt) is the same thing as 4 cups or 2 pints. If we still need more liquid we can switch to using gallons. A gallon (gal) is the same as 16 cups or 8 pints or 4 quarts. It is the largest liquid measurement.
## Is 4 quarts less than 1 gallon?
1 gallon equals 4 quarts because 1×4=4.
## How many quarts is 8 glasses of water?
8 cups equal 2 quarts.
## What is half of 9 cups?
Scale, Half and Double Quantity Amounts in a Recipe (Chart)
Original Recipe Measure Half Scaled Measure Double Scaled Measure
4 cups (2 pints, or 1 quart) 2 cups (1 pint) 8 cups (1/2 gal.)
4 1/2 cups 2 1/4 cups 9 cups
5 cups (1 1/4 quarts) 2 1/2 cups 10 cups (2 1/2 quarts)
5 1/2 cups 2 3/4 cups 11 cups
9 thg 10, 2008
## Does 1/8 tsp equal a pinch?
Dash: 1/8 tsp. Pinch: 1/16 tsp. Smidgen or Shake: 1/32 tsp.
## Is a gallon 16 cups?
There are 16 cups (C) in 1 gallon (gal). Cups and gallons are measurements of volume and capacity in the US customary and imperial systems of measurement. Both these systems use different definitions of the gallon, but the relationship between gallons and cups stays the same within each system.
## What is a quart of water?
The U.S. liquid quart is equal to two liquid pints, or one-fourth U.S. gallon (57.75 cubic inches, or 946.35 cubic cm); and the dry quart is equal to two dry pints, or 1/32 bushel (67.2 cubic inches, or 1,101.22 cubic cm).
## Which is bigger 1 liter or 1 quart?
An easy way to figure from liters to gallons, for example, is that a quart is a little less than a liter and 4 liters is a little more than 1 gallon. To be exact, 1 liter is 0.264 gallon (a little more than a quart), and 4 liters is 1.06 gallons. Q-I want to buy a 35 mm. camera for my wife, who loves to take pictures.
## Does 2 cups equal 1 pint?
If we remember, 8 ounces = 1 cup, 2 cups = 1 pint (or 16 ounces = 1 pint). There are generally 2 cups in 1 pint, however depending on the ingredient, this may change.
## How many C are in a fl oz?
Cup to Fluid Ounce Conversion Table
Cups Fluid Ounces
1 c 8 fl oz
2 c 16 fl oz
3 c 24 fl oz
4 c 32 fl oz
### How many cups are in a quart?
How many cups are in a quart?
How many cups are in a quart?
## What is Qt to C?
Quart to Cup Conversion Table
Quarts Cups
2 qt 8 c
3 qt 12 c
4 qt 16 c
5 qt 20 c
## How many ounces are in a C?
1 cup = 8 fluid ounces.
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https://mathnature.com/essential-ideas-12-3/
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text/html
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# Section 12.3: Counting Without Counting
## 12.3 Outline
1. Distinguish permutations and combinations
3. Which method?
## 12.3 Essential Ideas
Fundamental Counting Principle
If one task can be performed in m ways, and if, after that task is completed, a second task can be performed in n ways, then the total number of ways of ways for both tasks is found by multiplication: mn. Repetitions are allowed.
Permutations
A permutation of r elements selected from a set of n elements is an ordered arrangement of those r elements selected without repetitions. The order of selection is important.
Combinations
A combination of r elements selected from a set of n elements is an subset of r elements selected without repetitions. The order of selection is not important.
Counting Formulas
Factorial: n!= n(n − 1)(n − 2)(n − 3) …(3)(2)(1).
Count-down formula: n! =n(n − 1)!
Permutation formula: The number of ways of selecting r elements from a set with cardinality n in which the order of section is important is n!/(n −r)!
Combination formula: The number of ways of selecting r elements from a set with cardinality n in which the order of selection is not important is n!/r!(n −r)!.
Number of distinguishable permutations: If n objects are partitioned so that there are n1 of one kind, n2 of another, and n3 of still a third kind (so that n1 + n2 + n3 = n), then the number of ways that the n objects can be selected is n! divided by the product of (n1)!(n2)!(n3)!.
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https://betterlesson.com/lesson/576058/making-waves-with-pasta-day-2-of-2?from=mtp_lesson
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# Making Waves... with pasta! (Day 2 of 2)
1 teachers like this lesson
Print Lesson
## Objective
SWBAT generate the graphs of sine and cosine using the unit circle and find the domain, range, and intercepts for the sine and cosine graphs.
#### Big Idea
Students build the sine and cosine functions using the unit circle and pasta.
## Building the Sine and Cosine Curves
40 minutes
Students should pick up where they left off on building their sine and cosine functions. Students should be working through the Student Handout - making waves and using the Building waves templates as an aide. See the Making Waves with pasta Day 1 lesson for more details.
This is what one of my student's work looked like as he started day 2: Sine Curve in Progress
As students completed their work I asked them to hang it around the classroom. It was helpful to have these basic parent functions displayed around the classroom as a reference when we moved into shifting sine and cosine functions and as students started to learn how to write an equation to model various trigonometric situations.
If students finish early, I encourag them to get a head start on their homework for the night.
## Closure: Clicker Questions
10 minutes
In the last 10 minutes of class, I stopped my students and asked them to reflect on the four clicker questions on pages 2-5 of today’s Flipchart_ Making waves. Hopefully, students should be using their newly created graphs to identify the domain and range of the sine and cosine functions.
## Homework
As students finished or at the end of class I assigned Homework 6 - Trigonometric Functions.docx to be completed this evening.
Please note: I have continued numbering these homeworks from the last unit (Rotaitons and Cyclical Functions) as the topics are closley related and my district has us teach this as one large unit. So the first homework assignment in this mini-unit will be #6.
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https://howmanyis.com/length/117-yd-in-mi/94091-3-yards-in-miles
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How many is
Conversion between units of measurement
You can easily convert 3 yards into miles using each unit definition:
Yards
yard = 3 ft = 0.9144 m
Miles
5280 ft = 1609.344 m
With this information, you can calculate the quantity of miles 3 yards is equal to.
## ¿How many mi are there in 3 yd?
In 3 yd there are 0.0017045455 mi.
Which is the same to say that 3 yards is 0.0017045455 miles.
Three yards equals to zero miles. *Approximation
### ¿What is the inverse calculation between 1 mile and 3 yards?
Performing the inverse calculation of the relationship between units, we obtain that 1 mile is 586.66667 times 3 yards.
A mile is five hundred eighty-six times three yards. *Approximation
| 191
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×
×
# Group Project: Collect the shoe size of everyone in the
ISBN: 9780321629111 32
## Solution for problem 1.33 Chapter 1
Probability and Statistics for Engineers and the Scientists | 9th Edition
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5
Problem 1.33
Group Project: Collect the shoe size of everyone in the class. Use the sample means and variances and the types of plots presented in this chapter to summarize any features that draw a distinction between the distributions of shoe sizes for males and females. Do the same for the height of everyone in the class.
Step-by-Step Solution:
Step 1 of 3
Statistics 401 Notes Statistics is about collecting data, organizing it, summarization, and then analyzing it. Stats Collecting o Ex: Surveys Organization o Grouping the data using graphs or charts Summarization o Finding the average- median, mean, mode o (center and spread of data- Ex: Standard deviation) o...
Step 2 of 3
Step 3 of 3
##### ISBN: 9780321629111
The full step-by-step solution to problem: 1.33 from chapter: 1 was answered by , our top Statistics solution expert on 05/06/17, 06:21PM. Probability and Statistics for Engineers and the Scientists was written by and is associated to the ISBN: 9780321629111. This textbook survival guide was created for the textbook: Probability and Statistics for Engineers and the Scientists, edition: 9. Since the solution to 1.33 from 1 chapter was answered, more than 226 students have viewed the full step-by-step answer. The answer to “Group Project: Collect the shoe size of everyone in the class. Use the sample means and variances and the types of plots presented in this chapter to summarize any features that draw a distinction between the distributions of shoe sizes for males and females. Do the same for the height of everyone in the class.” is broken down into a number of easy to follow steps, and 55 words. This full solution covers the following key subjects: . This expansive textbook survival guide covers 18 chapters, and 1582 solutions.
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## Solve the exercises at the end of Chapter X (Chapter 10 Math 7 Connect) – Math Book
Solve the exercises at the end of Chapter X (Chapter 10 Math 7 Connect)
### Solution 10.20 page 102 Math 7 Textbook Connecting knowledge volume 2 – KNTT
A box is made in the form of a rectangular box out of cardboard with a length of 20 cm, a width of 14 cm and a height of 15 cm.
a) The volume of the box.
b) Calculate the area of the cover used to make the box.
Detailed instructions for solving problems 10.20
Solution method
Area of rectangle $${S_{xq}} = 2\left( {a + b} \right).c$$
Surrounding area of cube: $${S_{xq}} = 4{a^2}$$.
Volume of rectangular box $$V = abc$$.
Volume of cube $$V = {a^3}$$.
Detailed explanation
a) The volume of the box is:
20. 14. 15 = 4200 (cm3)
b) The area of the cover to make the box corresponds to the surrounding area and the area of the 2 bottom faces of the rectangular box
The area of the cover used to make the box is:
2. ( 14 + 20 ). 15 + 2. 20. 14 = 1580 (cm2)
–>
— *****
### Solve problem 10.21 page 102 Math 7 textbook Connecting knowledge volume 2 – KNTT
Calculate the volume, perimeter, and total area of the rectangular box and prism in Figure 10.43
Detailed instructions for solving problems 10.21
Solution method
– Area of rectangular box $${S_{xq}} = 2\left( {a + b} \right).c$$
– Volume of rectangular box $$V = abc$$.
– Surrounding area of triangular vertical prism, quadrilateral vertical prism: $${S_{xq}} = Ch$$
– Volume of triangular vertical prism, tetragonal vertical prism: $$V = {S_{day}}.h$$
Detailed explanation
The area around the rectangular box is:
2. (4 + 9). 9 = 234
The total area of the rectangular box is:
234 + 2 . 9 . 4 = 306
The volume of the rectangular box is:
9 . 4 . 9 = 324
The area around the prism is:
20 . ( 5 + 12 + 13 ) = 600
The total area of the prism is:
600 + 2 . $$\frac{1}{2}$$ . 5 . 12 = 660
The volume of the rectangular box is:
20 x $$\frac{1}{2}$$ x 5 x12 = 600
–>
— *****
### Solve problem 10.22 page 102 Math 7 textbook Connecting knowledge volume 2 – KNTT
A number of bricks are arranged in the shape of a rectangular box to form a cube of side 20 cm as shown in Figure 10.44.
a) Calculate the perimeter and total area of the cube.
b) Find the size of each brick.
Detailed instructions for solving Lesson 10.22
Solution method
– Surrounding area of cube: $${S_{xq}} = 4{a^2}$$.
– Volume of cube $$V = {a^3}$$.
– Area of rectangular box $${S_{xq}} = 2\left( {a + b} \right).c$$
– Volume of rectangular box $$V = abc$$.
– Surrounding area of triangular vertical prism, quadrilateral vertical prism: $${S_{xq}} = Ch$$
– Volume of triangular vertical prism, tetragonal vertical prism: $$V = {S_{day}}.h$$
Detailed explanation
a) The area around the cube block is:
4 . 202 = 1600 (cm2)
The area of the bottom surface of the cube is:
20 . 20 = 400 (cm2)
The total area of the cube is:
1600 + 2 . 400 = 2400 (cm2)
b) According to the figure, the width of the rectangular brick is equal to $$\frac{1}{2}$$ next to the cube
The width of the rectangular box is:
20 : 2=10 (cm)
The height of the brick is equal to $$\frac{1}{4}$$ side of the cube
The height of the brick is:
20:4=5 (cm)
So each brick has dimensions: length 20cm, width 10cm, height 5cm.
–>
— *****
### Solution 10.23 page 102 Math 7 Textbook Connecting knowledge volume 2 – KNTT
A rectangular room has a length of 5 m, a width of 4 m and a height of 3 m. People want to roll paint the walls and ceiling. Ask the area to be painted in square meters, knowing that the total area of the doors is 5.8 m2 ?
Detailed instructions for solving Lesson 10.23
Solution method
– Area of rectangular box $${S_{xq}} = 2\left( {a + b} \right).c$$
– Surrounding area of triangular vertical prism, quadrilateral vertical prism: $${S_{xq}} = Ch$$
Detailed explanation
The area around the room is:
2 . ( 5 + 4 ). 3 = 21 (m2)
The total area of the room is:
21 + 2 . 5 . 4 = 61 (m2)
The area to be painted is the total area of the room minus the area of the doors, so the paint roller area is:
61 – 5.8 = 55.2 (m2)
–>
— *****
### Solve problem 10.24 page 102 Math 7 Textbook Connecting knowledge volume 2 – KNTT
A rectangular fish tank made of glass (without lid) is 80cm long, 50cm wide, 45cm high. The initial water level in the tank is 35 cm high.
a) Calculate the area of glass used to make that fish tank
b) A decorative stone is placed in the tank completely submerged in water, the water level of the tank rises to 37.5 cm. Calculate the volume of the rock.
Detailed instructions for solving Lesson 10.24
Solution method
Area of rectangle $${S_{xq}} = 2\left( {a + b} \right).c$$
Surrounding area of cube: $${S_{xq}} = 4{a^2}$$.
Volume of rectangular box $$V = abc$$.
Volume of cube $$V = {a^3}$$.
Detailed explanation
a) The area around the fish tank is:
2 . (80 + 50) . 45 = 11700 (cm2)
The glass area needed to make the fish tank is the surrounding area and the area of one bottom surface of the rectangular box, so the required glass area is:
11700 + ( 80 . 50 ) = 15700 (cm2)
b)
The additional height of the water level is :
37.5 – 35 = 2.5 (cm)
The volume of water that rises after the stone is thrown will be equal to the volume of the stone, so the volume of the stone is:
4000 x 2.5 = 10000 (cm3 )
–>
— *****
### Solution 10.25 page 102 Math textbook 7 Connecting knowledge volume 2 – KNTT
A cylindrical cup filled with water. If you put 5 cubes of ice cubes of side 2 cm into the cup, how much water will come out?
Detailed instructions for solving Lesson 10.25
Solution method
Volume of cube $$V = {a^3}$$.
Detailed explanation
The volume of a stone is:
23 = 8 (cm3 )
The total volume of the 5 stones is:
8 . 5 = 40 (cm3 )
The volume of 5 ice cubes will be equal to the volume of water that rises after adding ice
=> The amount of water spilled will be 80 cm3 water.
–>
— *****
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https://www.sscadda.com/target-ssc-cgl-10000-questions-quant-questions-for-ssc-cgl-day-92/
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crawl-data/CC-MAIN-2023-14/segments/1679296945289.9/warc/CC-MAIN-20230324211121-20230325001121-00033.warc.gz
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Latest SSC jobs » Target SSC CGL | 10,000+ Questions...
# Target SSC CGL | 10,000+ Questions | Quant Questions For SSC CGL : Day 92
This is the new year with new goals, new experiences and lots of exams to be scheduled soon. SSC CGL has recently released the exam dates so now it is time to gear up your preparations and try hard to get success. ADDA247 never fails to deliver something new and fruitful for you all. This time also we are providing you the best study plan as well as a study material. We are here going to prepare your Quantitative Aptitude section for the SSC CGL. In this article, we are providing you the details that how we are going to help you to clear the examination this year. We ADDA247 is going to provide you daily tests for all the subjects. The topic-wise quiz will be done from January till April. This will help you to get a deeper knowledge of all the topics and will prepare you thoroughly.
Q1. If x runs are scored by A, y runs by B and z runs by C, then x : y = y : z = 3 : 2. If total number of runs scored by A, B and C is 342, the runs scored by each would be respectively
(a) 144, 96, 64
(b) 162, 108, 72
(c) 180, 120, 80
(d) 189, 126, 84
Q2. Rs 900 is divided among A, B, C; the division is such that 1/2 of A’s money =1/3rd of B’s money =1/4th of C’s money. Find the amount (in Rs) received by A, B, C.
(a) 300, 400, 200
(b) 350, 450, 100
(c) 200, 300, 400
(d) 400, 150, 350
Q3. If Rs 126.50 is divided among A, B and C in the ratio of 2 : 5 : 4, the share of B exceeds that of A by
(a) Rs 36.50
(b) Rs 35.50
(c) Rs 34.50
(d) Rs 33.50
Q4. The average of first three numbers is double of the fourth number. If the average of all the four numbers is 12, find the 4th number.
(a) 16
(b) 48/7
(c) 20
(d) 18/7
Q5. If the average of 6 consecutive even numbers is 25, the difference between the largest and the smallest number is
(a) 8
(b) 10
(c) 12
(d) 14
Q6. A train goes from Ballygunge to Sealdah at an average speed of 20 km/hour and comes back at an average speed of 30 km/hour. The average speed of the train for the whole journey is
(a) 27 km/hr
(b) 26 km/hr
(c) 25 km/hr
(d) 24 km/hr
Q7. The arithmetic mean of 100 observations is 24.6 is added to each of the observations and, then each of them is multiplied by 2.5. Find the new arithmetic mean.
(a) 30
(b) 75
(c) 35
(d) 60
Q8. Sachin Tendulkar has a certain average for 11 innings. In the 12th innings he scores 120 runs and thereby increases his average by 5 runs. His new average is
(a) 60
(b) 62
(c) 65
(d) 66
Q9. The average of 11 results is 50. If the average of the first six results is 49 and that of the last six is 52, the sixth result is
(a) 48
(b) 50
(c) 52
(d) 56
Q10. By selling 25 metres of cloth a trader gains the selling price of 5 metres of cloth. The gain of the trader in % is
(a) 25
(b) 20
(c) 28
(d) 29
Solutions
#### Congratulations!
General Awareness & Science Capsule PDF
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https://www.math.ucdavis.edu/~kouba/CalcTwoDIRECTORY/arclengthdirectory/solution08.html
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:browse confirm e Area Solution 8
SOLUTION 8: $\ \$ If $y = (2/3)(x^2+1)^{3/2}$ for $0 \le x \le 2$, then $$\displaystyle{ { dy \over dx} = (2/3)(3/2)(x^2+1)^{1/2}(2x) = 2x(x^2+1)^{1/2} }$$ so that $$ARC = \displaystyle{ \int_{0}^{2} \sqrt{ 1 + \Big({dy \over dx}\Big)^2 } \ dx }$$ $$= \displaystyle{ \int_{0}^{2} \sqrt{ 1 + ( 2x(x^2+1)^{1/2} )^2 } \ dx }$$ $$= \displaystyle{ \int_{0}^{2} \sqrt{ 1 + 2^2x^2(x^2+1) } \ dx }$$ $$= \displaystyle{ \int_{0}^{2} \sqrt{ 1 + 4x^4+4x^2 } \ dx }$$ $$= \displaystyle{ \int_{0}^{2} \sqrt{ (2x^2)^2+2(2x^2) + 1 } \ dx }$$ $$= \displaystyle{ \int_{0}^{2} \sqrt{ (2x^2+1)^2 } \ dx }$$ $$= \displaystyle{ \int_{0}^{2} (2x^2+1) \ dx }$$ $$= \displaystyle{ \Big( 2 {x^3 \over 3} +x \Big) \ \Big\vert_{0}^{2} }$$ $$= \displaystyle{ \Big( 2 {(2)^3 \over 3} + 2 \Big) - \Big( 2 {(0)^3 \over 3} + 0 \Big) }$$ $$= \displaystyle{ {16 \over 3} + { 6 \over 3 } }$$ $$= \displaystyle{ {22 \over 3} }$$
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http://mathhelpforum.com/differential-equations/167260-u_-xy-u_-yz-u_-zx-u-0-a.html
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# Thread: u_{xy}+u_{yz}+u_{zx}-u=0
1. ## u_{xy}+u_{yz}+u_{zx}-u=0
Using separation of variables
$u_{xy}+u_{yz}+u_{zx}-u=0$
$u(x,y,z)=\varphi(x)\psi(y)\omega(z)$
$\varphi'(x)\psi'(y)\omega(z)+\varphi(x)\psi'(y)\om ega'(z)+\varphi'(x)\psi(y)\omega'(z)-\varphi(x)\psi(y)\omega(z)=0$
$\varphi'(x)[\psi'(y)\omega(z)+\psi(y)\omega'(z)]+\varphi(x)[\psi'(y)\omega'(z)-\psi(y)\omega(z)]=0$
$\displaystyle\frac{\varphi'(x)}{\varphi(x)}=-\left(\frac{\psi'(y)\omega(z)+\psi(y)\omega'(z)}{\ psi'(y)\omega'(z)-\psi(y)\omega(z)}\right)$
Now what?
2. Well, pick a constant of separation:
$\displaystyle\frac{\varphi'(x)}{\varphi(x)}=\lambd a=-\left(\frac{\psi'(y)\omega(z)+\psi(y)\omega'(z)}{\ psi'(y)\omega'(z)-\psi(y)\omega(z)}\right).$
The first equality gives you a DE for $\varphi.$ The second equality gives you another equation that, I believe separates out thus:
$\psi'(y)\omega(z)+\psi(y)\omega'(z)=-\lambda(\psi'(y)\omega'(z)-\psi(y)\omega(z)).$
You can do the same sort of trick you just did. That is, solve for $\psi'(y)/\psi(y),$ and choose another separation constant.
You follow?
3. Originally Posted by dwsmith
$u(x,y,z)=\varphi(x)\psi(y)\omega(z)$
How do you know this?
4. $\varphi'(x)-\lambda\varphi(x)=0\Rightarrow m-\lambda=0\Rightarrow m=\lambda$
$\displaystyle\lambda=-\left(\frac{\psi'(y)\omega(z)+\psi(y)\omega'(z)}{\ psi'(y)\omega'(z)-\psi(y)\omega(z)}\right)}$
$\displaystyle\Rightarrow\lambda\psi'(y)\omega'(z)-\lamba\psi(y)\omega(z)}\right)+\psi'(y)\omega(z)+\ psi(y)\omega'(z)=0$
$\Rightarrow\psi'(y)[\lambda\omega'(z)+\omega(z)}]+\psi(y)[\omega'(z)-\lambda\omega(z)]=0$
$\displaystyle\Rightarrow\mu=\frac{\psi'(y)}{\psi(y )}=-\left(\frac{\omega'(z)-\lambda\omega(z)}{\lambda\omega'(z)+\omega(z)}\rig ht)$
$\psi'(y)-\mu\psi(y)=0\Rightarrow n=\mu$
$\displaystyle\Rightarrow\mu=-\left(\frac{\omega'(z)-\lambda\omega(z)}{\lambda\omega'(z)+\omega(z)}\rig ht)$
$\omega'(z)-\lambda\omega(z)+\mu\lambda\omega'(z)+\mu\omega(z) =0$
$\displaystyle\omega'(z)[1+\mu\lambda]+\omega(z)[\mu-\lambda]=0\Rightarrow a(1+\mu\lambda)+\mu-\lambda=0\Rightarrow a=\frac{\lambda-\mu}{1+\mu\lambda}$
$\displaystyle u(x,y,z;\lambda,\mu)=\exp{(x\lambda)}*\exp{(y\mu)} *\exp{\left(z\frac{\lambda-\mu}{1+\mu\lambda}\right)}$
I tried the substitution of $\displaystyle\mu=s \ \mbox{and} \ \lambda=\frac{1-\mu t}{\mu+t}$ in order to manipulate the exponential method to the separations method but it fell short.
Exponential solution:
$\displaystyle u(x,y,z;s,t)=\exp{\left(x\frac{1-st}{s+t}\right)}*\exp{(ys)}*\exp{(zt)}$
What substitution do I need to make?
Thanks.
5. Originally Posted by chiph588@
How do you know this?
This is from Elementary PDE by Berg and McGregor 1966.
...different technique for finding a particular solutions of homogeneous linear PDE is called the method of separation of variables. The exponential solution $\exp{(rx+sy)}=\exp{(rx)}*\exp{(sy)}$, are products of functions of the separable variables.... We seek a solution of the form $u(x,y)=\varphi(x)\psi(y)$ (pg. 14).
6. Originally Posted by dwsmith
$\varphi'(x)-\lambda\varphi(x)=0\Rightarrow m-\lambda=0\Rightarrow m=\lambda$
$\displaystyle\lambda=-\left(\frac{\psi'(y)\omega(z)+\psi(y)\omega'(z)}{\ psi'(y)\omega'(z)-\psi(y)\omega(z)}\right)}$
$\displaystyle\Rightarrow\lambda\psi'(y)\omega'(z)-\lamba\psi(y)\omega(z)}\right)+\psi'(y)\omega(z)+\ psi(y)\omega'(z)=0$
$\Rightarrow\psi'(y)[\lambda\omega'(z)+\omega(z)}]+\psi(y)[\omega'(z)-\lambda\omega(z)]=0$
$\displaystyle\Rightarrow\mu=\frac{\psi'(y)}{\psi(y )}=-\left(\frac{\omega'(z)-\lambda\omega(z)}{\lambda\omega'(z)+\omega(z)}\rig ht)$
$\psi'(y)-\mu\psi(y)=0\Rightarrow n=\mu$
$\displaystyle\Rightarrow\mu=-\left(\frac{\omega'(z)-\lambda\omega(z)}{\lambda\omega'(z)+\omega(z)}\rig ht)$
$\omega'(z)-\lambda\omega(z)+\mu\lambda\omega'(z)+\mu\omega(z) =0$
$\displaystyle\omega'(z)[1+\mu\lambda]+\omega(z)[\mu-\lambda]=0\Rightarrow a(1+\mu\lambda)+\mu-\lambda=0\Rightarrow a=\frac{\lambda-\mu}{1+\mu\lambda}$
$\displaystyle u(x,y,z;\lambda,\mu)=\exp{(x\lambda)}*\exp{(y\mu)} *\exp{\left(z\frac{\lambda-\mu}{1+\mu\lambda}\right)}$
I tried the substitution of $\displaystyle\mu=s \ \mbox{and} \ \lambda=\frac{1-\mu t}{\mu+t}$ in order to manipulate the exponential method to the separations method but it fell short.
Exponential solution:
$\displaystyle u(x,y,z;s,t)=\exp{x\left(\frac{1-st}{s+t}\right)}*\exp{(ys)}*\exp{(zt)}$
What substitution do I need to make?
Thanks.
Dear dwsmith,
Think about it like this. You have the two solutions,
$\displaystyle u(x,y,z;\lambda,\mu)=\exp{(x\lambda)}*\exp{(y\mu)} *\exp{\left(z\frac{\lambda-\mu}{1+\mu\lambda}\right)}$--------(Seperation method)
$\displaystyle u(x,y,z;s,t)=\exp{x\left(\frac{1-st}{s+t}\right)}*\exp{(ys)}*\exp{(zt)}$---------(Exponential method)
Consider the exponential part with y as the variable. Since the coefficient of y must be equal in both of these equations,
$\mu=s---------(1)$
Now move on to the exponential term with z as the variable. Using the same reasoning,
$t=\dfrac{\lambda-\mu}{1+\mu\lambda}\Rightarrow t=\dfrac{\lambda-s}{1+s\lambda}\Rightarrow \lambda=\dfrac{t+s}{1-ts}--------(2)$
(1) and (2) gives the substitutions necessary to convert the solution obtained from the seperation method to the solution obtained from the exponential method. Hope you understood.
7. By making that substitution, we don't obtain the same solutions.
$\displaystyle u(x,y,z;\lambda,\mu)=\exp{(x\lambda)}*\exp{(y\mu)} *\exp{\left(z\frac{\lambda-\mu}{1+\mu\lambda}\right)}$
$\displaystyle s=\mu, \ t=\frac{\lambda-\mu}{1+\mu\lambda}, \ \mbox{and} \ \lambda=\frac{t+s}{1-st}$
$\displaystyle u(x,y,z;s,t)\rightarrow u(x,y,z;\lambda,\mu)=\exp{\left(x\frac{1}{\lambda} \right)}*\exp{(\mu s)}*\exp{\left(z\frac{\lambda-\mu}{1+\mu\lambda}\right)}$
Any thoughts?
I could say $\displaystyle \frac{1}{\lambda}=\lambda_2$ but would this be ok?
8. Originally Posted by dwsmith
By making that substitution, we don't obtain the same solutions.
$\displaystyle u(x,y,z;\lambda,\mu)=\exp{(x\lambda)}*\exp{(y\mu)} *\exp{\left(z\frac{\lambda-\mu}{1+\mu\lambda}\right)}$
$\displaystyle s=\mu, \ t=\frac{\lambda-\mu}{1+\mu\lambda}, \ \mbox{and} \ \lambda=\frac{t+s}{1-st}$
$\displaystyle u(x,y,z;s,t)\rightarrow u(x,y,z;\lambda,\mu)=\exp{\left(x\frac{1}{\lambda} \right)}*\exp{(\mu s)}*\exp{\left(z\frac{\lambda-\mu}{1+\mu\lambda}\right)}$
Any thoughts?
I could say $\displaystyle \frac{1}{\lambda}=\lambda_2$ but would this be ok?
Dear dwsmith,
You should get the same results. The problem is that there is a mistake in your calculations. Its in your first post. The fifth line should be,
$\displaystyle\frac{\varphi'(x)}{\varphi(x)}=-\left(\frac{\psi'(y)\omega'(z)-\psi(y)\omega(z)}{\psi'(y)\omega(z)+\psi(y)\omega' (z)}\right)$
You will have to do it all over again.....
9. Originally Posted by Sudharaka
Dear dwsmith,
You should get the same results. The problem is that there is a mistake in your calculations. Its in your first post. The fifth line should be,
$\displaystyle\frac{\varphi'(x)}{\varphi(x)}=-\left(\frac{\psi'(y)\omega'(z)-\psi(y)\omega(z)}{\psi'(y)\omega(z)+\psi(y)\omega' (z)}\right)$
You will have to do it all over again.....
Thanks. I am not redoing the problem though haha.
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https://www.enotes.com/homework-help/expansion-2x-2-1-sqrt-x-n-where-n-positive-475102
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text/html
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crawl-data/CC-MAIN-2017-13/segments/1490218189771.94/warc/CC-MAIN-20170322212949-00469-ip-10-233-31-227.ec2.internal.warc.gz
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# The expansion of (2x^(2)-1/sqrt(x))^(n) where n is a positive integer,has a term that is independent of x.Find the smallest value of n.Thanks! (Using the binomial theorem if possible).
tiburtius | High School Teacher | (Level 2) Educator
Posted on
General term in expansion (using binomial theorem) would be
(-1)^k((n),(k))(2x^2)^(n-k)(1/(sqrt x))^k
However, we are interested only in term that is independent of x so we will disregard everything else. If the term is independent of x it means that both powers of x must be the same i.e.
x^(2(n-k))=(sqrt x )^k
x^(2(n-k))=x^(k/2) because sqrt x=x^(1/2)
2(n-k)=k/2
4(n-k)=k
4n-4k=k
4n=5k
n=(5k)/4
In order to find the smallest n we need to determine the smallest k>0 for which n is a whole number. Obviously we get that for k=4 (for k=1,2,3 n is not a whole number).
n=(5\cdot4)/4
n=5
The smallest value of n is 5.
Sources:
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# Measuring and drawing angles
In this lesson, you will learn how to use a protractor to measure and be able to draw angles.
Quiz:
# Intro quiz - Recap from previous lesson
Before we start this lesson, let’s see what you can remember from this topic. Here’s a quick quiz!
## Question 5
Q1.The mathematical object to measure an angle is...
1/5
Q2.A standard protractor (the one that was used in the video) measures from 0 degrees to...
2/5
Q3.If I accidentally measured an angle from 180 degrees instead of 0 degrees, and I noticed that it measured to 150 degrees, what would the angle truly measure?
3/5
Q4.The middle point of the mathematical object, where we line up the point at which the two lines meet, and where we measure an angle, is called...
4/5
Q5.I measure an angle accidentally from 85 degrees instead of 0 degrees and it measures 112 degrees. I have done this as accurately as possible. Does this mean that my angle is 27 degrees?
5/5
Quiz:
# Intro quiz - Recap from previous lesson
Before we start this lesson, let’s see what you can remember from this topic. Here’s a quick quiz!
## Question 5
Q1.The mathematical object to measure an angle is...
1/5
Q2.A standard protractor (the one that was used in the video) measures from 0 degrees to...
2/5
Q3.If I accidentally measured an angle from 180 degrees instead of 0 degrees, and I noticed that it measured to 150 degrees, what would the angle truly measure?
3/5
Q4.The middle point of the mathematical object, where we line up the point at which the two lines meet, and where we measure an angle, is called...
4/5
Q5.I measure an angle accidentally from 85 degrees instead of 0 degrees and it measures 112 degrees. I have done this as accurately as possible. Does this mean that my angle is 27 degrees?
5/5
# Video
Click on the play button to start the video. If your teacher asks you to pause the video and look at the worksheet you should:
• Click "Close Video"
• Click "Next" to view the activity
Your video will re-appear on the next page, and will stay paused in the right place.
# Worksheet
These slides will take you through some tasks for the lesson. If you need to re-play the video, click the ‘Resume Video’ icon. If you are asked to add answers to the slides, first download or print out the worksheet. Once you have finished all the tasks, click ‘Next’ below.
Quiz:
# Measuring and drawing angles
Don’t worry if you get a question wrong! Forgetting is an important step in learning. We will recap next lesson.
## Question 5
Q1.Which of the following best represents the definition of the word 'angle' in mathematics?
1/5
Q2.A full turn would represent how many degrees?
2/5
Q3.Three quarters of a turn would represent how many degrees?
3/5
Q4.Two thirds of a turn would represent how many degrees?
4/5
Q5.If I were to turn 0 degrees, would I change direction at all?
5/5
Quiz:
# Measuring and drawing angles
Don’t worry if you get a question wrong! Forgetting is an important step in learning. We will recap next lesson.
## Question 5
Q1.Which of the following best represents the definition of the word 'angle' in mathematics?
1/5
Q2.A full turn would represent how many degrees?
2/5
Q3.Three quarters of a turn would represent how many degrees?
3/5
Q4.Two thirds of a turn would represent how many degrees?
4/5
Q5.If I were to turn 0 degrees, would I change direction at all?
5/5
# Lesson summary: Measuring and drawing angles
### Time to move!
Did you know that exercise helps your concentration and ability to learn?
For 5 mins...
Move around:
Walk
On the spot:
Dance
### Take part in The Big Ask.
The Children's Commissioner for England wants to know what matters to young people.
| 911
| 3,734
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https://math.stackexchange.com/questions/2455249/asymptotic-distribution-of-the-maximum-of-partial-sums-of-iid-binary-random-vari
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text/html
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crawl-data/CC-MAIN-2019-22/segments/1558232256764.75/warc/CC-MAIN-20190522063112-20190522085112-00069.warc.gz
| 561,915,678
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# asymptotic distribution of the maximum of partial sums of iid binary random variables (Bernoulli)
I've encountered a problem when using the random walk to approximate a Brownian motion:
Consider a sequence of iid random variable ${X_{i}}\$ where $i=1,2,...\$ with binary outcomes such that: $X_{i}=1$ with probability $p$, and $X_{i}=-1$ with $1-p$.
Define the partial sums as $Y_{n}=X_{1}+X_{2}+...+X_{n}$.
And the maximum as $Z_{n}=\max_{1\leq i \leq n} {Y_{i}}$
Let $a\$ be a positive number. The problem is to find the value of
$$\lim_{n\rightarrow\infty}\Pr\lbrace Z_{n}\geq a\rbrace$$
I tried to consider an example where $a=0.5$, then $$\Pr\{Z_{1}\geq0.5\}=p$$ $$\Pr\{Z_{2}\geq0.5\}=\Pr\{\max_{i=1,2}Y_{i}\geq0.5\}=\Pr\{\max\{X_{1},X_{1}+X_2\}\geq0.5\}$$ $$=1-\Pr\{X_1<0.5, X_1+X_2<0.5\}=1-\Pr\{X_1+X_2<0.5\mid X_1<0.5\}\Pr\{X_1<0.5\}$$ $$=1-1*(1-p)=p$$ and $$\Pr\{Z_3\geq0.5\}=1-\Pr\{X_1<0.5, X_1+X_2<0.5, X_1+X_2+X_3<0.5\}$$ but it seems complicated, according to the answer for a similar question: Distribution of maximum of partial sums of independent random variables, Markov property doesn't hold here. And the result is: $$\Pr\{Z_3\geq0.5\}=p^2+p-p^3$$
So is there a closed-form expression for this binary case? Many thanks!
• For the closed-form, I don't know but I'd doubt. In any case, you don't need it to solve your problem. – user52227 Oct 3 '17 at 8:32
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https://ch.mathworks.com/matlabcentral/cody/problems/45431-pitting-corrosion-on-a-metal-plate-count-the-number-of-pits/solutions/2541186
| 1,603,281,850,000,000,000
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crawl-data/CC-MAIN-2020-45/segments/1603107876307.21/warc/CC-MAIN-20201021093214-20201021123214-00290.warc.gz
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Cody
# Problem 45431. Pitting corrosion on a metal plate: Count the number of pits
Solution 2541186
Submitted on 13 Jun 2020 by Antoni Prus
This solution is locked. To view this solution, you need to provide a solution of the same size or smaller.
### Test Suite
Test Status Code Input and Output
1 Pass
filetext = fileread('count_pits.m') assert(isempty(strfind(filetext, 'rand'))) assert(isempty(strfind(filetext, 'fileread'))) assert(isempty(strfind(filetext, 'assert')))
filetext = 'function y = count_pits(X) if ~any(X==1) y = 0 else n=size(X,1); m=size(X,2); cord = []; for a=1:n for b=1:m if X(a,b) == 1 cord = [cord; [a b]]; end end end con = []; for c=1:size(cord,1) same = max(abs(cord(c, 1)-cord(:,1)),abs(cord(c,2)-cord(:,2))) <= 1; con = [con same]; end cord = [cord zeros(size(cord,1),1)]; conLog = con == 1; n = 1; for d = 1:size(con,2) if max(cord(conLog(:,d),3)) == 0 cord(conLog(:,d),3) = n; n = n+1; elseif nnz(unique(cord(conLog(:,d),3))) <= 1 un1 = unique(cord(conLog(:,d),3)) cord(d,3) = max(cord(conLog(:,d),3)); else un = unique(cord(conLog(:,d),3)) un = un(un~=0) for f = 1:length(un) cord((cord(:,3) == un(f)),3) =n; end cord(d, 3) = n; n= n+1; end end B = zeros(n,m); for e = 1:size(cord,1) B(cord(e,1), cord(e,2)) = cord(e,3); end B; y = length(unique(cord(:,3))) end end %This code written by profile_id 15735109 '
2 Pass
x = 1; y_correct = 1; assert(isequal(count_pits(x),y_correct))
y = 1
3 Pass
x = 0; y_correct = 0; assert(isequal(count_pits(x),y_correct))
y = 0
4 Pass
x = [1 0 1 0 1]; y_correct = 3; assert(isequal(count_pits(x),y_correct))
y = 3
5 Pass
x = [0 0 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 1 0 0 0 0 1 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 1 0 0 0 1 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 1 0 0 0 0]; assert(isequal(count_pits(x),5))
un1 = 0 1 un1 = 0 1 un1 = 0 1 un1 = 0 1 un1 = 2 un1 = 0 1 un1 = 2 un1 = 2 un1 = 3 un1 = 0 4 un1 = 0 4 un1 = 0 4 un1 = 5 y = 5
6 Pass
x = [ 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 1 0 1 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 1 1 0 0 1 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 0 1 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0 1 0 1 0 0 1 1 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0]; assert(isequal(count_pits(x),26))
un1 = 4 un = 0 2 5 un = 2 5 un1 = 7 un1 = 4 un1 = 0 6 un1 = 0 6 un1 = 0 6 un1 = 0 9 un1 = 0 10 un1 = 0 6 un1 = 0 6 un1 = 0 9 un1 = 0 10 un1 = 0 9 un1 = 0 10 un1 = 0 9 un1 = 14 un1 = 0 9 un1 = 0 9 un1 = 15 un1 = 0 17 un1 = 0 17 un1 = 0 17 un1 = 0 21 un1 = 0 21 un1 = 0 24 un1 = 0 23 un1 = 0 21 un1 = 24 un1 = 0 23 un = 0 21 23 un = 21 23 un1 = 0 25 un1 = 0 26 un1 = 0 25 un1 = 0 26 un1 = 0 26 un1 = 0 27 un1 = 0 27 un1 = 28 un1 = 0 27 un1 = 0 27 un1 = 0 27 y = 26
7 Pass
x = [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 1 0 0 0 0 1 0 0 1 0 1 0 0 0 0 0 0 0 1 0 0 0 1 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 1 1 0 0 1 0 0 0 0 0 1 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 1 1 0 0 0 0 1 1 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 1 0]; assert(isequal(count_pits(x),25))
un1 = 1 un1 = 2 un1 = 2 un1 = 3 un1 = 0 6 un1 = 0 6 un1 = 6 un1 = 0 6 un1 = 9 un1 = 0 6 un1 = 0 9 un1 = 0 12 un = 0 6 12 un = 6 12 un1 = 0 11 un1 = 0 9 un = 0 9 13 un = 9 13 un1 = 14 un1 = 14 un1 = 0 14 un1 = 0 11 un1 = 0 15 un1 = 0 15 un = 0 15 16 un = 15 16 un1 = 18 un1 = 0 18 un1 = 0 18 un1 = 19 un1 = 23 un1 = 22 un1 = 28 y = 25
8 Pass
x = [1 0 1 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 1 1 1 0 1 0 0 0 0 1 0 0 0]; assert(isequal(count_pits(x),5))
un1 = 0 5 un = 0 5 6 un = 5 6 un1 = 0 7 un1 = 7 un1 = 0 7 y = 5
9 Pass
x = eye(20); assert(isequal(count_pits(x),1))
un1 = 0 1 un1 = 0 1 un1 = 0 1 un1 = 0 1 un1 = 0 1 un1 = 0 1 un1 = 0 1 un1 = 0 1 un1 = 0 1 un1 = 0 1 un1 = 0 1 un1 = 0 1 un1 = 0 1 un1 = 0 1 un1 = 0 1 un1 = 0 1 un1 = 0 1 un1 = 0 1 un1 = 0 1 y = 1
10 Pass
x = [1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0]; assert(isequal(count_pits(x),9))
un1 = 3 un1 = 5 un1 = 9 y = 9
11 Pass
x = [0 0 0 1 0 0 0 0 0 1 0 1 1 0 0 0 0 0 1 0 0 0 0 0 1 1 1 0 0 1 0 1 0 0 0 1 0 0 1 0 1 0 0 0 0 0 1 0 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 1 1 0 0 0 0 0 0 0 0 1 0 0 0 1 1 0 1 0 0 0 1 0 0 0 0 0 1 0 0 0 0 1 0 0 0 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 1]; assert(isequal(count_pits(x),19))
un1 = 0 2 un1 = 0 2 un1 = 1 un1 = 0 2 un1 = 0 2 un1 = 0 2 un1 = 0 2 un1 = 0 2 un1 = 0 4 un1 = 5 un1 = 0 2 un1 = 0 4 un1 = 0 2 un1 = 8 un1 = 8 un1 = 0 9 un1 = 0 9 un1 = 11 un1 = 11 un1 = 13 un1 = 0 14 un1 = 14 un = 14 16 un = 14 16 un1 = 19 y = 19
12 Pass
x = [0 0 0 0 0 0 0 1 1 0 0 0 1 0 0 0 0 0 1 1 0 1 0 0 1 0 0 0 0 1 0 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 1 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 1 0 0 0 0 1 1 0 1 1 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 1 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 1 1 0 1 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 1 0 1 0 0 1 0 1 0 1 0 1 0 0 0 0 0 1 1 0 0 1 0 0 0 0 0 0 1 1 0 1 1 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 0 1 0 0 1 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0]; assert(isequal(count_pits(x),21))
un1 = 0 1 un1 = 3 un1 = 0 1 un1 = 0 2 un1 = 3 un = 0 1 2 un = 1 2 un1 = 0 7 un1 = 6 un1 = 13 un1 = 14 un1 = 0 11 un1 = 12 un1 = 0 13 un1 = 0 11 un1 = 15 un1 = 0 13 un1 = 0 13 un1 = 0 15 un1 = 0 13 un1 = 16 un1 = 0 15 un = 13 16 un = 13 16 un1 = 0 17 un1 = 0 17 un1 = 0 18 un1 = 0 15 un1 = 0 15 un1 = 0 17 un = 0 17 18 un = 17 18 un = 0 15 20 un = 15 20 un1 = 0 24 un1 = 21 un1 = 21 un1 = 0 23 un1 = 0 23 un1 = 0 24 un1 = 0 23 un1 = 26 un1 = 27 un1 = 28 y = 21
13 Pass
x = repmat([0 1;1 0],5,7); assert(isequal(count_pits(x),1))
un1 = 0 1 un1 = 0 2 un1 = 0 3 un1 = 0 1 un1 = 0 1 un = 0 1 2 un = 1 2 un1 = 0 5 un = 0 3 5 un = 3 5 un1 = 0 6 un = 0 4 6 un = 4 6 un1 = 0 7 un1 = 0 7 un1 = 0 7 un1 = 0 7 un1 = 0 7 un1 = 0 7 un1 = 0 7 un1 = 0 7 un1 = 0 7 un1 = 0 7 un1 = 0 7 un1 = 0 7 un1 = 0 7 un1 = 0 7 un1 = 0 7 un1 = 0 7 un1 = 0 7 un1 = 0 7 un1 = 0 7 un1 = 0 7 un1 = 0 7 un1 = 0 7 un1 = 0 7 un1 = 0 7 un1 = 0 7 un1 = 0 7 un1 = 0 7 un1 = 0 7 un1 = 0 7 un1 = 0 7 un1 = 0 7 un1 = 0 7 un1 = 0 7 un1 = 0 7 un1 = 0 7 un1 = 0 7 un1 = 0 7 un1 = 0 7 un1 = 0 7 un1 = 0 7 un1 = 0 7 un1 = 0 7 un1 = 0 7 un1 = 0 7 un1 = 0 7 un1 = 0 7 un1 = 0 7 un1 = 0 7 un1 = 0 7 un1 = 0 7 un1 = 0 7 un1 = 0 7 un1 = 0 7 un1 = 0 7 un1 = 0 7 un1 = 0 7 y = 1
14 Pass
x = repmat([0 1;0 0],5,7); assert(isequal(count_pits(x),35))
y = 35
15 Pass
x = [0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 1 0 1 1 0 0 1 0 1 1 0 0 0 0 1 0 1 0 1 0 0 1 1 0 1 0 1 1 0 0 0 0 1 0 0 0 0 0 0 0 1 0 1 0 1 1 0 0 1 0 0 0 1 1 1 0 1 0 0 0 0 0 1 0 0 1 1 0 0 0 1 1 0 0 0 1 1 0 0 0 0 0 0 0 1 1 0 1 1 1 0 0 0 1 1 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 0 0 0 0 0 0 1 1 1 0 0 0 1 0 0 0 0 0 1 1 1 0 1 0 0 0 0 0 1 0 0 0 1 0 1 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 1 0 0 1 1 0 0 0 1 0 0 1 0 0 1 0 0 0 1 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 1 1 1 0 0 0 1 0 0 1 1 0 0 1 0 0 1 0 0 1 0 0 0 1 0 0 1 0 0 0 0 1 1 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 1 0 1 0 0 0 0 1 0 0 0 1 0 0 0 1 1 0 1 0 0 1 0]; assert(isequal(count_pits(x),15))
un1 = 0 3 un1 = 0 4 un = 0 2 5 un = 2 5 un1 = 0 6 un1 = 0 3 un1 = 0 3 un = 3 4 un = 3 4 un = 0 6 7 un = 6 7 un1 = 0 8 un1 = 0 8 un1 = 0 8 un1 = 0 8 un1 = 0 8 un1 = 0 8 un1 = 0 8 un1 = 0 8 un1 = 0 8 un1 = 0 10 un1 = 0 10 un1 = 0 8 un1 = 9 un1 = 0 8 un1 = 0 8 un1 = 0 10 un1 = 0 10 un1 = 0 11 un1 = 0 8 un1 = 0 8 un1 = 0 10 un1 = 0 10 un1 = 0 10 un1 = 0 11 un1 = 0 11 un = 0 8 10 un = 8 10 un1 = 0 12 un1 = 0 11 un1 = 0 13 un1 = 0 13 un1 = 0 12 un1 = 14 un1 = 13 un1 = 0 12 un1 = 0 14 un1 = 0 14 un1 = 0 14 un1 = 0 19 un1 = 0 14 un1 = 18 un1 = 0 19 un1 = 0 21 un1 = 0 14 un1 = 0 19 un1 = 0 19 un1 = 0 19 un1 = 0 21 un1 = 0 14 un1 = 0 14 un1 = 0 19 un1 = 0 19 un1 = 0 21 un1 = 0 14 un1 = 0 22 un = 0 19 22 un = 19 22 un1 = 0 23 un1 = 0 21 un1 = 0 23 un1 = 0 23 un1 = 0 21 un1 = 0 23 un = 21 26 un = 21 26 un = 24 27 un = 24 27 y = 15
16 Pass
x = [0 0 1 1 1 0 1 0 0 1 0 0 0 0 0 0 0 0 1 0 1 0 1 1 0 0 0 0 0 0 1 0 0 1 0 1 1 1 1 0 1 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 1 0 1 0 0 1 0 0 1 0 0 0 0 0 0 0 1 0 1 1 1 0 0 1 1 0 0 0 0 1 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 1 1 0 1 0 1 0 0 1 0 0 1 0 1 1 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0 0]; assert(isequal(count_pits(x),13))
un1 = 0 1 un1 = 0 1 un1 = 0 1 un1 = 0 1 un1 = 0 2 un1 = 0 2 un1 = 0 1 un1 = 0 1 un1 = 0 1 un = 0 1 2 un = 1 2 un1 = 0 5 un1 = 0 5 un1 = 0 5 un1 = 6 un1 = 7 un1 = 9 un1 = 10 un = 10 11 un = 10 11 un1 = 0 13 un1 = 0 12 un1 = 0 12 un1 = 0 13 un = 0 12 13 un = 12 13 un1 = 0 15 un1 = 0 15 un1 = 0 14 un1 = 14 un1 = 0 15 un1 = 0 15 un1 = 0 18 un = 0 14 18 un = 14 18 un1 = 17 un1 = 0 15 un1 = 0 19 un1 = 0 19 un1 = 0 15 y = 13
17 Pass
x = zeros(5); assert(isequal(count_pits(x),0))
y = 0
18 Pass
x = [ 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 1 0 0 0 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 1 0 0 0 1 0 0 0 0 0 0 0 0 1 1 0 0 0 1 0 1 0 0 1 0 0 0 0 1 1 0 0 0 1 0 1 0 0 0 0 1 1 0 1 0 0 1 0 0 0 1 1 0 0 0 1 1 1 0 1 0 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 1 1 1 0 0 0 0 1 0 0 0 0 1 0 1 1 1 1 0 0 0 1 1 0 1 1 0 1 1 0 1 1 0 0 0 1 0 0 0 1 0 1 0 1 0 0 0 1 0 0 0 1 0 0 1 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 1 0 1 1 0 0 0 0 0 0 0 1 0 0 1 0 1 0 0 0 1 1 1 0 1 0 0 0 0 0 1 0 0 0 0 0 1 1 1 0 0 0 0 1 1 1 0 0 0 1 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 1 0 1 0 1 0 0 1 0 0 0 1 0 0 0 1 1 0 0 0 0 1 0 0 0 1 0 1 0 1 0 1 0 1 1 0 0 0 1 1 0 0 0 0 1 0 0 0 1 0 0 0 0 1 1 1 1 0 0 0 0 0 0 1 1 0 1 0 0 0 1 0 1 0 0 0 0 1 0 0 0 1 1 1 0 0 0 0 0 0 1 0 1 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 1 0 1 1 0 1 0 0 1 0 0 0 0 0 0 0 0]; assert(isequal(count_pits(x),24))
un1 = 0 3 un1 = 0 4 un1 = 0 4 un1 = 0 4 un1 = 0 5 un1 = 0 6 un1 = 0 3 un1 = 0 3 un1 = 0 4 un1 = 0 5 un1 = 0 6 un1 = 0 7 un1 = 0 3 un1 = 0 3 un1 = 0 5 un1 = 0 5 un1 = 0 6 un1 = 0 7 un1 = 0 7 un1 = 0 3 un1 = 0 3 un1 = 0 3 un1 = 0 5 un = 0 6 8 un = 6 8 un1 = 0 10 un1 = 0 3 un1 = 0 9 un1 = 0 10 un1 = 0 10 un1 = 0 10 un1 = 0 3 un1 = 0 9 un1 = 0 9 un1 = 12 un1 = 0 10 un1 = 0 10 un1 = 13 un1 = 0 11 un1 = 0 11 un = 3 14 un = 3 14 un1 = 0 9 un1 = 0 10 un = 0 10 13 un = 10 13 un1 = 0 11 un1 = 0 15 un1 = 0 9 un1 = 0 17 un1 = 0 15 un1 = 0 9 un1 = 0 9 un1 = 0 17 un1 = 0 19 un1 = 0 15 un = 0 9 15 un = 9 15 un1 = 0 21 un1 = 0 17 un1 = 0 19 un1 = 0 19 un1 = 0 19 un1 = 0 21 un1 = 0 22 un1 = 0 22 un1 = 0 17 un1 = 0 19 un1 = 0 22 un1 = 0 17 un1 = 0 19 un1 = 0 19 un1 = 0 17 un1 = 27 un = 19 24 un = 19 24 un1 = 0 25 un1 = 26 un1 = 0 17 un1 = 0 27 un1 = 0 27 un1 = 30 un1 = 0 25 un1 = 29 un1 = 0 27 un1 = 0 27 un1 = 0 27 un1 = 0 27 un1 = 0 25 un1 = 0 25 un1 = 0 27 un1 = 0 27 un = 0 25 33 un = 25 33 un1 = 0 34 un1 = 0 32 un1 = 0 27 un = 0 27 32 un = 27 32 un = 0 35 38 un = 35 38 un1 = 39 un1 = 39 y = 24
19 Pass
x = [0 1 0 1 1 1 0 0 0 1 0 0 1 0 0 1 1 0 0 1 0 0 0 1 1 1 0 1 1 1 0 0 0 0 1 1 1 0 0 0 0 1 0 1 1 0 0 1 1 1 0 1 1 0 1 1 1 0 0 0 1 1 1 0 1 1 1 0 1 1 0 0 1 0 1 1 0 1 1 0 1 0 0 0 1 1 1 0 0 1 1 1 1 1 1 1 1 1 0 0 1 0 1 1 1 0 0 0 0 1 1 1 1 1 1 0 0 1 0 1 1 1 0 1 1 1 0 1 0 1 1 0 0 0 0 1 1 0 1 0 0 1 0 1 1 0 1 0 0 0 1 0 0 0 1 0 1 0 0 1 1 0 1 1 1 0 0 0 1 1 0 1 0 1 1 1 1 1 0 0 0 0 0 1 1 0 0 0 1 0 1 1 1 1 0 0 0 0 1 0 0 0 1 1 0 1 1 1 1 0 1 0 0 0 0 0 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 0 1 1 0 0 1 1 0 1 1 0 1 0 0 1 0 1 0 0 0 0 0 1 0 0 0 1 1 1 0 1 1 1 1 1 0 0 1 1 1 1 1 1 0 1 0 0 1 0 1 0 1 0 0 1 0 0 0 0 0 0 1 1 0 1 1 1 0 0 1 0 0 1 0 1 1 1 1 1 0 1 0 0 0 0 0 0 1 1 0 0 0 0 1 1 1 0 1 0 0 0 1 0 0 1 0 1 1 0 0 1 1 1 1 0 1 1 1 0 1 0 0 1 0 1 1 0 1 1 0 0 0 0 1 1 0 0 0 1 1 0 1 0 1 1 1 0 1 1 1 1 1 0 0 0 1 0 0 0 0 1 0 0 1 0 1 1 1 0]; assert(isequal(count_pits(x),5))
un1 = 0 2 un1 = 0 2 un1 = 5 un1 = 0 2 un1 = 0 2 un1 = 0 2 un1 = 0 3 un1 = 0 3 un1 = 0 3 un1 = 0 5 un1 = 0 5 un1 = 0 5 un = 0 2 7 un = 2 7 un1 = 0 8 un1 = 0 3 un1 = 0 3 un1 = 0 3 un1 = 9 un1 = 0 5 un1 = 0 5 un1 = 0 5 un1 = 0 8 un1 = 0 8 un1 = 8 un1 = 0 8 un1 = 0 8 un = 0 3 8 un = 3 8 un1 = 0 10 un1 = 0 10 un1 = 0 9 un1 = 0 5 un1 = 0 5 un1 = 0 5 un1 = 0 5 un1 = 0 10 un1 = 0 10 un1 = 0 10 un1 = 0 10 un1 = 0 10 un1 = 0 10 un = 0 9 10 un = 9 10 un1 = 0 11 un = 0 5 11 un = 5 11 un1 = 0 12 un1 = 0 12 un1 = 0 12 un1 = 0 12 un1 = 0 12 un = 0 12 13 un = 12 13 un1 = 0 14 un1 = 0 14 un1 = 0 14 un1 = 0 14 un1 = 0 14 un1 = 0 14 un1 = 0 14 un1 = 0 14 un1 = 0 14 un1 = 0 14 un1 = 0 14 un1 = 0 14 un1 = 0 14 un1 = 0 14 un1 = 0 14 un1 = 0 14 un1 = 0 14 un = 0 14 15 un = 14 15 un1 = 0 17 un1 = 0 17 un1 = 0 17 un = 16 17 un = 16 17 un1 = 0 18 un1 = 0 18 un1 = 0 18 un1 = 0 18 un1 = 0 18 un1 = 0 18 un1 = 0 18 un1 = 0 18 un = 0 18 19 un = 18 19 un1 = 0 20 un1 = 0 20 un1 = 0 20 un1 = 0 20 un1 = 0 20 un1 = 0 20 un1 = 0 20 un1 = 0 20 un1 = 0 20 un1 = 0 20 un1 = 0 20 un1 = 0 20 un1 = 0 20 un1 = 0 20 un1 = 0 20 un1 = 0 20 un1 = 0 20 un1 = 0 20 un1 = 0 20 un1 = 0 20 un1 = 0 20 un = 0 20 21 un = 20 21 un1 = 0 22 un1 = 0 22 un1 = 0 22 un1 = 0 22 un1 = 0 22 un1 = 0 22 un1 = 0 22 un1 = 0 22 un1 = 0 22 un1 = 0 22 un1 = 0 22 un1 = 24 un1 = 0 22 un1 = 0 22 un1 = 0 22 un1 = 0 23 un1 = 0 22 un1 = 0 22 un1 = 0 22 un1 = 0 24 un1 = 0 22 un1 = 0 22 un1 = 0 22 un = 0 22 23 un = 22 23 un1 = 0 25 un1 = 0 25 un1 = 0 25 un1 = 0 25 un1 = 0 25 un1 = 0 25 un1 = 0 25 un1 = 0 25 un = 0 24 25 un = 24 25 un1 = 0 26 un1 = 0 26 un1 = 0 26 un1 = 0 26 un1 = 0 26 un1 = 0 26 un1 = 0 26 un1 = 27 un1 = 0 27 un = 0 26 27 un = 26 27 un1 = 0 28 un1 = 0 28 un1 = 0 28 un1 = 0 28 un1 = 0 28 un1 = 0 28 un1 = 0 28 un1 = 0 28 un1 = 0 28 un1 = 0 28 un1 = 0 28 un1 = 0 28 un1 = 0 28 un1 = 0 28 un1 = 0 28 un = 0 28 29 un = 28 29 un1 = 0 30 un1 = 0 31 un = 0 30 31 un = 30 31 un1 = 0 32 un1 = 0 32 un1 = 0 32 un1 = 0 32 un1 = 0 32 un1 = 0 32 un1 = 0 32 un1 = 0 32 un1 = 0 32 un1 = 0 32 un1 = 0 32 un = 32 33 un = 32 33 un1 = 0 34 un1 = 0 34 un1 = 0 34 un1 = 0 34 un1 = 0 34 un1 = 0 34 un1 = 0 34 un1 = 0 35 un1 = 0 35 un1 = 0 34 un1 = 0 34 un1 = 0 34 un1 = 0 34 un1 = 0 34 un1 = 0 34 y = 5
20 Pass
x = [ 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 1 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]; assert(isequal(count_pits(x),30))
un1 = 3 un1 = 0 2 un1 = 0 2 un1 = 5 un1 = 5 un1 = 7 un1 = 0 8 un1 = 0 8 un1 = 0 12 un1 = 0 12 un1 = 18 un1 = 0 22 un1 = 23 un1 = 0 22 un1 = 0 22 un1 = 24 un1 = 24 un1 = 29 y = 30
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] |
https://rutumulkar.com/blog/2014/all-about-that-bayes-intro-to-probability/
| 1,571,200,436,000,000,000
|
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# All about that Bayes - An Intro to Probability
RANDOM VARIABLES
In this world things keep happening around us. Each event occurring is a Random Variable. A Random Variable is an event, like elections, snow or hail. Random variables have an outcome attached them - the value of which is between 0 and 1. This is the likelihood of that event happening. We hear the outcomes of random variables all the time - There is a 50% chance or precipitation, The Seattle Seahawks have a 90% chance of winning the game.
SIMPLE PROBABILITY
Where do we get these numbers from? From past data.
Year 2008 2009 2010 2011 2012 2013 2014 2015 Rain Rainy Dry Rainy Rainy Rainy Dry Dry Rainy
PROBABILITY OF 2 EVENTS
What is the probability that event A and Event B happening together? Consider the following table, with data about the Rain and Sun received by Seattle for the past few years.
Year 2008 2009 2010 2011 2012 2013 2014 2015 Rain Rainy Dry Rainy Rainy Rainy Dry Dry Rainy Sun Sunny Sunny Sunny Cloudy Cloudy Cloudy Sunny Sunny
Using the above information, can you compute what is the probability that it will be Sunny and Rainy in 2016?
We can get this number easily from the Joint Distribution
RAIN Rainy Dry SUN Sunny 3/8 2/8 Cloudy 2/8 1/8
In 3 out of the 8 examples above, it is Sunny and Rainy at the same time. Similarly, in 1 out of 8 times it is Cloudy and it is Dry. So we can compute the probability of multiple events happening at the same time using the Joint Distribution. If there are more than 2 variables, the table will be of a higher dimension
We can extend this table further include Marginalization. Marginalization is just a fancy word for adding up all the probabilities in each row, and the probabilities in each column respectively.
RAIN Rainy Dry Margin SUN Sunny 0.375 0.25 0.625 Cloudy 0.25 0.125 0.375 Margin 0.625 0.375 1
Why are margins helpful? They remove the effects of one of the two events in the table. So, if we want to know the probability that it will rain (irrespective of other events), we can find it from the marginal table as 0.625. From Table 1, we can confirm this by computing all the individual instances that it rains - 5/8 = 0.625
CONDITIONAL PROBABILITY
What do we do when one of the outcomes is already given to us? On this new day in 2016, it is very sunny, but what is the probability that it will rain?
which is read as - probability that it will rain, given that there is sun.
This is computed in the same way as we compute normal probability, but we will just look at the cases where Sun = Sun from Table 1. There are 5 instances of Sun = Sun in Table 1, and in 3 of those cases Rain = Rain. So the probability of
We can also compute this from Table 3. Total probability of Sun = 0.625 (Row 1 Marginal probability). Probability of Rain and Sun = 0.375
Probability of Rain given Sun = 0.375/0.625 = 0.6
DIFFERENCE BETWEEN CONDITIONAL AND JOINT PROBABILITY
Conditional and Joint probability are often mistaken for each other because of the similarity in their naming convention. So what is the difference between: $P(AB)$ and $P(A \mid B)$
The first is Joint Probability and the second is Conditional Probability.
Joint probability computes the probability of 2 events happening together. In the case above - what is the probability that Event A and Event B both happen together? We do not know whether either of these events actually happened, and are computing the probability of both of them happening together.
Conditional probability is similar, but with one difference - We already know that one of the events (e.g. Event B) did happen. So we are looking for the probability of Event A, when we know the Event B already happened or that the probability of Event B is 1. This is a subtle but a significantly different way of looking at things.
BAYES RULE
Equating (4) and (5)
This is the Bayes Rule.
Bayes Rule is interesting, and significant, because we can use it to discover the conditional probability of something, using the conditional probability going the other direction. For example: to find the probability $P(death \mid smoking)$ , we can get this unknown from $P(smoking \mid death)$, which is much easier to collect data for, as it is easier to find out whether the person who died was a smoker or a non smoker.
Lets look at some real examples of probability in action. Consider a prosecutor, who wants to know whether to charge someone with a crime, given the forensic evidence of fingerprints, and town population.
The data we have is the following:
• One person in a town of 100,000 committed a crime. The probability that is he guilty $P(G) = 0.00001$, where $P(G)$ is the probability of a person being guilty of having committed a crime
• The forensics experts tell us, that if someone commits a crime, then they leave behind fingerprints 99% of the time. $P(F \mid G) = 0.99$, where $P(F \mid G)$ is the probability of fingerprints, given crime is commited
• There are usually 3 people’s fingerprints in any given location. So $P(F) = 3 * 0.00001 = 0.00003$. This is because only 1 in 100,000 people could have their fingerprints
We need to compute:
Using Bayes Rule we know that:
Plugging in the values that we already know:
This is a good enough probability to get in touch with the suspect, and get his side of the story. However, when the prosecutor talks to the detective, the detective points out that the suspects actually lives at the scrime scene. This makes it highly likely to find the suspect’s fingerprints in that location. And the new probability of finding fingerprints becomes : $P(F) = 0.99$
Plugging in those values again into (9), we get:
So it completely changes the probability of the suspect being guilty.
This example is interesting because we computed the probability of a $P(G \mid F)$ using the probability of $P(F \mid G)$. This is because we have more data from previous solved crimes about how many peple actually leave fingerprints behind, and the correlation of that with them being guilty.
Another motivation for using conditional probability, is that conditional probability in one direction is often less stable that the conditional probability in the other direction. For example, the probability of disease given a symptom $P(D \mid S)$ is less stable as compared to probability of symptom given disease $P(S \mid D)$
So, consider a situation where you think that you might have a horrible disease Severenitis. You know that Severenitis is very rare and the probability that someone actually has it is 0.0001. There is a test for it that is reasonably accurate 99%. You go get the test, and it comes back positive. You think, “oh no! I am 99% likely to have the disease”. Is this correct? Lets do the Math.
Let $P(H \leftarrow w)$ be the probability of Health being well, and $P(H \leftarrow s)$ be the probability of Health being sick. Let and $P(T \leftarrow p)$ be the probability of the Test being positive and $P(T \leftarrow n)$ be the probability of the Test being negative.
We know that the probability you have the disease is low $P(H \leftarrow s) = 0.0001$. We also know that the test is 99% accurate. What does this mean? It means that if you are sick, then the test will accurately predict it by 99%
$P(T \leftarrow n \mid H \leftarrow w) = 0.99$
$P(T \leftarrow n \mid H \leftarrow s) = 0.01$
$P(T \leftarrow p \mid H \leftarrow w) = 0.01$
$P(T \leftarrow p \mid H \leftarrow s) = 0.99$
We need to find out the probability that you are sick given that the test is positive or $P(H \leftarrow s \mid T \leftarrow p)$
Using Bayes Rule:
We know the numerator, but not the denominator. However, it is easy enough to compute the denominator using some clever math!
We know that the total probability of
$P(H \leftarrow s \mid T \leftarrow p) + P(H \leftarrow w \mid T \leftarrow p) = 1$
Adding (16) and (17), and equating with (15) we get:
Therefore:
Substituting (7) into (4) we get:
This is the reason why doctors are hesitant to order expensive tests if it is unlikely tht you have the disease. Even though the test is accurate, rare diseases are so rare that the very rarity dominates the accuracy of the test.
NAIVE BAYES
When someone applies Naive Bayes to a problem, they are assuming conditional independence of all the events. This means:
When this is plugged into Bayes Rules:
1. Let $P(BCD…) = \alpha$ which is the normalization constant. Then,
What we have done here, is assumed that the events A, B, C etc. are not dependent on each other, thereby reducing a very high dimensional table into several low dimensional tables. If we have 100 features, and each feature can take 2 values, then we would have a table of size $2^{100}$. However, assuming independence of events we reduce this to one hundred 4 element tables.
Naive bayes is rarely ever true, but it often works because we are not interested in the right probability, but the fact that the correct class has the highest probability.
References:
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Home »» Placement Papers »» All Papers »» TCS »» Placement Paper
### TCS latest Pattern Questions with Explanations - 1
1) The water from one outlet, flowing at a constant rate, can fill the swimming pool in 9 hours. The water from second outlet, flowing at a constant rate can fill up the same pool in approximately in 5 hours. If both the outlets are used at the same time, approximately what is the number of hours required to fill the pool?
Ans: Assume tank capacity is 45 Liters. Given that the first pipe fills the tank in 9 hours. So its capacity is 45 / 9 = 5 Liters/ Hour. Second pipe fills the tank in 5 hours. So its capacity is 45 / 5 = 9 Liters/Hour. If both pipes are opened together, then combined capacity is 14 liters/hour. To fill a tank of capacity 45 liters, Both pipes takes 45 / 14 = 3.21 Hours.
2) If 75 % of a class answered the first question on a certain test correctly, 55 percent answered the second question on the test correctly, and 20 percent answered neither of the questions correctly, what percentage answered both correctly?
It is a problem belongs to sets. We use the following formula n(A∪" role="presentation" style="display: inline; line-height: normal; font-size: 13.65px; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; color: rgb(34, 34, 34); font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; position: relative;">B) = n(A) + n(B) - n(A∩" role="presentation" style="display: inline; line-height: normal; font-size: 13.65px; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; color: rgb(34, 34, 34); font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; position: relative;">B)
Here n(A∪" role="presentation" style="display: inline; line-height: normal; font-size: 13.65px; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; color: rgb(34, 34, 34); font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; position: relative;">B) is the people who answered atleast one of the questions.
It was given that 20% answered neither question then the students who answered atleast one question is 100% - 20% = 80%
Now substituting in the formula we get 80% = 75% + 55% - n(A∩" role="presentation" style="display: inline; line-height: normal; font-size: 13.65px; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; color: rgb(34, 34, 34); font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; position: relative;">B)
⇒" role="presentation" style="display: inline; line-height: normal; font-size: 13.65px; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; color: rgb(34, 34, 34); font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; position: relative;"> n(A∩" role="presentation" style="display: inline; line-height: normal; font-size: 13.65px; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; color: rgb(34, 34, 34); font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; position: relative;">B) = 50%
3) A student's average ( arithmetic mean) test score on 4 tests is 78. What must be the students score on a 5th test for the students average score on the 5th test to be 80?
Ans: We know that Average =Sum of the observations No of observations" role="presentation" style="display: inline; line-height: normal; font-size: 13.65px; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; color: rgb(34, 34, 34); font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; position: relative;">=Sum of the observations No of observations=Sum of the observations No of observations
So Sum of 4 test scores = 78×" role="presentation" style="display: inline; line-height: normal; font-size: 13.65px; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; color: rgb(34, 34, 34); font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; position: relative;">××4=312
Sum of 5 tests scores = 80×" role="presentation" style="display: inline; line-height: normal; font-size: 13.65px; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; color: rgb(34, 34, 34); font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; position: relative;">××5=400
⇒" role="presentation" style="display: inline; line-height: normal; font-size: 13.65px; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; color: rgb(34, 34, 34); font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; position: relative;"> 5th test score=400-312=88
Alternative method: If the student scores 78 in the fifth test also, what could be his average? No change. Is it not?
But to bring the average to 80, he must have scored enough marks extra so that each of the five subject scores increase upto 80. i.e., he should have scored 2 x 5 = 10 runs extra in the fifth subject. So 5th subject score is 78 + 10 = 88
4) Rural households have more purchasing power than do urban households at the same income level, since some of the income urban and suburban households use for food and shelter can be used by the rural households for other needs. Which of the following inferences is best supported by the statement made above?
(A) The average rural household includes more people than does the average urban or suburban household.
(B) Rural households have lower food and housing costs than do either urban or suburban households.
(C) Suburban households generally have more purchasing power than do either rural or urban households.
(D) The median income of urban and suburban households is generally higher than that of rural households.
(E) All three types of households spend more of their income on housing than on all other purchases combined.
Ans: If average rural household includes more people, then how come they have more purchasing power? Infact, they have less purchasing power as they have to feed more people. Option A ruled out.
Option C does not explain why rural households have more purchasing power than urban. Ruled out.
If median income of urban and suburban households is generally higher than rural households they are likely to have more purchasing power, assuming other parameters constant. But this does not explain why rural households have more purchasing power. Options D ruled out.
Option E does not provide any explanation why rural households have more purchasing power. Ruled out.
Option B is correct as, If rural households spend less income on food and shelter due to less prices they definitely have more disposable income to spend.
5) Jose is a student of horticulture in the University of Hose. In a horticultural experiment in his final year, 200 seeds were planted in plot I and 300 were planted in plot II. If 57% of the seeds in plot I germinated and 42% of the seeds in plot II germinated, what percent of the total number of planted seeds germinated?
Ans: Total seeds germinated in Plot I = 57% of 200 = 114
Total seeds germinated in Plot II = 42% of 300 = 126
Total germinated seeds = 114 + 126 = 240
The percentage of germinated seeds of the total seeds = 240500×100" role="presentation" style="display: inline; line-height: normal; font-size: 13.65px; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; color: rgb(34, 34, 34); font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; position: relative;">240500×100240500×100 = 48%
6) A closed cylindrical tank contains 36π" role="presentation" style="display: inline; line-height: normal; font-size: 13.65px; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;">ππ cubic feet of water and its filled to half its capacity. When the tank is placed upright on its circular base on level ground, the height of water in the tank is 4 feet. When the tank is placed on its side on level ground, what is the height, in feet, of the surface of the water above the ground?
Ans: We know that the volume of cylinder = πr2h" role="presentation" style="display: inline; line-height: normal; font-size: 13.65px; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; color: rgb(34, 34, 34); font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; position: relative;">πr2hπr2h
Given tank hight = 4ft.
⇒" role="presentation" style="display: inline; line-height: normal; font-size: 13.65px; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; color: rgb(34, 34, 34); font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; position: relative;">πr24" role="presentation" style="display: inline; line-height: normal; font-size: 13.65px; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; color: rgb(34, 34, 34); font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; position: relative;">πr24πr24 = 36π" role="presentation" style="display: inline; line-height: normal; font-size: 13.65px; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; color: rgb(34, 34, 34); font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; position: relative;">ππ
⇒" role="presentation" style="display: inline; line-height: normal; font-size: 13.65px; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; color: rgb(34, 34, 34); font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; position: relative;"> r = 3
So the radius is 3 which means the diameter is 6.
As the cylinder is filled to initially exactly half of the capacity, When this cylinder is placed on its side, Water comes upto the height of the radius.
So water comes upto 3 ft.
7) The present ratio of students to teachers at a certain school is 30 to 1. If the student enrollment were to increase by 50 students and the number of teachers were to increase by 5, the ratio of the teachers would then be 25 to 1 What is the present number of teachers?
Assume the present students and teachers are 30K, K
After new recruitments of students and teachers the strength becomes 30K + 50, K + 5 respectively. But given that this ratio = 25 : 1
⇒30K+50K+5=251" role="presentation" style="display: inline; line-height: normal; font-size: 13.65px; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; color: rgb(34, 34, 34); font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; position: relative;">30K+50K+5=251⇒30K+50K+5=251
Solving we get K = 15
So present teachers are 15.
8) College T has 1000 students. Of the 200 students majoring in one or more of the sciences,130 are majoring in Chemistry and 150 are majoring in Biology. If at least 30 of the students are not majoring in either Chemistry or Biology, then the number of students majoring in both Chemistry and Biology could be any number from
If we assume exactly 30 students are not majoring in any subject then the students who take atleast one subject = 200 - 30 = 170
We know that n(A∪" role="presentation" style="display: inline; line-height: normal; font-size: 13.65px; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; color: rgb(34, 34, 34); font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; position: relative;">B) = n(A) + n(B) - n(A∩" role="presentation" style="display: inline; line-height: normal; font-size: 13.65px; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; color: rgb(34, 34, 34); font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; position: relative;">B)
⇒" role="presentation" style="display: inline; line-height: normal; font-size: 13.65px; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; color: rgb(34, 34, 34); font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; position: relative;"> 170 = 130 + 150 - n(A∩" role="presentation" style="display: inline; line-height: normal; font-size: 13.65px; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; color: rgb(34, 34, 34); font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; position: relative;">B)
Solving we get n(A∩" role="presentation" style="display: inline; line-height: normal; font-size: 13.65px; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; color: rgb(34, 34, 34); font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; position: relative;">B) = 110.
i.e., Students who can take both subjects are 110
But If more than 30 students are not taking any subject, what can be the maximum number of students who can take both the subjects?
As there are 130 students are majoring in chemistry, assume these students are taking biology also. So maximum students who can take both the subjects is 130
So the number of students who can take both subjects can be any number from 110 to 130.
9) Kelly and Chris are moving into a new city. Both of them love books and thus packed several boxes with books. If Chris packed 60% of the total number of boxes, what was the ratio of the number of boxes Kelly packed to the number of boxes Chris packed?
Simple questions. If chris packs 60% of the boxes, kelly packs remaining 40%
So Kelly : Chris = 40% : 60% = 2 : 3
10) Among a group of 2500 people, 35 percent invest in municipal bonds, 18 percent invest in oil stocks, and 7 percent invest in both municipal bonds and oil stocks. If 1 person is to be randomly selected from 2500 people, what is the probability that the person selected will be one who invests in municipal bonds but not in oil stocks?
Ans: Here 2500 is redundant
From the diagram we know that only ones who invested in municipal bonds are 28%, the probability is 28 / 100 = 7/25
11) Machine A produces bolts at a uniform rate of 120 every 40 second, and Machine B produces bolts at a uniform rate of 100 every 20 seconds. If the two machines run simultaneously, how many seconds will it take for them to produce a total of 200 bolts?
Ans: Machine A produces 120/40 = 3 bolts in 1 second and machine B produces 100/20 = 5 bolts in one second.
Hence, both of them will produce 8 bolts per second.
Hence, they wil take 200/8 = 25 seconds to produce 200 bolts.
12) How many prime numbers between 1 and 100 are factors of 7150?
Ans: 7, 150 = 2×52×11×13" role="presentation" style="display: inline; line-height: normal; font-size: 13.65px; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; color: rgb(34, 34, 34); font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; position: relative;">52×11×132×52×11×13
So there are 4 distinct prime numbers that are below 100
13) Analyzing the good returns that Halocircle Insurance Pvt Ltd was giving, Ratika bought a 1-year, Rs 10,000 certificate of deposit that paid interest at an annual rate of 8% compounded semi-annually.What was the total amount of interest paid on this certificate at maturity?
This is a question on compound interest to be calculated semi annually.
In the case of semi annual compounding, Interest rate becomes half and Number of periods becomes 2 per year.
So A = P(1+R100)n" role="presentation" style="display: inline; line-height: normal; font-size: 13.65px; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; color: rgb(34, 34, 34); font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; position: relative;">(1+R100)n(1+R100)n
⇒A=10,000(1+4100)2=10,000×2625" role="presentation" style="display: inline; line-height: normal; font-size: 13.65px; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; color: rgb(34, 34, 34); font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; position: relative;">⇒A=10,000(1+4100)2=10,000×2625⇒A=10,000(1+4100)2=10,000×2625
= 10,816
Interest = A - P = 10, 816 - 10,000 = 816
14) Juan is a gold medalist in athletics. In the month of May, if Juan takes 11 seconds to run y yards, how many seconds will it take him to run x yards at the same rate?
Ans: If juan takes 11 seconds to run Y yards, for 1 yard he will take 11 / y seconds. To run x yards his time will be 11 / y ×" role="presentation" style="display: inline; line-height: normal; font-size: 13.65px; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; color: rgb(34, 34, 34); font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; position: relative;">×× x = 11x/ y
15) A certain company retirement plan has a rule of 70 provision that allows an employee to retire when the employee's age plus years of employment with the company total at least 70. In what year could a female employee hired in 1986 on her 32nd birthday first be eligible to retire under this provision?
Assume it has taken x years to the female employee to reach the rule of 70.
So her age should be 32 + x. Also she gains x years of experience.
⇒" role="presentation" style="display: inline; line-height: normal; font-size: 13.65px; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; color: rgb(34, 34, 34); font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; position: relative;"> (32 + x) + x = 70
⇒" role="presentation" style="display: inline; line-height: normal; font-size: 13.65px; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; color: rgb(34, 34, 34); font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; position: relative;"> x = 19.
Her age at the time of retirement = 1986 + 19 = 2005
16) Of the following, which is the closest approximation of (50.2*0.49)/199.8 ?
ans: For approximation (50.2×" role="presentation" style="display: inline; line-height: normal; font-size: 13.65px; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; color: rgb(34, 34, 34); font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; position: relative;">××0.49)/199.8 can be taken as
50×" role="presentation" style="display: inline; line-height: normal; font-size: 13.65px; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; color: rgb(34, 34, 34); font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; position: relative;">××0.5/200 = 25/200 = 1/8 = 0.125
17) Andalusia has been promoting the importance of health maintenance. From January 1,1991 to January 1,1993, the number of people enrolled in health maintenance organizations increased by 15 percent. The enrollment on January 1,1993 was 45 million. How many million people(to the nearest million) was enrolled in health maintenance organizations on January 1,1991?
Ans: If a number K is to be increased by x % it should be multiplied by (100+x)100" role="presentation" style="display: inline; line-height: normal; font-size: 13.65px; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; color: rgb(34, 34, 34); font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; position: relative;">(100+x)100(100+x)100
So When the enrollment in January 1, 1991 is multiplied by (100+x)100" role="presentation" style="display: inline; line-height: normal; font-size: 13.65px; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; color: rgb(34, 34, 34); font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; position: relative;">(100+x)100(100+x)100 we got 45 million.
K×(100+15)100=45" role="presentation" style="display: inline; line-height: normal; font-size: 13.65px; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; color: rgb(34, 34, 34); font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; position: relative;">(100+15)100=45K×(100+15)100=45
K = 45×100115" role="presentation" style="display: inline; line-height: normal; font-size: 13.65px; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; color: rgb(34, 34, 34); font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; position: relative;">45×10011545×100115 = 39.13
18) What is the lowest possible integer that is divisible by each of the integers 1 through 7, inclusive?
Ans: If a number has to be divisible by each number from 1 to 7, that number should be L.C.M of(1,2,3,4,5,6,7) = 420
19) If the area of a square region having sides of length 6 cms is equal to the area of a rectangular region having width 2.5 cms, then the length of the rectangle, in cms, is
Ans: Given Area of the square = Area of rectangle
⇒a2=l.b" role="presentation" style="display: inline; line-height: normal; font-size: 13.65px; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; color: rgb(34, 34, 34); font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; position: relative;">a2=l.b⇒a2=l.b
Substituting the above values in the formula
⇒62=l.2.5" role="presentation" style="display: inline; line-height: normal; font-size: 13.65px; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; color: rgb(34, 34, 34); font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; position: relative;">62=l.2.5⇒62=l.2.5
⇒" role="presentation" style="display: inline; line-height: normal; font-size: 13.65px; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; color: rgb(34, 34, 34); font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; position: relative;"> l = 14.4 cm
20) A tank contains 10,000 gallons of a solution that is 5 percent sodium chloride by volume. If 2500 gallons of water evaporate from the tank, the remaining solution will be approximately what percentage of sodium chloride?
Ans: Sodium chloride in the original solution = 5% of 10,000 = 500
Water in the original solution = 10,000 - 500 = 9,500
If 2,500 Liters of the water is evaporated then the remaining water = 9,500 - 2,500 = 7,000
Sodium chloride concentration = 500500+7000×100" role="presentation" style="display: inline; line-height: normal; font-size: 13.65px; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; color: rgb(34, 34, 34); font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; position: relative;">500500+7000×100500500+7000×100 = 6.67 %
(concentration should be calculated always on the total volume)
21) After loading a dock, each worker on the night crew loaded 3/4 as many boxes as each worker on the day of the crew. If the night crew has 4/5 as many workers as the day crew, what fraction of all the boxes loaded by two crews did the day crew load?
Assume the number of boxes loaded in dayshift is equal to 4, then the number of boxed loaded in night shift = 3
Assume the worked on dayshift = 5, then workers on night shift = 4
So boxes loaded in day shift = 4 x 5 = 20, and boxes loaded in night shift = 3 x 4 = 12
so fraction of boxes loaded in day shift = 2020+12=58" role="presentation" style="display: inline; line-height: normal; font-size: 13.65px; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; color: rgb(34, 34, 34); font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; position: relative;">2020+12=582020+12=58
22) A bakery opened yesterday with its daily supply of 40 dozen rolls. Half of the rolls were sold by noon and 80 % of the remaining rolls were sold between noon and closing time. How many dozen rolls had not been sold when the bakery closed yesterday?
Ans: If half of the rolls were sold by noon, the remaining are 50 % (40) = 20.
Given 80% of the remaining were sold after the noon to closing time
⇒" role="presentation" style="display: inline; line-height: normal; font-size: 13.65px; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; color: rgb(34, 34, 34); font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; position: relative;"> 80% (20) = 16
Unsold = 20 - 16 = 4
23) If N=4P, where P is a prime number greater than 2, how many different positive even divisors does n have including n?
Ans: N = 22×P1" role="presentation" style="display: inline; line-height: normal; font-size: 13.65px; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; color: rgb(34, 34, 34); font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; position: relative;">22×P122×P1
We know that total factors of a number which is in the format of aP×bQ×cR..." role="presentation" style="display: inline; line-height: normal; font-size: 13.65px; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; color: rgb(34, 34, 34); font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; position: relative;">aP×bQ×cR...aP×bQ×cR... = (P + 1). (Q + 1). (R + 1) .... = (2 + 1).(1 + 1) = 6
Also odd factors of any number can be calculated easily by not taking 2 and its powers.
So odd factors of 22×P1" role="presentation" style="display: inline; line-height: normal; font-size: 13.65px; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; color: rgb(34, 34, 34); font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; position: relative;">22×P122×P1 = the factors of P1" role="presentation" style="display: inline; line-height: normal; font-size: 13.65px; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; color: rgb(34, 34, 34); font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; position: relative;">P1P1 = (1 + 1) = 2
Even factors of the number = 6 - 2 = 4
24) A dealer originally bought 100 identical batteries at a total cost of q rupees. If each battery was sold at 50 percent above the original cost per battery, then, in terms of q, for how many rupees was each battery sold?
Ans: Per battery cost = q / 100
If each battery is sold for 50% gain, then selling price = CostPrice×(100+Gain100)" role="presentation" style="display: inline; line-height: normal; font-size: 13.65px; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; color: rgb(34, 34, 34); font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; position: relative;">CostPrice×(100+Gain100)CostPrice×(100+Gain100)
⇒" role="presentation" style="display: inline; line-height: normal; font-size: 13.65px; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; color: rgb(34, 34, 34); font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; position: relative;">q100×(100+50100)=3q200" role="presentation" style="display: inline; line-height: normal; font-size: 13.65px; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; color: rgb(34, 34, 34); font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; position: relative;">q100×(100+50100)=3q200q100×(100+50100)=3q200
25) The price of lunch for 15 people was 207 pounds, including a 15 percent gratuity of service. What was the average price per person, EXCLUDING the gratuity?
Ans: Let the net price excluding the gratuity of service = x pounds
Then, total price including 15% gratuity of service = x×(100+15100)" role="presentation" style="display: inline; line-height: normal; font-size: 13.65px; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; color: rgb(34, 34, 34); font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; position: relative;">(100+15100)x×(100+15100) = 1.15 x pounds
So, 1.15 x = 207 pounds
⇒" role="presentation" style="display: inline; line-height: normal; font-size: 13.65px; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; color: rgb(34, 34, 34); font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; position: relative;"> x = 207 / 1.15 = 180 pounds
Net price of lunch for each person = 180 / 15 = 12 pounds
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Solutions
Mechanics of Materials
# Mechanics of Materials (3rd Edition)Solutions for Chapter 3.10
• 1302 step-by-step solutions
• Solved by publishers, professors & experts
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Chapter: Problem:
SAMPLE SOLUTION
Chapter: Problem:
• Step 1 of 11
Draw free body diagram of a truss.
• Step 2 of 11
Resolve forces and in the direction of -axis.
Here, is the force acting member and is the force acting in member .
…… (1)
Resolve forces and in the direction of y-axis.
……. (2)
• Step 3 of 11
Determine the magnitude of the force .
Substitute for and for in equation (1).
• Step 4 of 11
Determine the magnitude of the force .
Substitute for in equation (1).
• Step 5 of 11
(a)
Calculate axial stress of bar (1).
Here force acting on bar 1 is , area of bar (1) is , and stress acting on bar 1 is .
Substitute for and for .
Thus, the stress acting on bar 1 is tensile.
• Step 6 of 11
Calculate axial stress of bar (2).
Here force acting on bar 2 is , area of bar (2) is , and stress acting on bar 2 is .
Substitute for and for .
Thus, the stress acting on bar 2 is compressive.
• Step 7 of 11
(b)
Calculate axial elongation of bar 1.
Here, axial elongation of bar (1) is, length of bar (1) is , and elastic modulus of bar (1) is.
Substitute for, for, for , and for .
• Step 8 of 11
Calculate axial elongation of bar 2.
Here, axial elongation of bar (2) is , length of bar (2) is , and elastic modulus of bar (2) is.
Substitute for, for, for , and for .
Here, indicates compression and it acts opposite to axial force in rod (2).
• Step 9 of 11
Draw the schematic diagram representing axial elongations and vertical and horizontal displacements of rods (1) and (2).
• Step 10 of 11
Resolve the eleongations in the direction of x-axis.
Here, is the horizontal displacment of joint C.
Substitute for and for .
Thus, the horizontal displacment of joint C is .
• Step 11 of 11
Resolve the eleongations in the direction of y-axis.
Here, is the horizontal displacment of joint C.
Substitute for and for .
Thus, thevertical displacment of joint C is .
Corresponding Textbook
Mechanics of Materials | 3rd Edition
9780470481813ISBN-13: 0470481811ISBN: Roy R CraigAuthors:
Alternate ISBN: 9780470912003, 9781118136331, 9781118136348
| 670
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GeoGebra Classroom
Graphing Inequalities
Use the bottom two sliders to create a solid or dashed line and to change the direction of the shading. Move the two points on the grid to graph the inequality. Move the top slider to see if your answer matches the actual graph.
1. Graphing inequalities with x
Move the points and drag the sliders to match the graph
2. Graphing inequalities with x
Move the points and drag the sliders to match the graph
3. Graphing inequalities with y
Move the points and drag the sliders to match the graph for the inequality.
4. Graphing inequalities with y
Move the points and drag the sliders to match the graph for the inequality.
5. Graph the 2-variable inequality
Drag the points and move the sliders to graph the inequality.
6. Graph the 2-variable inequality
Drag the points and move the sliders to graph the inequality.
7. Graph the 2-variable inequality
Drag the points and move the sliders to graph the inequality.
8. Graph the 2-variable inequality
Drag the points and move the sliders to graph the inequality.
9. Graph the 2-variable inequality
Drag the points and move the sliders to graph the inequality.
10. Graph the 2-variable inequality
Drag the points and move the sliders to graph the inequality.
11. Graph the 2-variable inequality
Drag the points and move the sliders to graph the inequality.
12. Graph the 2-variable inequality
Drag the points and move the sliders to graph the inequality.
13. Graph the 2-variable inequality
Drag the points and move the sliders to graph the inequality.
14. Graph the 2-variable inequality
Drag the points and move the sliders to graph the inequality.
15. Graph the 2-variable inequality (Standard Form)
Solve the equation for y to put it into slope-intercept form. Then drag the points and move the sliders to graph the inequality. You could also find the x- and y-intercepts of the equation to graph it.
| 413
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# I really do not understand entropy - entropy increases wrt T
• barnflakes
In summary, the conversation discusses the relationship between heat energy, temperature, and entropy in a monatomic gas. While the equation for entropy suggests that increasing the temperature also increases the number of microstates, the speaker is struggling to understand how this applies to a gas with no internal degrees of freedom. However, the other participant provides an example with atoms and explains that adding energy can create more microstates and increase temperature.
barnflakes
Every explanation of this I have read has been extremely poor.
Imagine we have a MONATOMIC gas, with no internal degrees of freedom. The gas is confined to a box of volume V, and this volume is constant and is not allowed to increased upon adding heat energy.
We add an infinitesimal amount of heat energy to the box, delta Q. Now, there is an equation which tells me that the entropy just increased:
\delta S = \delta Q/T
However, let's think about the statistical mechanical definition of entropy, which is that the entropy is proportional to the number of microstates that the system can occupy for a given energy.
If the entropy increases, the number of microstates that give the same total energy must have increased.
I cannot for the life of me see how increasing the temperature increases the number of microstates of a monatomic gas.
The heated gas has no additional translational degrees of freedom compared with before, and it has no rotational or vibrational degrees of freedom since it is monatomic so that doesn't count either.
So where are these additional microstates coming from?
The higher the temperature the more ways you can reproduce the observed heat through motion.
Let a system with 3 atoms and only one of these have speed v0. The total energy is $$E_0=\frac{1}{2}mv_0^2$$ and, for easy to compute, let energy quantum step is e0 = E0/2.
With no energy addition, after some time we have these probabilities: $$(2e_0,0,0)\times3,\,(e_0,e_0,0)\times3 = 6 \text{states}$$
Adding energy e0 to the system we have: $$(3e_0,0,0)\times3,\,(2e_0,e_0,0)\times3,\,(e_0,e_0,e_0)\times3 =9 \text{states}$$
and so on. So, energy addition make more microstates. See that the new system have more energy, so more temperature. That is the meaning of T over the fraction. In high temperature the same energy addition make less entropy addition.
In some systems energy is fragment above, so energy addition over a limit makes less microstates, and for these cases the entropy move to less but the system is not alone.
## 1. What is entropy and how does it relate to temperature?
Entropy is a measure of the disorder or randomness in a system. The higher the temperature, the more energy is available to increase the disorder in the system, thus leading to an increase in entropy.
## 2. Why does entropy increase with temperature?
As temperature increases, the molecules in a system gain more kinetic energy and are able to move around and interact with each other more, leading to a greater degree of disorder and an increase in entropy.
## 3. How is entropy related to the second law of thermodynamics?
The second law of thermodynamics states that the total entropy of a closed system will always increase over time. This means that, in any natural process, the total amount of disorder or randomness in the system will increase, and this is reflected in the increase of entropy with temperature.
## 4. Does entropy always increase with temperature?
In most cases, yes. However, there are some exceptions, such as when a substance undergoes a phase change (e.g. from solid to liquid), where the increase in entropy is not directly related to temperature but rather to the change in the physical state of the substance.
## 5. How does entropy affect the behavior of a system?
The increase in entropy with temperature leads to a tendency for a system to move towards a state of maximum disorder. This can affect the behavior of the system, as it may lead to changes in pressure, volume, and other properties in order to increase the overall entropy of the system.
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] |
https://www.jiskha.com/questions/1006595/ethan-ordered-4-sub-sandwiches-for-a-party-each-1-2-sandwich-is-one-serving-does-he-have
| 1,576,486,962,000,000,000
|
text/html
|
crawl-data/CC-MAIN-2019-51/segments/1575541318556.99/warc/CC-MAIN-20191216065654-20191216093654-00060.warc.gz
| 749,590,479
| 5,256
|
# Math
Ethan ordered 4 sub sandwiches for a party. Each 1/2 sandwich is one serving. Does he have enough to serve 7 friends? How much is leftover or how much more is needed? Explain.
1. 👍 0
2. 👎 0
3. 👁 100
1. 4*2 = 8 servings
1. 👍 0
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posted by Damon
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3. 7
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4. ggcggfhgcghvjh
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posted by fh
## Similar Questions
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http://mathhelpforum.com/algebra/187566-completing-square-using-fractions.html
| 1,529,363,294,000,000,000
|
text/html
|
crawl-data/CC-MAIN-2018-26/segments/1529267861456.51/warc/CC-MAIN-20180618222556-20180619002556-00022.warc.gz
| 202,873,187
| 10,593
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# Thread: Completing the Square using Fractions
1. ## Completing the Square using Fractions
$\displaystyle 2x^2 -7x +3 = 0$
$\displaystyle 2x^2 -7x = -3$
$\displaystyle \frac{\2x^2}{2} -\frac{\7}{2}x = -\frac{\3}{2}$
Take one half of -$\displaystyle \frac{\7}{2}x$ and then square that answer which = $\displaystyle \frac{\49}{16}$ and add to both sides
$\displaystyle x^2 -\frac{\7}{2}$ +$\displaystyle \frac{\49}{16}$= -$\displaystyle \frac{\3}{2}$ + $\displaystyle \frac{\49}{16}$
Combine like terms on the right to get
$\displaystyle \frac{\25}{16}$
and then subtract that from both sides to get
$\displaystyle x^2 -\frac{\7}{2} + \frac{\24}{16}=0$
$\displaystyle \frac{\24}{16}$ can be reduced to $\displaystyle \frac{\3}{2}$ so that
$\displaystyle x^2 -\frac{\7}{2} + \frac{\3}{2}=0$
Eliminate the fractions by multiplying it all by 2 and then instead of completing a square I've completed a large useless circle as I arrive back at the original equation.
$\displaystyle 2x^2 -7x +3=0$
Now, I also used $\displaystyle \frac{\49}{4}$ as the square of half of $\displaystyle -\frac{\7}{2}$ but eventually it also ends up as
$\displaystyle \frac{\3}{2}$ on the left hand side of the equation.
I did this problem with the quadratic formula in about 45 seconds but I have to learn it this way. If nothing else my LaTex skills have been markedly improved in this post.
Any help is appreciated. Thanks.
2. ## Re: Completing the Square using Fractions
Originally Posted by Ingersoll
$\displaystyle 2x^2 -7x +3 = 0$
$\displaystyle 2x^2 -7x = -3$
$\displaystyle \frac{\2x^2}{2} -\frac{\7}{2}x = -\frac{\3}{2}$
Take one half of -$\displaystyle \frac{\7}{2}x$ and then square that answer which = $\displaystyle \frac{\49}{16}$ and add to both sides
$\displaystyle x^2 -\frac{7}{2}x +\frac{49}{16}$= $\displaystyle -\frac{3}{2}+\frac{49}{16}$
Combine like terms on the right to get
$\displaystyle \frac{\25}{16}$ ... ok
and then subtract that from both sides to get ... no
$\displaystyle x^2 -\frac{7}{2}x+\frac{49}{16}= \frac{25}{16}$
$\displaystyle \left(x - \frac{7}{4}\right)^2 = \frac{25}{16}$ this is why they call it "completing" the square
"un" square both sides ...
$\displaystyle x - \frac{7}{4} = \pm \frac{5}{4}$
$\displaystyle x = \frac{7 \pm 5}{4}$
btw ... one open tex at the beginning of a line ... one close /tex at the end of a line
3. ## Re: Completing the Square using Fractions
$\displaystyle x^2 -\frac{\7}{2}x +\frac{\49}{16}= -\frac{\3}{2}+ \frac{\49}{16}$
Until this point, you had done everything right. Without doing anything else, the left-hand (only) can be factored into a product of squares. The right is a number, which we can find the square root of. First I'm going to rewrite this after simplifying the right-hand side.
$\displaystyle x^2 -\frac{\7}{2}x$ +$\displaystyle \frac{\49}{16}$= $\displaystyle \frac{\25}{16}$
Now we can factor the left-hand side (AS IT IS RIGHT NOW).
$\displaystyle \left(x-\frac{7}{4}\right)^2=\frac{\25}{16}$
Now that it is in this form, we can take the square root of both sides. We are left with
$\displaystyle x-\frac{7}{4}=\pm \frac{5}{4}$ remember that every number has two square roots...
$\displaystyle x=\frac{7}{4}\pm \frac{5}{4}$
If we subtract, $\displaystyle x=\frac{2}{4}=\frac{1}{2}$. If we add, we have $\displaystyle x=\frac{12}{4}=3$. Thus $\displaystyle x \in \{\frac{1}{2}, 3\}$
I go over the conceptual logic of completing the square in this video.
4. ## Re: Completing the Square using Fractions
Thank you both so much. It's always right in front of my face. Thanks again.
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https://comidoc.net/udemy/motion-class-9-science-ncert
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text/html
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crawl-data/CC-MAIN-2023-14/segments/1679296949678.39/warc/CC-MAIN-20230331175950-20230331205950-00278.warc.gz
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# Motion Class 9 Science NCERT
Motion Class 9 Science NCERT
4.75 (2 reviews)
Udemy
platform
English
language
Science
category
456
students
2 hours
content
Sep 2021
last update
FREE
regular price
## Description
In everyday life, we see some objects at rest and others in motion. Birds fly, fish swim, blood flows through veins and arteries, and cars move. Atoms, molecules, planets, stars and galaxies are all in motion. We often perceive an object to be in motion when its position changes with time. However, there are situations where the motion is inferred through indirect evidences. For example, we infer the motion of air by observing the movement of dust and the movement of leaves and branches of trees. What causes the phenomena of sunrise, sunset and changing of seasons? Is it due to the motion of the earth? If it is true, why don’t we directly perceive the motion of the earth? An object may appear to be moving for one person and stationary for some other. For the passengers in a moving bus, the roadside trees appear to be moving backwards. A person standing on the road–side perceives the bus alongwith the passengers as moving. However, a passenger inside the bus sees his fellow passengers to be at rest. What do these observations indicate? Most motions are complex. Some objects may move in a straight line, others may take a circular path. Some may rotate and a few others may vibrate. There may be situations involving a combination of these. In this chapter, we shall first learn to describe the motion of objects along a straight line. We shall also learn to express such motions through simple equations and graphs. Later, we shall discuss ways of describin.
We describe the location of an object by specifying a reference point. Let us understand this by an example. Let us assume that a school in a village is 2 km north of the railway station. We have specified the position of the school with respect to the railway station. In this example, the railway station is the reference point. We could have also chosen other reference points according to our convenience. Therefore, to describe the position of an object we need to specify a reference point called the origin
The rate of motion of an object can be more comprehensive if we specify its direction of motion along with its speed. The quantity that specifies both these aspects is called velocity. Velocity is the speed of an object moving in a definite direction. The velocity of an object can be uniform or variable. It can be changed by changing the object’s speed, direction of motion or both. When an object is moving along a straight line at a variable speed, we can express the magnitude of its rate of motion in terms of average velocity. It is calculated in the same way as we calculate average speed. In case the velocity of the object is changing at a uniform rate, then average velocity is given by the arithmetic mean of initial velocity and final velocity for a given period of time.
## Content
Introduction
### Distance, Displacement, velocity, speed, acceleration, perimeter
Distance, Displacement, velocity, speed, acceleration, perimeter
### Numericals of distance time , v-t , speed time graph, three equations of motion
Numericals of distance time , v-t , speed time graph, three equations of motion
### Finding distance & displacement, average speed and velocity, velocity time graph
Finding distance & displacement, average speed and velocity, velocity time graph
### Distance time graph, speed time graph etc.
Distance time graph, speed time graph, calculating distance in case of v-t graph
## Related Topics
4308815
udemy ID
9/20/2021
course created date
9/27/2021
course indexed date
Bot
course submited by
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# Evaluating Returns & Cash Flow Streams Solve nine problems...
## Question
Evaluating Returns & Cash Flow Streams
Solve nine problems addressing a range of issues related to valuation of stocks, bonds, annuities, and cash flow streams.
The result of a financial manager's efforts is ultimately reflected in stock price; maximizing shareowner wealth is what finance is all about. This assessment examines the classic financial tradeoff of risk versus reward.
Assignment Instructions:
Complete Problems 1–9 to apply the necessary knowledge to assess returns and cash flow streams. You may solve the problems algebraically, or you may use a financial calculator or an Excel spreadsheet. In addition to your solution to each computational problem, you must show the supporting work leading to your solution to receive credit for your answer. Note the following:
• You may need an HP 10B II business calculator.
• You may use Word or Excel, but you will find Excel to be most helpful for creating spreadsheets.
• If you choose to solve the problems algebraically, be sure to show your computations.
• If you use a financial calculator, show your input values.
• If you use an Excel spreadsheet, show your input values and formulas.
Problem 1: Portfolio Required Return
You are the money manager of a \$10 million investment fund, which consists of four stocks. This fund has the following investments and betas:
Stock Investment Beta
A \$3,000,000 1.50
B \$1,000,000 (0.50)
C \$2,000,000 1.25
D \$4,000,000 0.75
If the market's required rate of return is 12 percent, and the risk-free rate is 4 percent, what is the fund's required rate of return?
Problem 2: Required Rate of Return
• Stock R's beta = 1.5
• Stock S's beta = 0.75
Consider that the required return on an average stock is 14 percent. The risk-free rate of return is 6 percent. If this is so, the required return on the riskier stock exceeds the required return on the less risky stock by how much?
Problem 3: CAPM and Required Return
Calculate the required rate of return for XYZ Inc. using the following information:
• The investors expect a 3.0 percent rate of inflation.
• The real risk-free rate is 2.0 percent.
• The market risk premium is 6.0 percent.
• XYZ Inc. has a beta of 1.7.
• Over the past 5 years, the realized rate of return has averaged 13.0 percent.
Problem 4: Bond Valuation
You have two bonds in your portfolio. Each bond has a face value of \$1000 and pays an 8 percent annual coupon. Bond X matures in 1 year, and Bond Y matures in 15 years.
1. If the going interest rate is 4 percent, 9 percent, and 14 percent, what will the value of each bond be? Assume Bond X only has one more interest payment to be made at maturity. Assume there are 15 more payments to be made on Bond Y.
2. The longer-term bond's price varies more than the shorter-term bond's price when interest rates change. Explain why.
Problem 5: Yield to Call
Five years ago, XYZ Inc. issued 20-year bonds with a 12 percent annual coupon rate at their \$1,000 par value. The bonds had 5 years of call protection and an 8 percent call premium. Yesterday, XYZ Inc. called the bonds.
For this problem, imagine that the investor who purchased the bonds when they were issued held them until they were called. Considering this, compute the realized rate of return. Should the investor be happy with XYZ Inc. calling the bonds? Why or why not?
Problem 6: Yield to Maturity
XYZ Inc. bonds have 5 years left to maturity. Interest is paid annually, and the bonds have a \$1,000 par value and a coupon rate of 8 percent.
1. What is the yield to maturity at a current market price of (1) \$800 and (2) \$1,200?
2. If a "fair" market interest rate for such bonds was 12 percent—that is, is rd=12%—would you pay \$800 for each bond? Why or why not?
Problem 7: After-Tax Cost of Debt
The XYZ Inc.'s currently outstanding bonds have a 10 percent yield to maturity and an 8 percent coupon. It can issue new bonds at par that would provide a similar yield to maturity. If its marginal tax rate is 40 percent, what is XYZ's after-tax cost of debt?
Problem 8: Present Value of an Annuity
Find the present values of the following ordinary annuities if discounting occurs once a year:
1. \$300 per year for 10 years at 10 percent.
2. \$150 per year for 5 years at 5 percent.
3. \$350 per year for 5 years at 0 percent.
Problem 9: Uneven Cash Flow Stream
Use the table below to answer the following:
1. What are the present values of the following cash flow streams if they are compounded at 5 percent annually?
2. What are the PVs of the streams at 0 percent compounded annually?
0 1 2 3 4 5
Stream A \$0 \$100 \$400 \$400 \$400 \$300
Stream B \$0 \$300 \$400 \$400 \$400 \$100
## Solution Preview
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# what is co-axial circle how to determine its equation and also tell its figure
A system of circles, every pair of which has the same radical axis is called a coaxial system of circle. The radical axis of two circle is the locus of a point which moves in such a way that the length of the tangents drawn from it to the circles are equal.
The equation of a system of coaxial circle which the equation of the radical axis and one of the circle of the system are given.
S = x2 + y2 + 2gx + 2fy + c = o be the circle and L = lx + my + n = 0 be the radical axis, then
S + λ L = 0, λ is arbitrary constant, represents the coaxial system of circle.
The equation of any two circle of the system are given.
S1 = x2 + y2 + 2g1 x + 2f1y + c1 = 0 and S2 = x2 + y2 + 2g2x + 2f2y + c2 = 0 be two circles of the system, then
S1+λ S2 = 0 (λ = – 1) represent the coaxial system of circle.
x2 + y2 + 2gx + c = 0, where g is a variable and c is constant is the simple form of the equation of coaxial system of circle. The common radical axis of the system is y-axis.
• -2
What are you looking for?
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# Maharashtra Board Class 6 Maths Solutions Chapter 5 Decimal Fractions Practice Set 17
## Maharashtra State Board Class 6 Maths Solutions Chapter 5 Decimal Fractions Practice Set 17
Question 1.
Carry out the following divisions.
i. 4.8÷2
ii. 17.5÷5
iii. 20.6÷2
iv. 32.5÷25
Solution:
i. 4.8÷2
ii. 17.5÷5
iii. 20.6÷2
iv. 32.5÷25
Question 2.
A road is 4 km 800 m long. If trees are planted on both its sides at intervals of 9.6 m, how many trees were planted?
Solution:
Length of road = 4 km 800 m
= 4 × 1000 m + 800 m
= 4000 m + 800 m
= 4800 m
Number of trees on one side = 4800 ÷ 9.6
= 500
∴ Number of trees on both sides = 2 x number of trees on one side
= 2 x 500 = 1000
If the trees are planted at the beginning of the road, then
Total number of trees = 1000 + 2 = 1002
∴ Total number of trees planted is 1000 or 1002.
Question 3.
Pradnya exercises regularly by walking along a circular path on a field. If she walks a distance of 3.825 km in 9 rounds of the path, how much does she walk in one round?
Solution:
Total distance walked in 9 rounds = 3.825 km
∴Distance walked in 1 round = 3.825 4 ÷ 9
= 0.425 km
∴ Total distance walked in 1 round is 0.425 km.
Question 4.
A pharmaceutical manufacturer bought 0.25 quintal of hirada, a medicinal plant, for Rs 9500. What is the cost per quintal of hirada? (1 quintal = 100 kg)
Solution:
Cost of 0.25 quintal of hirada = Rs 9500
∴ Cost of 1 quintal of hirada = 9500 ÷ 0.25
= Rs 38,000
∴ Cost per quintal of hirada is Rs 38,000.
#### Maharashtra Board Class 6 Maths Chapter 4 Operations on Fractions Practice Set 17 Intext Questions and Activities
Question 1.
Maths is fun! (Textbook pg. no. 34)
1. Consider any three digit number (say 527).
2. Multiply the number by 7. Then multiply the product obtained by 13, and this product by 11.
3. The find product is 5,27,527.
Take two or three other numbers. Do the same multiplication and find out how it is done.
Solution:
7 × 13 × 11 = 1001
∴ 527 × 1001 = 527 × (1000+ 1)
= (527 × 1000) + (527 × 1)
= 527000 + 527 = 527527
Thus, when any three digit number is multiplied with 1001, the product obtained is a six digit number in which the original three digit number written back to back twice.
(Students may consider any other three digit numbers and verify the property.)
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# Understanding the assumptions of Linear Regression
I have been studying the four assumptions of linear regression and different sources give different interpretations of the same.
The four assumptions are :
a) Linearity - Existence of linear relationship between the dependent and the independent variable
b) Normality - The residual values must be normally distributed ( But, some sources say that the Response variable must be normally distributed. I don't know which one is true)
c) Independence - The residuals must be independent of each other
d) Homoscedasticity - The residuals have constant variance for any value of X.
I have understood the first assumption , i.e, the Linearity. But, I'm finding it hard to understand "Why" the remaining three assumptions must hold true for a better model. I haven't found a proper reasoning for this anywhere. Even the standard textbooks do not go deep into this topic.
Can someone help me understand "why" the b), c) and d) assumptions must be true ? Also, I am not sure about the normality condition that if the response variable must be normal OR the residuals should be normally distributed. ( Different sources say different things).
• Commented Feb 23, 2021 at 8:50
• Those are assumptions of the so-called "classical linear regression model", but by no means are necessary for linear regression to work in general. Commented Feb 23, 2021 at 12:04
• You could call these a list of assumptions for ordinarily least squares regression. However, regression is generally much broader: It is a model for the conditional distributions of $Y$ given $X$. So in general, the assumptions of regression are this: The potentially observable data from reality (as tapped through your design and measurement) are reasonably well matched (in a frequency sense) to the data produced by whatever model you choose. Commented Feb 23, 2021 at 13:13
## Condition 1
You seem to understand this one. This assumption can be written as $$f(x_i) = \beta^\mathrm{T}x_i$$ and in matrix form as $$\mathbf{f} = \mathbf{X}\beta$$
## Condition 2
In short, you need that $$\mathbb{P}(y\mid x) = \mathcal{N}(\mu(x), \sigma^2)$$ where $$\mu(x)$$ is a function of $$X$$. This assumption is met when the residuals are normal since $$y = f(x) + \varepsilon$$ and $$\varepsilon\sim\mathcal{N}(0,\sigma^2)$$ means that $$y \mid x \sim \mathcal{N}(f(x),\sigma^2)$$. So yes, you could say that
The response variable $$y$$ needs to follow a normal distribution
but most of the time it's because the residuals $$\varepsilon$$ are normal.
## Condition 3
This condition is needed to justify the OLS loss which is $$\mathcal{L}(\hat{f}, \mathcal{D}) \propto \sum_{i=1}^n\,\lVert y_i-\hat{f}(x_i)\rVert^2 = \lVert \mathbf{y}-\hat{\mathbf{f}}\rVert^2$$
If you want to avoid this condition you then need to change your loss to $$(\mathbf{y}-\hat{\mathbf{f}})^\mathrm{T}\mathbf{W}(\mathbf{y}-\hat{\mathbf{f}})$$ where $$\mathbf{W} = \pmb{\Sigma}^{-1}$$ is the covariance of the noise terms. This is called generalized least squares.
## Condition 4
This condition means that the variance $$\mathrm{Var}[Y\mid X=x]$$ is constant. This is what is needed to have the loss $$\mathcal{L}(\hat{f}, \mathcal{D}) = \frac{1}{2\sigma^2}\sum_{i=1}^n\,\lVert y_i-\hat{f}(x_i)\rVert^2$$ If instead you want to model a noise variance which changes as $$X$$ takes different values, you get the loss $$\mathcal{L}(\hat{f}, \mathcal{D}) = \sum_{i=1}^n\,\frac{1}{2\sigma^2(x_i)}\lVert y_i-\hat{f}(x_i)\rVert^2 = \sum_{i=1}^n\,w(x_i)\lVert y_i-\hat{f}(x_i)\rVert^2$$ which weights each residuals according to a function $$w(x_i)$$ of the input value $$x_i$$. This is similar to LOESS regression.
Just to add to the discussion.
Assumption 1. Linearity
This assumption says that we believe the true population model is linear in parameters (not in explanatory variables). Thus: $$y = a_0 + a_1 x + a_2 x^2 + a_3 ln(x) +e$$ Is a linear model. However, you can also interpret it in a slighly different way. the linearity assumption says that the conditional mean is a linear function of parameters: $$E(y|x) = = a_0 + a_1 x + a_2 x^2 + a_3 ln(x)$$ Which is a bit more flexible, because we are basically averaging out the error.
This assumption basically imposes restrictions on how the error affects the model and allows us to estimate a Linear Regression model, usually via OLS. There are other important assumptions left tho.
1a) Random sampling: Your sample is representative of the population you want to study or say something about.
1b) the Var(X)>>0 and/or no multicollinearity. You want your data to have variation (which is used to identify coefficients in the model, but that variation has to be unique that that variable. For example, if $$X1 = X2$$, then you cannot include both in a model because when controlling for one, the other variable does not vary.
1c) you want X to be independent of the population unobservables $$e$$. Otherwise, your model and coefficients cannot be interpreted as causal relationships.
ADDED: Keep in mind that once you estimate your model, the errors $$\hat{e}$$ are by construction linearly independent with X. However, they are not the same as the population unobservables $$e$$.
Now why would this allow you to estimate causal relationships? Lets consider the assumptions again. Model is linear: $$y=a_0+a_1*x +e$$. If X is independent of e, and we can observe both, the causal effect of $$x$$ on $$y$$ could be done as follows: $$y^1=a_0+a_1*(x+1) +e$$ $$y^0=a_0+a_1*(x) +e$$ Thus the effect of a 1 unit change in $$x$$ can be is just $$y^1-y^0$$. This is possible because we can "assume" e is constant.
However, When X and e are correlated, if $$X$$ changes, then $$e$$ may change too for unknown reasons: $$y^1=a_0+a_1*(x+1) +e + \Delta e$$ $$y^0=a_0+a_1*(x) +e$$ In this case $$y^1-y^0$$ is not the effect of a change in X, because it also includes a change in $$e$$ we cannot explain. (technically we do not even know it is there)
When X and $$e$$ are correlated in the population model, and you estimate it via OLS, $$\hat{a_1}$$ will be a combination of $$a_1$$ and the correlation between $$X$$ and $$e$$.
The rest of the assumptions do not have to do with the estimation of the coefficient in a Linear regression, but with the estimation of the standard errors of those coefficients, and the efficiency of information use. Thus they are important, but you can live without them.
Assumption 2. Normality
This is not a "MUST". the LR regression model in fact imposes no assumption on the errors. your errors could be poisson, uniform, chi2, etc etc etc, and you could still estimate the LR using OLS. Now, if you want to estimate the model with other method, then yes, you need to impose a distributional assumption on the errors.
Then why is it added as an assumption in some texts?. One reason is as follows. When you estimate your LR using OLS, you find the following solution for $$\beta s$$: $$\hat{\beta}=(X'X)^{-1} X'Y$$ $$\hat{\beta}=(X'X)^{-1} X'(X\beta+e)$$ $$\hat{\beta}=\beta+(X'X)^{-1} X'e$$ So the estimated coefficients are a function of the errors $$e$$. Now, if you want to run tests on $$\beta s$$, you need to know something about their distribution (is it symmetrical, or asymmetrical, or flat, or all bundle up with a few extreme values). If the errors are normal, however, $$\beta s$$ are also normal, so we can use the family of Normalbased distributions (t-test, F-test, Chi2, z etc), to do inference.
In other words, the assumption of normality just makes life easier because it warrants the estimated coefficients are also normal.
What if $$e$$ is not normal? When the sample is large enough...(no strict criteria that i know to decide what is large enough), the normality of the errors are no longer needed, and one relies on the Central limit theory. This basically implies that $$\beta$$ still distributions as normal, even if the errors $$e$$ do not.
So, $$e$$ doesnt need to be normal, but its a good assumption that makes it easy to accept that $$\hat{\beta}$$ is normally distributed (and you can use standard tests).
Assumption 3. Independence
This assumption is usually taken as given. When you have crossection data, and you get a random sample of the population (truly random), it is easy to assume that those unobservables we leave in $$e$$ are independent of each other. When data is collected from "clusters", this may not be true.
For example, if you get info from families, it is very likely that some of those "errors" are common among all family members, although they are independent across different families.
When this happens, the estimation of standard errors of the $$\beta s$$ will be incorrect. The standard formulas assume errors are independent, but if they are indeed correlated, using standard formulas will most likely understate the true variation of the betas.
In terms of the LR. Independence of the errors is a property that simplifies the estimation of standard errors, but it is not necessary, since you can "correct" for it.
Assumption 4. Homoskedasticity
So this assumption means variation of $$e$$ does not change with X. In other words, every single observation has the same amount of information for the estimation of $$\beta s$$ because the error variation is constant.
When there is heteroskedasticity, this is no longer the case. $$Var(e|X)$$ may increase or decrease with X.
Now, if this happens, some observations may have better information than others to estimate the coefficients. Observations with large conditional variance ($$Var(e|X)$$ is high) will be less precise for estimating Betas (thus should receive less weight in your estimation), on the other hand, those with low variance will be more precise and receive more weight.
Standard OLS assumes all observations are equally important for estimating $$\beta s$$, so if your model is heteroskedastic, this equal weight assumption will be incorrect, and that will be reflected in the precision of your coefficients. from the analytical point of view you will need to at the very least correct standard errors (other methods for the estimation) if you want to make any time of statistical inference.
HTH
• (+1) You say you want X to be independent of the error. Isn't the error term always going to be uncorrelated to the explanatory variables as a consequence of the model fitting? And your model and coefficients cannot be interpreted as causal relationships could you elaborate on this? I thought you cannot say anything about causal relationships between x and y regardless. Commented Feb 23, 2021 at 13:34
• Hi, I added some information to my answer but. 1) the population error (unobservables) is never "observed", and it could or could not be correlated with X. The sample error $\hat{e}$ is by construction uncorrelated. 2) if unobservables are uncorrelated with X, you can use the though experiment, what would happen if X changes...because everything else is "constant". This is what you could interpret as causal effect. If the error is correlated, if X change , $e$ will also change (and Y will change). So youcannot get a causal relationship, since because changes in Y will also contain $\Delta e$ Commented Feb 23, 2021 at 14:41
• So, for the causal effect question. We can say something about causality if the assumptions we use are credible. Commented Feb 23, 2021 at 14:46
• If all I am interested in is the p-values (and not the coefficients size), do I still need the Normality assumption? Does the calculation of the p-values rely on the distribution?
– Sam
Commented Oct 11, 2021 at 9:34
• This may be a good way to remember objection.lol/objection/1897686 Commented Oct 15, 2021 at 11:45
I'm going to answer to test my understanding since the literature on the topic is huge and lots of people on this forum can give better explanations.
I assume we are talking about the ordinary linear regression, not the generalized linear models.
b) Normality - The residual values must be normally distributed ( But, some sources say that the Response variable must be normally distributed. I don't know which one is true)
I also found this confusing. You want the response to be normal conditional to the explanatory variable(s) and this is equivalent to the residuals being normal.
For example, you want to model adult human height as a function of sex. This is considered a well-behaved trait satisfying the linear model assumptions. However, adult height as a whole has a slightly bimodal distribution (one peak for males, one for females) but within males and within females ( i.e. conditional to the explanatory variable sex) the distribution is normal and the assumption satisfied.
c) Independence - The residuals must be independent of each other
Each observation is supposed to give information about the mean of the (normal) distribution it comes from. If observations are correlated, then, intuitively, you are not getting useful information.
Returning to the human height example, if you measure the same male person multiple times you are not getting independent measures for the mean of males group. Consequently, your estimate will be biased by the height of that person. (There are variations of the linear model that account for such repeated measures).
d) Homoscedasticity - The residuals have constant variance for any value of X.
If the variance changes with the mean of X, than the fit will be dominated by observation will large X. For example, you model the weight of different animal species that vary a lot in size, say cats and elephants. The variance within elephants is huge compared to cats' even if the elephants are very similar in size relative to each other. This means that the elephant datapoint have a large influence on the slope of the regression line.
I will here try to answer the "why":
You have decided to fit a linear model to your observations using OLS method, and with only one dependent variable you now have the slope, $$\hat{\beta}$$, for that variable. Pay attention to the "hat", which means that $$\hat{\beta}$$ is just the sample estimate of the underlying (unknown) population slope, $$\beta$$. From $$\hat{\beta}$$ you have quantified the linear relationship between the indpendent and the dependent variable in your sample, but you are probably interested in the linear relationship in the underlying population.
(1) If the above conditions are met, you suddenly have knowledge about how $$\hat{\beta}$$ relates to $$\beta$$, and you can quantify the uncertainty, usually done through a confidence interval (CI). A standard textbook will tell you how. If the above conditions are not met, the confidence interval calculation is not valid, and hence, your conclusions about the population is not valid.
In summary, for a CI to be valid, you need to make sure that the conditions for that specific CI calculation are met. If the conditions are not met, there might be another CI calculation that fits the conditions that your observations actually meet, but maybe not.
(2) There are other desirable properties of $$\hat{\beta}$$ that depends on the method of estimation and whether the various conditions are met. For example, $$\hat{\beta}$$ is an unbiased estimator of $$\beta$$, meaning that the expected value of $$\hat{\beta}$$ is $$\beta$$. It is also consistent and efficient, both desirable properties.
To sum up: If none of the conditions are met, you only have a $$\hat{\beta}$$ but you don't know anything about its uncertainty and how it relates to $$\beta$$. If all the conditions are met, you know that $$\hat{\beta}$$ is an unbiased, consistent and efficient estimator for $$\beta$$, and you can quantify the uncertainty by calculating a CI.
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4725,
2794,
279,
98640,
1288,
387,
14614,
4332,
13,
320,
34496,
8336,
2019,
2204,
2574,
3677,
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12535,
291,
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220,
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11,
220,
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16,
520,
220,
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198,
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527,
32946,
315,
279,
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330,
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31649,
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498,
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555,
912,
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527,
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369,
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311,
990,
304,
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13,
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291,
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220,
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11,
220,
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264,
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315,
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13,
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11,
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374,
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25,
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311,
264,
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86,
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14415,
315,
279,
1988,
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87,
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3,
13244,
1115,
374,
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311,
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382,
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430,
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279,
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374,
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320,
1962,
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570,
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25,
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1101,
14532,
433,
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264,
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398,
2204,
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13,
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430,
279,
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3152,
374,
264,
13790,
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315,
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25,
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284,
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18,
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810,
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389,
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323,
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264,
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11,
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374,
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315,
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477,
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16,
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### I Need Help With A Complex Formula
• Follow
```I have been trying to make a formula that will out put a certain percentage
for the numbers I input. for example if i put 2,000,000 in Cell A16 I want it
give me 3% in Cell B16. This is the complex formula that i came up with
=IF(A16>=2000000,3%,IF(AND(A16>=150000,A16<2000000),4%,IF(AND(A16>=1000000,A16<1500000),4.5%,IF(AND(A16>=600000,A16<1000000),5%,IF(AND(A16>=450000,A16<600000),5.5%,IF(A16<450000,6%,0))))))
Every time i use this formula, I doesn't give me the right percentages if I
input anything under 150,000,000. For example if I input the number 600,000
then i should return me the percentage of 5%. However when i attempt this,
the output percentage will always be 4%. to me this means that only
this portion is functional:
=IF(A16>=2000000,3%,IF(AND(A16>=150000,A16<2000000),4%
If anyone can find out what I'm doing wrong. I would mean a world of help
for me.
```
0
```=IF(A16="","",IF(A16>=2000000,3%,IF(AND(A16>=1500000,A16<2000000),4%,IF(AND(A16>=1000000,A16<1500000),4.5%,IF(AND(A16>=600000,A16<1000000),5%,IF(AND(A16>=450000,A16<600000),5.5%,IF(A16<450000,6%,0)))))))
Remember to Click Yes, if this post helps!
--------------------
(Ms-Exl-Learner)
--------------------
"Jonathan Cheek" wrote:
> I have been trying to make a formula that will out put a certain percentage
> for the numbers I input. for example if i put 2,000,000 in Cell A16 I want it
> give me 3% in Cell B16. This is the complex formula that i came up with
>
> =IF(A16>=2000000,3%,IF(AND(A16>=150000,A16<2000000),4%,IF(AND(A16>=1000000,A16<1500000),4.5%,IF(AND(A16>=600000,A16<1000000),5%,IF(AND(A16>=450000,A16<600000),5.5%,IF(A16<450000,6%,0))))))
>
> Every time i use this formula, I doesn't give me the right percentages if I
> input anything under 150,000,000. For example if I input the number 600,000
> then i should return me the percentage of 5%. However when i attempt this,
> the output percentage will always be 4%. to me this means that only
> this portion is functional:
>
> =IF(A16>=2000000,3%,IF(AND(A16>=150000,A16<2000000),4%
>
> If anyone can find out what I'm doing wrong. I would mean a world of help
> for me.
```
0
```Hi Jonathin,
I think you have a zero missing from your second IF thingy.
You have 150000, and I think it should be 1500000
Regards - Dave.
"Jonathan Cheek" wrote:
> I have been trying to make a formula that will out put a certain percentage
> for the numbers I input. for example if i put 2,000,000 in Cell A16 I want it
> give me 3% in Cell B16. This is the complex formula that i came up with
>
> =IF(A16>=2000000,3%,IF(AND(A16>=150000,A16<2000000),4%,IF(AND(A16>=1000000,A16<1500000),4.5%,IF(AND(A16>=600000,A16<1000000),5%,IF(AND(A16>=450000,A16<600000),5.5%,IF(A16<450000,6%,0))))))
>
> Every time i use this formula, I doesn't give me the right percentages if I
> input anything under 150,000,000. For example if I input the number 600,000
> then i should return me the percentage of 5%. However when i attempt this,
> the output percentage will always be 4%. to me this means that only
> this portion is functional:
>
> =IF(A16>=2000000,3%,IF(AND(A16>=150000,A16<2000000),4%
>
> If anyone can find out what I'm doing wrong. I would mean a world of help
> for me.
```
0
```There seem to be a number of unnecessary tests there.
You've tested for >=2000000, so you don't then need to test for <2000000, &
similarly for the later tests.
You can simplify
=IF(A16="","",IF(A16>=2000000,3%,IF(AND(A16>=1500000,A16<2000000),4%,IF(AND(A16>=1000000,A16<1500000),4.5%,IF(AND(A16>=600000,A16<1000000),5%,IF(AND(A16>=450000,A16<600000),5.5%,IF(A16<450000,6%,0))))))) to =IF(A16="","",IF(A16>=2000000,3%,IF(A16>=1500000,4%,IF(A16>=1000000,4.5%,IF(A16>=600000,5%,IF(A16>=450000,5.5%,6%))))))--David Biddulph"Ms-Exl-Learner" <[email protected]> wrote in messagenews:[email protected]...>=IF(A16="","",IF(A16>=2000000,3%,IF(AND(A16>=1500000,A16<2000000),4%,IF(AND(A16>=1000000,A16<1500000),4.5%,IF(AND(A16>=600000,A16<1000000),5%,IF(AND(A16>=450000,A16<600000),5.5%,IF(A16<450000,6%,0)))))))>> Remember to Click Yes, if this post helps!>> --------------------> (Ms-Exl-Learner)> -------------------->>> "Jonathan Cheek" wrote:>>> I have been trying to make a formula that will out put a certainpercentage>> for the numbers I input. for example if i put 2,000,000 in Cell A16 Iwant it>> give me 3% in Cell B16. This is the complex formula that i came up with>>>>=IF(A16>=2000000,3%,IF(AND(A16>=150000,A16<2000000),4%,IF(AND(A16>=1000000,A16<1500000),4.5%,IF(AND(A16>=600000,A16<1000000),5%,IF(AND(A16>=450000,A16<600000),5.5%,IF(A16<450000,6%,0))))))>>>> Every time i use this formula, I doesn't give me the right percentages ifI>> input anything under 150,000,000. For example if I input the number600,000>> then i should return me the percentage of 5%. However when i attemptthis,>> the output percentage will always be 4%. to me this means that only>> this portion is functional:>>>> =IF(A16>=2000000,3%,IF(AND(A16>=150000,A16<2000000),4%>>>> If anyone can find out what I'm doing wrong. I would mean a world of help>> for me.
```
0
```Yes David Sir you are right, I have just modified the OP's formula and given
the same. After seeing your post only I come to know that it can be
simplified.
--------------------
(Ms-Exl-Learner)
--------------------
"David Biddulph" wrote:
> There seem to be a number of unnecessary tests there.
> You've tested for >=2000000, so you don't then need to test for <2000000, &
> similarly for the later tests.
>
> You can simplify
> =IF(A16="","",IF(A16>=2000000,3%,IF(AND(A16>=1500000,A16<2000000),4%,IF(AND(A16>=1000000,A16<1500000),4.5%,IF(AND(A16>=600000,A16<1000000),5%,IF(AND(A16>=450000,A16<600000),5.5%,IF(A16<450000,6%,0))))))) to =IF(A16="","",IF(A16>=2000000,3%,IF(A16>=1500000,4%,IF(A16>=1000000,4.5%,IF(A16>=600000,5%,IF(A16>=450000,5.5%,6%))))))--David Biddulph"Ms-Exl-Learner" <[email protected]> wrote in messagenews:[email protected]...>=IF(A16="","",IF(A16>=2000000,3%,IF(AND(A16>=1500000,A16<2000000),4%,IF(AND(A16>=1000000,A16<1500000),4.5%,IF(AND(A16>=600000,A16<1000000),5%,IF(AND(A16>=450000,A16<600000),5.5%,IF(A16<450000,6%,0)))))))>> Remember to Click Yes, if this post helps!>> --------------------> (Ms-Exl-Learner)> -------------------->>> "Jonathan Cheek" wrote:>>> I have been trying to make a formula that will out put a certainpercentage>> for the numbers I input. for example if i put 2,000,000 in Cell A16 Iwant it>> give me 3% in Cell B16. This is
the complex formula that i came up with>>>>=IF(A16>=2000000,3%,IF(AND(A16>=150000,A16<2000000),4%,IF(AND(A16>=1000000,A16<1500000),4.5%,IF(AND(A16>=600000,A16<1000000),5%,IF(AND(A16>=450000,A16<600000),5.5%,IF(A16<450000,6%,0))))))>>>> Every time i use this formula, I doesn't give me the right percentages ifI>> input anything under 150,000,000. For example if I input the number600,000>> then i should return me the percentage of 5%. However when i attemptthis,>> the output percentage will always be 4%. to me this means that only>> this portion is functional:>>>> =IF(A16>=2000000,3%,IF(AND(A16>=150000,A16<2000000),4%>>>> If anyone can find out what I'm doing wrong. I would mean a world of help>> for me.
>
> .
>
```
0
```Try
=IF(A16="","",LOOKUP(A16,{0,450000,600000,1000000,1500000,2000000},{0.06,0.055,0.05,0.045,0.04,0.03}))
---
HTH
Bob Phillips
"Jonathan Cheek" <Jonathan [email protected]> wrote in message
news:[email protected]...
>I have been trying to make a formula that will out put a certain percentage
> for the numbers I input. for example if i put 2,000,000 in Cell A16 I want
> it
> give me 3% in Cell B16. This is the complex formula that i came up with
>
> =IF(A16>=2000000,3%,IF(AND(A16>=150000,A16<2000000),4%,IF(AND(A16>=1000000,A16<1500000),4.5%,IF(AND(A16>=600000,A16<1000000),5%,IF(AND(A16>=450000,A16<600000),5.5%,IF(A16<450000,6%,0))))))
>
> Every time i use this formula, I doesn't give me the right percentages if
> I
> input anything under 150,000,000. For example if I input the number
> 600,000
> then i should return me the percentage of 5%. However when i attempt this,
> the output percentage will always be 4%. to me this means that only
> this portion is functional:
>
> =IF(A16>=2000000,3%,IF(AND(A16>=150000,A16<2000000),4%
>
> If anyone can find out what I'm doing wrong. I would mean a world of help
> for me.
```
0
```Thank You all! All your solutions worked very well.
```
0
6 Replies
128 Views
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7/27/2012 12:35:53 AM
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http://forums.wolfram.com/mathgroup/archive/2011/Mar/msg00851.html
| 1,713,606,330,000,000,000
|
text/html
|
crawl-data/CC-MAIN-2024-18/segments/1712296817576.41/warc/CC-MAIN-20240420091126-20240420121126-00620.warc.gz
| 10,893,696
| 7,637
|
Re: how do I solve for this
• To: mathgroup at smc.vnet.net
• Subject: [mg117637] Re: how do I solve for this
• From: Peter Breitfeld <phbrf at t-online.de>
• Date: Tue, 29 Mar 2011 06:51:20 -0500 (EST)
• References: <imfa6a\$io5\[email protected]>
```thinktank1985 wrote:
First make your variables constant (I use n instead of N, because N is a
function, which should not be used as a Variable):
SetAttributes[{p,n,kb,b,a},Constant]
eq = p == (n kb T)/(V - n b) - (a n^2)/V^2
Calculate the total derivative:
totDiff=Dt[eq,T]
Out=
0 == (kb n)/(-b n + V) + (2 a n^2 Dt[V, T])/V^3 -
(kb n T Dt[V, T])/(-b n + V)^2
Solve for Dt[V,T]:
Solve[totDiff,Dt[V,T]] // Together
Out= {{Dt[V, T] -> (kb n (b n - V) V^3)/
(2 a b^2 n^4 - 4 a b n^3 V + 2 a n^2 V^2 - kb n T V^3)}}
> Say I have
>
> p=N*kb*T/(V-N*b)-a*N^2/V^2
>
> I want to evaluate the derivative of V with respect to T, keeping
> p,N,kb,b,a constant.
>
> I understand that this can be done by hand. I just want to know
> whether this can be done by using mathematica. I couldnt understand
> how to use Solve to do this. maybe there is something else I am not
> aware of
>
--
_________________________________________________________________
Peter Breitfeld, Bad Saulgau, Germany -- http://www.pBreitfeld.de
```
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| 503
| 1,478
|
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| 3.671875
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CC-MAIN-2024-18
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latest
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https://www.gradesaver.com/textbooks/science/physics/college-physics-4th-edition/chapter-12-problems-page-462/8
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## College Physics (4th Edition)
We can use $340~m/s$ as the speed of sound. Since the time it takes the light to reach us is negligible, we can ignore the travel time of the light. We can find the time it tkes a sound wave to travel a distance of $1.6~km$: $t = \frac{d}{v} = \frac{1600~m}{340~m/s} = 4.7~s$ The rule that five seconds elapse for each mile of distance is approximately correct.
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• 0
# The cost of digging a well after every metre of digging, when it costs Rs 150 for the first metre and rises by Rs 50 for each subsequent metre. Find wether the question is an A.P or not.
• 0
The question given is from class 10th NCERT book of Excersice no. 5.1 and question number 1( iii ). In the given question you have to find whether it is an A.P or not . Give the solution of the question.
Share
1. Sol.
Cost to dig a well for first metre = Rs.150
Cost to dig a well for first 2 metres = Rs.150+50 = Rs.200
Cost to dig a well for first 3 metres = Rs.200+50 = Rs.250
Cost to dig a well for first 4 metres =Rs.250+50 = Rs.300
Cost to dig …………………………… so. on
Here we can see that 150, 200, 250, 300 … forms an A.P. with a common difference of 50 between them.
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Science, Maths & Technology
### Become an OU student
Everyday maths 2 (Wales)
Start this free course now. Just create an account and sign in. Enrol and complete the course for a free statement of participation or digital badge if available.
# 1.1 Expressing a remainder as a decimal
To split a prize of £125 between 5 friends you would do this calculation:
• £125 ÷ 5 and get the answer £25.
This is a convenient, exact amount of money. However, often when you perform calculations, especially those involving division, you do not always get an answer that is suitable for the question.
For example, if there were 4 friends who shared the same prize we would do the calculation £125 ÷ 4 and get the answer £31 remainder £1. If we did the same calculation on a calculator you would get the answer £31.25, the remainder has been converted into a decimal. Let’s look at how to express the remainder as a decimal.
You can write one hundred and twenty five pounds in two different ways: £125 or £125.00. Both ways show the same amount but the second way allows you to continue the calculation and express it as a decimal.
Figure 2 Expressed as a decimal: 125 ÷ 4
We can use the same principal with any whole number, adding as many zeros after the decimal point as required. Look at the following example.
A teacher wants to share 35 kg of clay between 8 groups of students. How much clay will each group get?
Figure 3 Expressed as a decimal: 35 ÷ 8
You can see that each group would get 4.375 kg of clay.
## Activity 3: Expressing a remainder as a decimal
Work out the answers to the following without using a calculator.
1. 178 ÷ 4
2. 212 ÷ 5
3. 63 ÷ 8
4. 227 ÷ 4
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Linear Response Function
Get Linear Response Function essential facts below. View Videos or join the Linear Response Function discussion. Add Linear Response Function to your PopFlock.com topic list for future reference or share this resource on social media.
Linear Response Function
A linear response function describes the input-output relationship of a signal transducer such as a radio turning electromagnetic waves into music or a neuron turning synaptic input into a response. Because of its many applications in information theory, physics and engineering there exist alternative names for specific linear response functions such as susceptibility, impulse response or impedance, see also transfer function. The concept of a Green's function or fundamental solution of an ordinary differential equation is closely related.
## Mathematical definition
Denote the input of a system by ${\displaystyle h(t)}$ (e.g. a force), and the response of the system by ${\displaystyle x(t)}$ (e.g. a position). Generally, the value of ${\displaystyle x(t)}$ will depend not only on the present value of ${\displaystyle h(t)}$, but also on past values. Approximately ${\displaystyle x(t)}$ is a weighted sum of the previous values of ${\displaystyle h(t')}$, with the weights given by the linear response function ${\displaystyle \chi (t-t')}$:
${\displaystyle x(t)=\int _{-\infty }^{t}dt'\,\chi (t-t')h(t')+\dots \,.}$
The explicit term on the right-hand side is the leading order term of a Volterra expansion for the full nonlinear response. If the system in question is highly non-linear, higher order terms in the expansion, denoted by the dots, become important and the signal transducer cannot adequately be described just by its linear response function.
The complex-valued Fourier transform ${\displaystyle {\tilde {\chi }}(\omega )}$ of the linear response function is very useful as it describes the output of the system if the input is a sine wave ${\displaystyle h(t)=h_{0}\cdot \sin(\omega t)}$ with frequency ${\displaystyle \omega }$. The output reads
${\displaystyle x(t)=|{\tilde {\chi }}(\omega )|\cdot h_{0}\cdot \sin(\omega t+\arg {\tilde {\chi }}(\omega ))\,,}$
with amplitude gain ${\displaystyle |{\tilde {\chi }}(\omega )|}$ and phase shift ${\displaystyle \arg {\tilde {\chi }}(\omega )}$.
## Example
Consider a damped harmonic oscillator with input given by an external driving force ${\displaystyle h(t)}$,
${\displaystyle {\ddot {x}}(t)+\gamma {\dot {x}}(t)+\omega _{0}^{2}x(t)=h(t).\,}$
The complex-valued Fourier transform of the linear response function is given by
${\displaystyle {\tilde {\chi }}(\omega )={\frac {{\tilde {x}}(\omega )}{{\tilde {h}}(\omega )}}={\frac {1}{\omega _{0}^{2}-\omega ^{2}+i\gamma \omega }}.\,}$
The amplitude gain is given by the magnitude of the complex number ${\displaystyle {\tilde {\chi }}(\omega ),}$ and the phase shift by the arctan of the imaginary part of the function, divided by the real one.
From this representation, we see that for small ${\displaystyle \gamma }$ the Fourier transform ${\displaystyle {\tilde {\chi }}(\omega )}$ of the linear response function yields a pronounced maximum ("Resonance") at the frequency ${\displaystyle \omega \approx \omega _{0}}$. The linear response function for a harmonic oscillator is mathematically identical to that of an RLC circuit. The width of the maximum ${\displaystyle ,\Delta \omega ,}$ typically is much smaller than ${\displaystyle \omega _{0},}$ so that the Quality factor ${\displaystyle Q:=\omega _{0}/\Delta \omega }$ can be extremely large.
## Kubo formula
The exposition of linear response theory, in the context of quantum statistics, can be found in a paper by Ryogo Kubo.[1] This defines particularly the Kubo formula, which considers the general case that the "force" h(t) is a perturbation of the basic operator of the system, the Hamiltonian, ${\displaystyle {\hat {H}}_{0}\to {\hat {H}}_{0}-h(t'){\hat {B}}(t')\,}$ where ${\displaystyle {\hat {B}}}$ corresponds to a measurable quantity as input, while the output x(t) is the perturbation of the thermal expectation of another measurable quantity ${\displaystyle {\hat {A}}(t)}$. The Kubo formula then defines the quantum-statistical calculation of the susceptibility ${\displaystyle \chi (t-t')}$ by a general formula involving only the mentioned operators.
As a consequence of the principle of causality the complex-valued function ${\displaystyle {\tilde {\chi }}(\omega )}$ has poles only in the lower half-plane. This leads to the Kramers-Kronig relations, which relates the real and the imaginary parts of ${\displaystyle {\tilde {\chi }}(\omega )}$ by integration. The simplest example is once more the damped harmonic oscillator.[2]
## References
1. ^ Kubo, R., Statistical Mechanical Theory of Irreversible Processes I, Journal of the Physical Society of Japan, vol. 12, pp. 570-586 (1957).
2. ^ De Clozeaux,Linear Response Theory, in: E. Anton?ik et al., Theory of condensed matter, IAEA Vienna, 1968
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Author Message
Post subject: Help: Age questionPosted: Thu Dec 13, 2012 7:58 pm
Joined: Wed Nov 23, 2011 12:52 pm
Posts: 603
Location: Shamballa
Hi.What is the best way of explaining the answer to dd.
"Peter's grandma is 5 times as old as Peter was 3 years ago.
If Peter's grandma is 55,how old is Peter ?"
_________________
"To err is human;to forgive ,divine"
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Post subject: Re: Help: Age questionPosted: Thu Dec 13, 2012 8:26 pm
Joined: Wed Jan 18, 2012 11:41 am
Posts: 4601
Location: Essex
Gran's current age = 55 = 5 times something
55 divided by 5 = 11
therefore Peter was 11 three years ago
therefore he is (11+3) years old now i.e. 14.
_________________
Outside of a dog, a book is a man's best friend. Inside of a dog it's too dark to read.Groucho Marx
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Post subject: Re: Help: Age questionPosted: Thu Dec 13, 2012 8:28 pm
Joined: Tue Jul 21, 2009 9:56 pm
Posts: 8228
We know that Peter's grandma is 55. If you divide this by 5, because she is 5 times as old as Peter was 3 years ago, you get 11. This is the age Peter was 3 years ago. So how old is he now? 11 plus 3 = 14. He is 14.
The greatest explaining you need to do is how a she became a grandmother at the age of 39.
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Post subject: Re: Help: Age questionPosted: Thu Dec 13, 2012 8:33 pm
Joined: Wed Nov 23, 2011 12:52 pm
Posts: 603
Location: Shamballa
Thanks v much.Thought I was going mad as the answer was "13" !
_________________
"To err is human;to forgive ,divine"
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Post subject: Re: Help: Age questionPosted: Thu Dec 13, 2012 8:41 pm
Joined: Wed Jan 18, 2012 11:41 am
Posts: 4601
Location: Essex
mystery wrote:
The greatest explaining you need to do is how a she became a grandmother at the age of 39.
- or at 41...
Either way, I'm afraid in some places I have worked in, either would have been regarded as an abnormally long intergenerational gap and prompted demands for investigation of infertility
(MIL had DH at 19; by choice, we started our family in our late 30s. MIL complained that she was too young to be a grandmother . DD (12) had a friend in yr6 whose mum is even now 13 years younger than I was when I had DD!).
_________________
Outside of a dog, a book is a man's best friend. Inside of a dog it's too dark to read.Groucho Marx
Last edited by ToadMum on Thu Dec 13, 2012 8:59 pm, edited 1 time in total.
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Post subject: Re: Help: Age questionPosted: Thu Dec 13, 2012 8:47 pm
Joined: Mon Jun 18, 2007 2:32 pm
Posts: 6966
Location: East Kent
Quote:
MIL complained that she was too young to be a grandmother
my Dad complained that , at 68, he was far too young to be a grandfather...
I am 55..and am hoping that it will be a long while before I am a grandmother!
Top
Post subject: Re: Help: Age questionPosted: Thu Dec 13, 2012 9:02 pm
Joined: Mon Jul 04, 2011 1:47 pm
Posts: 2151
Location: Warwickshire
I was too old to be a mother at 41. That's not totally true; it is just my ds is so wild. If he had been a quiet, well behaved, er, girl, life would have been a lot easier.
I wouldn't have wanted dc in my twenties or early thirties. But I didn't plan one at 40. I thought it was impossible to get pregnant at 40! How stupid!
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Post subject: Re: Help: Age questionPosted: Thu Dec 13, 2012 9:28 pm
Joined: Mon Feb 12, 2007 1:21 pm
Posts: 11952
When I worked elsewhere in the country one grandma was 31, her daughter was in Year 10 when baby arrived.
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Post subject: Re: Help: Age questionPosted: Thu Dec 13, 2012 9:30 pm
Joined: Mon Jul 04, 2011 1:47 pm
Posts: 2151
Location: Warwickshire
That is shocking.
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Post subject: Re: Help: Age questionPosted: Thu Dec 13, 2012 10:16 pm
Joined: Tue Jul 21, 2009 9:56 pm
Posts: 8228
Oops! I can't subtract properly. Must be because I am granny age.
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https://www.vedantu.com/question-answer/find-the-selling-price-when-cprs650-and-loss8-class-9-maths-cbse-60ab923e30cccf5f3f22a893
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Last updated date: 09th Dec 2023
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# Find the selling price when CP=Rs650 and loss=8%.
Answer
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279.9k+ views
Hint: We need to find the selling price when we are given the cost price and also we are given the loss at which the item is sold. For this, we need to be aware about the terms such as cost price, selling price. And also, we should be aware about the formulae related to the calculation of Selling price when cost price is given.
Complete step by step answer:
We will first look at some definitions:
Cost Price: The price at which goods are or have been bought by a merchant or retailer is known as cost price.
Selling Price: It is the price at which a good or commodity is sold by a shopkeeper to a customer.
If a shopkeeper sells the commodity at a price less than the actual value or at a price less than the price at which he bought it from another shopkeeper, he will face a loss.
The loss percentage is calculated as follows:
Loss = Cost Price - Selling Price$=CP-SP$
Loss Percent = $\dfrac{CP-SP}{CP}\times 100%$
Here, the Loss percentage is 8% as given. And CP is also given to be Rs650. Plugging these values we get:
$8=\dfrac{650-SP}{650}\times 100$
$\implies \dfrac{8}{100}=1-\dfrac{SP}{650}$
$\implies \dfrac{SP}{650}=1-\dfrac{8}{100}$
$\implies \dfrac{SP}{650}=\dfrac{92}{100}$
$\implies SP=\dfrac{92}{100}\times 650$
$\implies SP=\dfrac{13}{2}\times 92$
$\implies SP=Rs.598$
Note: Don’t get confused between percentage and fraction. For example, 3% means 0.03 times the quantity given and 0.3% means $0.3\times \dfrac{1}{100}$ i.e. 0.003 times the quantity given. Percent will always mean $\dfrac{1}{100}$, so calculate accordingly without mixing both of these together.
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# 32700 in Words
The number 32700 in words is written as Thirty-Two Thousand Seven Hundred. For example, if you bought a refrigerator worth Rs. 32700, then you can say, “I bought a refrigerator worth Rupees Thirty-Two Thousand Seven Hundred”. We know that 32700 is a cardinal number as it shows some quantity. Learn how to spell and write the number 32700 in English in this article.
32700 in Words Thirty-Two Thousand Seven Hundred Thirty-Two Thousand Seven Hundred in numerical form 32700
## 32700 in English Words
Generally, we express numbers in words using the English alphabet. Hence, we can read the number 32700 in English as Thirty-Two Thousand Seven Hundred.
## How to Write 32700 in Words?
Place value chart is important to write the number 32700 in words. So, let us create a table of 5 columns to represent the place value chart for the number 32700 as it is a five-digit number.
Ten Thousands Thousands Hundreds Tens Ones 3 2 7 0 0
Thus, we can write the expanded form as:
3 x Ten Thousand + 2 x Thousand + 7 x Hundred + 0 x Ten + 0 x One
= 3 x 10000 + 2 x 1000 + 7 x 100 + 0 x 10 + 0 x 1
= 30000 + 2000 + 700 + 0 + 0
= 30000 + 2000 + 700
= 32700
= Thirty-Two Thousand Seven Hundred
Therefore, 32700 in words is written as Thirty-Two Thousand Seven Hundred
Interesting way of writing 32700 in words
3 = Three
32 = Thirty-Two
327 = Three Hundred and Twenty-Seven
3270 = Three Thousand Two Hundred Seventy
32700 =Thirty-Two Thousand Seven Hundred
Thus, the word form of the number 32700 is Thirty-Two Thousand Seven Hundred
32700 is a natural number that is the successor of 32699 and the predecessor of 32701
• 32700 in words – Thirty-Two Thousand Seven Hundred
• Is 32700 an odd number? – No
• Is 32700 an even number? – Yes
• Is 32700 a perfect square number? – No
• Is 32700 a perfect cube number? – No
• Is 32700 a prime number? – No
• Is 32700 a composite number? – Yes
## Frequently Asked Questions on 32700 in Words
### Write 32700 in words.
32700 in words is written as Thirty-Two Thousand Seven Hundred.
### Simplify 42700 – 10000, and express in words.
Simplifying 42700 – 10000, we get 32700. Hence, 32700 in words is Thirty-Two Thousand Seven Hundred.
### 32700 is an odd number. True or False.
False, 32700 is not an odd number.
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https://convertoctopus.com/6719-deciliters-to-quarts
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text/html
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crawl-data/CC-MAIN-2023-50/segments/1700679100258.29/warc/CC-MAIN-20231130225634-20231201015634-00743.warc.gz
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## Conversion formula
The conversion factor from deciliters to quarts is 0.10566882049662, which means that 1 deciliter is equal to 0.10566882049662 quarts:
1 dL = 0.10566882049662 qt
To convert 6719 deciliters into quarts we have to multiply 6719 by the conversion factor in order to get the volume amount from deciliters to quarts. We can also form a simple proportion to calculate the result:
1 dL → 0.10566882049662 qt
6719 dL → V(qt)
Solve the above proportion to obtain the volume V in quarts:
V(qt) = 6719 dL × 0.10566882049662 qt
V(qt) = 709.98880491681 qt
The final result is:
6719 dL → 709.98880491681 qt
We conclude that 6719 deciliters is equivalent to 709.98880491681 quarts:
6719 deciliters = 709.98880491681 quarts
## Alternative conversion
We can also convert by utilizing the inverse value of the conversion factor. In this case 1 quart is equal to 0.0014084729126358 × 6719 deciliters.
Another way is saying that 6719 deciliters is equal to 1 ÷ 0.0014084729126358 quarts.
## Approximate result
For practical purposes we can round our final result to an approximate numerical value. We can say that six thousand seven hundred nineteen deciliters is approximately seven hundred nine point nine eight nine quarts:
6719 dL ≅ 709.989 qt
An alternative is also that one quart is approximately zero point zero zero one times six thousand seven hundred nineteen deciliters.
## Conversion table
### deciliters to quarts chart
For quick reference purposes, below is the conversion table you can use to convert from deciliters to quarts
deciliters (dL) quarts (qt)
6720 deciliters 710.094 quarts
6721 deciliters 710.2 quarts
6722 deciliters 710.306 quarts
6723 deciliters 710.411 quarts
6724 deciliters 710.517 quarts
6725 deciliters 710.623 quarts
6726 deciliters 710.728 quarts
6727 deciliters 710.834 quarts
6728 deciliters 710.94 quarts
6729 deciliters 711.045 quarts
| 560
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Blog
# How to Convert Seconds to Minutes in Excel?
Are you looking to quickly and easily convert seconds to minutes in Excel? You’ve come to the right place! In this article, we’ll show you the simple steps to convert seconds to minutes in Excel, and explain why this formula is useful. Whether you’re a data analyst, accountant, financial advisor, or just someone looking to improve their spreadsheet skills, this guide will help you learn how to convert seconds to minutes in Excel. So let’s get started!
## How to Convert Seconds to Minutes in Microsoft Excel
Microsoft Excel is a powerful tool that can be used for a variety of purposes. One of the most useful functions of Excel is its ability to convert seconds to minutes. This tutorial will show you how to quickly convert seconds to minutes in Excel.
The first step is to open Microsoft Excel and create a new spreadsheet. You can do this by clicking on the “File” menu and then “New”. Once you have created your spreadsheet, you will need to enter the number of seconds that you want to convert. It is important to note that you must use the correct formatting for the seconds. For example, if you want to convert 10 seconds, you will need to enter “10” in the cell and not “10s” or anything else.
### Using Excel Formulas
Once you have entered the number of seconds, you can use an Excel formula to convert the seconds to minutes. The formula is simple: divide the number of seconds by 60 and the result will be the number of minutes. For example, if you enter 10 seconds in cell A1, the formula in cell B1 would be “=A1/60”. This will give you the result of 0.16666666667 minutes.
If you want the result to be displayed as a whole number, you can use the “ROUND” function to round the result to the nearest whole number. For example, if you use the formula “=ROUND(A1/60, 0)” in cell B1, the result will be 1 minute.
Another way to convert seconds to minutes in Excel is to use the Formatting menu. You can do this by selecting the cell that contains the number of seconds and then clicking on the “Format” menu. From here, you can select the “Number” tab and then select the “Minutes” option. This will format the cell to display the result as minutes, rather than seconds.
### Using the Built-In Function
If you are using Microsoft Excel 2007 or later, you can also use the built-in “MINUTE” function to convert seconds to minutes. This function takes the number of seconds as an argument and returns the number of minutes. For example, if you enter the formula “=MINUTE(A1)” in cell B1, the result will be 0.16666666667 minutes.
### Using the Time Function
Finally, you can also use the “TIME” function to convert seconds to minutes. This function takes three arguments: the hours, minutes and seconds. For example, if you enter the formula “=TIME(0, A1, 0)” in cell B1, the result will be 0.16666666667 minutes.
## Conclusion
In this tutorial, we have shown you how to convert seconds to minutes in Microsoft Excel. We have shown you how to use formulas, the Formatting menu, the built-in function and the Time function to convert seconds to minutes.
## Top 6 Frequently Asked Questions
### Question 1: What is the formula for converting seconds to minutes in Excel?
Answer: The formula for converting seconds to minutes in Excel is to divide the seconds by 60. For example, if you had 120 seconds you would divide 120 by 60 to get 2 minutes. In Excel this would be expressed as =120/60.
### Question 2: How do I format the minutes after the conversion?
Answer: After the conversion from seconds to minutes, the minutes can be formatted a few different ways. If you want to keep the minutes as a decimal value, you can format the cell as a number with two decimal places. If you want to show the minutes in a time format, you can format the cell as a Time format with the option of showing the seconds. You can also change the Time format to only show minutes and hours, but no seconds.
### Question 3: What should I do if the seconds value is a negative number?
Answer: If the seconds value is a negative number, you should still use the same formula to convert seconds to minutes. This formula will work regardless of whether the seconds value is positive or negative. When you format the minutes, the negative sign will be maintained.
### Question 4: Can I convert minutes to seconds in Excel?
Answer: Yes, you can convert minutes to seconds in Excel. To do this, you would use the same formula as converting seconds to minutes, but instead you would multiply the minutes by 60. For example, if you had 2 minutes you would multiply 2 by 60 to get 120 seconds. In Excel this would be expressed as =2*60.
### Question 5: How do I handle time values that include both hours and minutes?
Answer: To handle time values that include both hours and minutes, you can use the HOUR and MINUTE functions in Excel. For example, if you had 2 hours and 30 minutes you would use the formula =HOUR(2:30)*60+MINUTE(2:30). The HOUR function will return 2, which is multiplied by 60 to get 120, and the MINUTE function will return 30, which is then added to the result from the HOUR function to get a total of 150 seconds.
### Question 6: Can I use the formula to convert time values that are greater than 24 hours?
Answer: Yes, you can use the formula to convert time values that are greater than 24 hours. You would just need to make sure you format the cell as a Time format and specify the number of hours you want to display. For example, if you had 48 hours you would format the cell as a Time format and set the number of hours to 48. This would allow you to see the result as 48 hours, instead of 2 days.
### How to convert Seconds to Time in Excel 2013
Excel is a powerful tool that can help you quickly and easily convert seconds to minutes. By following the steps outlined in this article, you can save time and energy when converting seconds to minutes in Excel. With a few clicks of your mouse, you can easily convert seconds to minutes and make your data more organized and accessible. So what are you waiting for? Go ahead and give it a try!
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+0
# Factorials
0
148
2
Find the value of n that satisfies 2(n+1)!+6n!=4(n+1)!, where n! = n\cdot (n-1)\cdot (n-2) \cdots 2\cdot 1.
May 1, 2022
#1
+118220
0
Use the latex button and enter your question properly.
May 1, 2022
#2
+2540
0
We have the equation: $$2(n+1)!+6n!=4(n+1)!$$
We can subtract $$2(n+1)!$$ from both sides, which yields: $$6n!=2(n+1)!$$
Note that $$(n+1)! = n! \times (n+1)$$
Substituting this gives us: $$6n!=2(n+1)n!$$
Now, we just have to divide by $$n!$$, then solve for n.
Can you take it from here?
May 1, 2022
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# 3 thousand exercise books are arranged into 3 piles. The first pile has 10 more books than the second pile. The number of books in the second pile is twice the numbers of books in the third pile. How many books are there in the third pile?
1
by alialfonso
2014-11-03T15:19:10+08:00
Let 'x' be the number of books on the first pile
'y' be the number of books on the second pile
'z' be the number of books on the third pile
x + y + z = 3000 -----equation 1
x = y + 10 -----equation 2
y = 2z -----equation 3
from equation 3
y = 2z
z = y/2 -----equation 3'
substitute equations 3' and 2 to equation 1
x + y + z =3000
(y+10) + y + (y/2) = 3000
y + 10 + y + y/2 = 3000
2y + y/2 = 3000-10
(4y+y)/2 = 2990
5y = 2990(2)
5y = 5980
y=1196
substitute y=1196 to equation 2
x = y + 10
x = 1196 +10
x = 1206
substitute y=1196 to equation 3'
z = y/2
z = 1196/2
z = 598
therefore there are 598 books in the third pile.
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# College Physics
posted by on .
A rock is dropped from a sea cliff and the sound of it striking the ocean is heard 6.4 s later. If the speed of sound is 340 m/s, how high is the cliff?
• College Physics - ,
The time of 6.4 s is the sum of the time required for the rock to hit the sea, sqrt (2H/g), and the time for the sound to get back, (H/V).
H = cliff height
V = speed of sound
g = acceleration of gravity
Solve this equation for H:
6.4 = H/340 + sqrt(H/4.9)
That can be turned into a quadratic equation. Only one root will be positive.
• College Physics - ,
172.35
• College Physics - ,
356.8
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When a variable loses its scope, its data value is not lost. True or False?
When a variable loses its scope, its data value is not lost. False.
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Asked 2/22/2013 11:54:03 AM
Updated 1/23/2017 9:26:50 AM
Flagged by matahari [1/23/2017 5:09:15 AM], Unflagged by jeifunk [1/23/2017 9:26:48 AM], Edited by jeifunk [1/23/2017 9:26:50 AM]
s
Rating
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When a variable loses its scope, its data value is not lost. False.
Added 1/23/2017 5:09:14 AM
This answer has been confirmed as correct and helpful.
Confirmed by jeifunk [1/23/2017 9:26:51 AM]
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https://www.aakash.ac.in/ncert-solutions/class-7/maths/chapter-4-simple-equations
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crawl-data/CC-MAIN-2024-22/segments/1715971058830.51/warc/CC-MAIN-20240525161720-20240525191720-00200.warc.gz
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| 34,588
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# NCERT Solutions for Class 7 Maths Chapter 4: Simple Equations
An equation is a mathematical statement on a variable where two expressions on either side of the equality sign should have equal values. The simple equation represents the relationships of two expressions on either side of the sign. It has one variable. The expressions on two sides are called LHS(left-hand side) and RHS(right-hand side). A variable takes on a different numeric value, and a constant has a fixed value. In Ex 4.1, students will learn about what equation means. There is an equality sign in any equation that shows the value of the expression to the left sign of the equation is equal to the equation's right side. In Ex 4.2, you will learn how to calculate or solve an equation.
In an equation, both the sides should be equal. An equation remains unchanged when Lhs and Rhs are interchanged, when the same number is added to both the sides when the same number is subtracted from both the sides, the same number is multiplied from both sides the Rhs, and Lhs is divided by the same number. In Ex 4.3, students will get to know about moving a number from one side to another instead of adding and subtracting it from both sides of the equation. In Ex 4.4, you will learn how to use Simple equations in the practical Questions. Here you will be given a practical question according to which you have to form equations and solve it. Some Important topic for this chapter are:
• What is an Equation
• Solving an equation
• More Equation
• From solution to the equation
• Application of simple equations to the practical equation.
Also See NCERT Solutions for Class 7 Maths Chapter 1 Integers NCERT Solutions for Class 7 Maths Chapter 2 Fractions and Decimals NCERT Solutions for Class 7 Maths Chapter 3 Data Handling NCERT Solutions for Class 7 Maths Chapter 5 Lines and Angles NCERT Solutions for Class 7 Maths Chapter 6 Triangles NCERT Solutions for Class 7 Maths Chapter 7 Congruence of Triangles NCERT Solutions for Class 7 Maths Chapter 8 Comparing Quantities NCERT Solutions for Class 7 Maths Chapter 9 Rational Numbers NCERT Solutions for Class 7 Maths Chapter 10 Practical Geometry NCERT Solutions for Class 7 Maths Chapter 11 Perimeter and Area NCERT Solutions for Class 7 Maths Chapter 12 Algebraic Expressions NCERT Solutions for Class 7 Maths Chapter 13 Exponents and Powers NCERT Solutions for Class 7 Maths Chapter 14 Symmetry NCERT Solutions for Class 7 Maths Chapter 15 Visualizing Solid Shapes
### NCERT Solutions For Other Class
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# Standard deviation
```Data
Collection and
Descriptive
Statistics
2011 Pearson Prentice Hall, Salkind.
Explain the steps in the data collection
process.
Construct a data collection form and code
data collected.
Identify 10 “commandments” of data
collection.
Define the difference between inferential and
descriptive statistics.
Compute the different measures of central
tendency from a set of scores.
Explain measures of central tendency and
when each one should be used.
2011 Pearson Prentice Hall, Salkind.
Compute the range, standard
deviation, and variance from a set of
scores.
Explain measures of variability and
when each one should be used.
Discuss why the normal curve is
important to the research process.
Compute a z-score from a set of
scores.
Explain what a z-score means.
2011 Pearson Prentice Hall, Salkind.
The Data Collection Process
Descriptive Statistics
◦ Measures of Central Tendency
◦ Measures of Variability
Understanding Distributions
2011 Pearson Prentice Hall, Salkind.
2011 Pearson Prentice Hall, Salkind.
Constructing a data collection form
Establishing a coding strategy
Collecting the data
Entering data onto the collection form
2011 Pearson Prentice Hall, Salkind.
2.00
gender
Total
4.00
6.00
10.00
Total
male
20
16
23
19
95
female
19
21
18
16
105
39
37
41
35
200
2011 Pearson Prentice Hall, Salkind.
2011 Pearson Prentice Hall, Salkind.
Begins with raw data
◦ Raw data are unorganized data
2011 Pearson Prentice Hall, Salkind.
One column for each variable
ID
Gender
Building
Score
Mathematics
Score
1
2
3
4
5
2
2
1
2
2
8
2
8
4
10
1
6
6
6
6
55
41
46
56
45
60
44
37
59
32
One row for each subject
2011 Pearson Prentice Hall, Salkind.
If subjects choose from several
responses, optical scoring sheets
might be used
Scoring is fast
Scoring is accurate
Additional analyses are easily done
Expense
2011 Pearson Prentice Hall, Salkind.
Variable
Range of Data Possible
Example
ID Number
001 through 200
Gender
1 or 2
2
1, 2, 4, 6, 8, or 10
4
Building
1 through 6
1
1 through 100
78
Mathematics Score
1 through 100
69
138
Use single digits when possible
Use codes that are simple and
unambiguous
Use codes that are explicit and discrete
2011 Pearson Prentice Hall, Salkind.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
Get permission from your institutional review board to
collect the data
Think about the type of data you will have to collect
Think about where the data will come from
Be sure the data collection form is clear and easy to
use
Make a duplicate of the original data and keep it in a
separate location
Ensure that those collecting data are well-trained
Cultivate sources for finding participants
Follow up on participants that you originally missed
Don’t throw away original data
2011 Pearson Prentice Hall, Salkind.
Descriptive statistics—basic measures
◦ Average scores on a variable
◦ How different scores are from one another
Inferential statistics—help make
◦ Null and research hypotheses
◦ Generalizing from sample to population
2011 Pearson Prentice Hall, Salkind.
2011 Pearson Prentice Hall, Salkind.
Distributions of Scores
• Comparing Distributions of Scores
2011 Pearson Prentice Hall, Salkind.
Mean—arithmetic average
Median—midpoint in a distribution
Mode—most frequent score
2011 Pearson Prentice Hall, Salkind.
What it is
◦ Arithmetic average
◦ Sum of
scores/number of
scores
How to compute it
◦ X = X
n
1.
2.
= summation sign
X = each score
n = size of sample
scores
Divide the total by
the number of
scores
2011 Pearson Prentice Hall, Salkind.
What it is
◦ Midpoint of
distribution
◦ Half of scores
above and half of
scores below
How to compute it
when n is odd
1.
2.
3.
Order scores from
lowest to highest
Count number of
scores
Select middle score
How to compute it
when n is even
1.
2.
3.
Order scores from
lowest to highest
Count number of
scores
Compute X of two
middle scores
2011 Pearson Prentice Hall, Salkind.
What it is
◦ Most frequently
occurring score
What it is not!
◦ How often the most
frequent score
occurs
2011 Pearson Prentice Hall, Salkind.
Measure of
Central
Tendency
Level of
Measurement
Use When
Examples
Mode
Nominal
Data are
categorical
Eye color,
party
affiliation
Median
Ordinal
Data include
extreme
scores
Rank in class,
birth order,
income
Mean
Interval and
ratio
You can, and
the data fit
Speed of
response, age
in years
2011 Pearson Prentice Hall, Salkind.
Variability is the degree of spread or
dispersion in a set of scores
Range—difference between highest and
lowest score
Standard deviation—average difference of
each score from mean
2011 Pearson Prentice Hall, Salkind.
s
◦
◦
◦
◦
= (X – X)2
n-1
= summation sign
X = each score
X = mean
n = size of sample
2011 Pearson Prentice Hall, Salkind.
X
13
14
1.
List scores and
compute mean
15
12
13
14
13
16
15
9
X = 13.4
2011 Pearson Prentice Hall, Salkind.
X
(X-X)
13
-0.4
14
0.6
15
1.6
12
-1.4
13
-0.4
14
0.6
13
-0.4
16
2.6
15
1.6
9
-4.4
X = 13.4
1.
2.
List scores and
compute mean
Subtract mean
from each
score
X = 0
2011 Pearson Prentice Hall, Salkind.
(X – X)
X
(X – X)2
13
-0.4
0.16
14
0.6
0.36
15
1.6
2.56
12
-1.4
1.96
13
-0.4
0.16
14
0.6
0.36
13
-0.4
0.16
16
2.6
6.76
15
1.6
2.56
9
-4.4
19.36
X =13.4
X=0
1.
2.
3.
List scores and
compute mean
Subtract mean
from each score
Square each
deviation
2011 Pearson Prentice Hall, Salkind.
(X – X)
X
(X – X)2
13
-0.4
0.16
14
0.6
0.36
15
1.6
2.56
12
-1.4
1.96
13
-0.4
0.16
14
0.6
0.36
13
-0.4
0.16
16
2.6
6.76
15
1.6
2.56
9
-4.4
19.36
X =13.4
X=0
X2 =
34.4
1.
2.
3.
4.
List scores and
compute mean
Subtract mean
from each score
Square each
deviation
Sum squared
deviations
2011 Pearson Prentice Hall, Salkind.
(X – X)
X
(X – X)2
13
-0.4
0.16
14
0.6
0.36
15
1.6
2.56
12
-1.4
1.96
13
-0.4
0.16
14
0.6
0.36
13
-0.4
0.16
16
2.6
6.76
15
1.6
2.56
9
-4.4
19.36
X =13.4
X=0
X2 =
34.4
1.
2.
3.
4.
5.
6.
List scores and
compute mean
Subtract mean from
each score
Square each deviation
Sum squared deviations
Divide sum of squared
deviation by n – 1
• 34.4/9 = 3.82 (= s2)
Compute square root
of step 5
• 3.82 = 1.95
2011 Pearson Prentice Hall, Salkind.
2011 Pearson Prentice Hall, Salkind.
Mean = median = mode
Tails approach X axis, but do not touch
2011 Pearson Prentice Hall, Salkind.
2011 Pearson Prentice Hall, Salkind.
The normal curve is symmetrical
One standard deviation to either side of the mean contains
34% of area under curve
68% of scores lie within ± 1 standard deviation of mean
2011 Pearson Prentice Hall, Salkind.
Standard scores have been “standardized”
SO THAT
Scores from different distributions have
◦ the same reference point
◦ the same standard deviation
Computation
Z = (X – X)
s
–Z = standard score
–X = individual score
–X = mean
–s = standard deviation
2011 Pearson Prentice Hall, Salkind.
Standard scores are used to compare scores
from different distributions
Class
Mean
Sara
Micah
90
90
Class
Standard
Deviation
2
4
Student’s Student’s
Raw
z Score
Score
92
1
92
.5
2011 Pearson Prentice Hall, Salkind.
Because
◦ Different z scores represent different
locations on the x-axis, and
◦ Location on the x-axis is associated
with a particular percentage of the
distribution
z scores can be used to predict
◦ The percentage of scores both above
and below a particular score, and
◦ The probability that a particular score
will occur in a distribution
2011 Pearson Prentice Hall, Salkind.
Explain the steps in the data collection process?
Construct a data collection form and code data collected?
Identify 10 “commandments” of data collection?
Define the difference between inferential and descriptive statistics?
Compute the different measures of central tendency from a set of
scores?
Explain measures of central tendency and when each one should be
used?
Compute the range, standard deviation, and variance from a set of
scores?
Explain measures of variability and when each one should be used?
Discuss why the normal curve is important to the research process?
Compute a z-score from a set of scores?
Explain what a z-score means?
2011 Pearson Prentice Hall, Salkind.
```
| 2,871
| 8,123
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https://mathhelpboards.com/threads/among-2n-1-integers-summation-of-some-n-of-these-is-divisible-by-n.614/
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# Among 2n-1 integers summation of some n of these is divisible by n.
#### caffeinemachine
##### Well-known member
MHB Math Scholar
Let $k$ be a positive integer. Let $n=2^{k-1}$. Prove that, from $2n-1$ positive integers, one can select $n$ integers, such that their sum is divisible by $n$.
#### CaptainBlack
##### Well-known member
Let $k$ be a positive integer. Let $n=2^{k-1}$. Prove that, from $2n-1$ positive integers, one can select $n$ integers, such that their sum is divisible by $n$.
Suppose this true for some $$k$$.
Now consider the case for $$k+1$$, then $$n_{k+1}=2n_k$$ and any set of $$2n_{k+1}-1$$ positive integers. Now split the $$2n_{k+1}-1$$ positive integers into three sets two of size $$2n_k-1$$, and one of size $$1$$.
We can select $$n_k$$ integers from the first set with sum divisible by $$n_k$$, and a set of $$n_k$$ integers from the second set with sum divisible by $$n_k$$. So we have a combined set of $$2n_k=n_{k+1}$$ integers from the set of 2n_{k+1}-1 integers with sum equal to 2n_k=n_{k+1}.
The rest of the details for a proof by induction I leave to the reader.
CB
#### caffeinemachine
##### Well-known member
MHB Math Scholar
Suppose this true for some $$k$$.
We can select $$n_k$$ integers from the first set with sum divisible by $$n_k$$, and a set of $$n_k$$ integers from the second set with sum divisible by $$n_k$$. So we have a combined set of $$2n_k=n_{k+1}$$ integers from the set of $2n_{k+1}-1$ integers with sum equal to $2n_k=n_{k+1}$.
I think "equal to" is a typo. What you probably intended was "divisible by". But even that is not guaranteed (at least I am not convinced). Divisibility by $n_k$ is guaranteed though.
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https://www.physicsforums.com/threads/writing-equations-for-position-velocity-and-accelerations-as-functions-of-time.286833/
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# Writing Equations for Position, Velocity, and Accelerations as functions of time
• Jordash
In summary, the conversation discusses a projectile motion problem where a projectile is launched from a cliff onto a flat valley floor. The equations for position, velocity, and acceleration as functions of time are requested for this problem, with the assumption of constant gravity and no aerodynamic effects. The concept of a projectile and the use of kinematics formulas for constant acceleration are also mentioned.
Jordash
## Homework Statement
A projectile is launched with a speed of 50.0 m/s and an angle of 40.0o above the horizontal, from the top of a 75.0 m high cliff onto a flat valley floor at the base of the cliff. Assume that g = 10.0 m/s2 and ignore aerodynamic effects.
Write equations for position, velocity, and acceleration as function of time for the projectile.
## The Attempt at a Solution
I've been looking through the time functions but I can't seem to figure out how to write these equations. Any help would be greatly appreciated.
It's a projectile motion problem. What is a projectile? An object that, once set into motion (launched) travels solely under the influence of gravity. Gravity is a constant force (it isn't really, but it is for our purposes, unless you're traveling really large vertical distances). So you already know that the acceleration is constant (doesn't change with time). From that you can either work backwards using calculus to find the velocity and position. OR, if you haven't studied calculus, then your teacher will have taught you kinematics formulas that *describe* the motion of an object under constant acceleration (formulas that are derived from calculus), and you can just *use* them.
I didn't really understand our teacher when he taught those formulas, is there a place I can look which teaches the formulas i'll need to use?
accident put wrong post in this forum
Last edited:
## 1. What is the difference between position, velocity, and acceleration?
Position refers to an object's location in space, velocity refers to the rate of change of an object's position, and acceleration refers to the rate of change of an object's velocity. In other words, position is where the object is, velocity is how fast the object is moving, and acceleration is how much the object's speed is changing.
## 2. How do I write an equation for position as a function of time?
The equation for position as a function of time is x(t) = x0 + v0t + ½at2, where x0 is the initial position, v0 is the initial velocity, a is the acceleration, and t is time. This equation can be used to calculate an object's position at any given time if its initial position, initial velocity, and acceleration are known.
## 3. How do I write an equation for velocity as a function of time?
The equation for velocity as a function of time is v(t) = v0 + at, where v0 is the initial velocity, a is the acceleration, and t is time. This equation can be used to calculate an object's velocity at any given time if its initial velocity and acceleration are known.
## 4. How do I write an equation for acceleration as a function of time?
The equation for acceleration as a function of time is a(t) = a0, where a0 is the initial acceleration. This equation assumes that the acceleration is constant over time. If the acceleration is not constant, a more complex equation must be used.
## 5. Can I use these equations for objects moving in a circular motion?
Yes, these equations can be used for objects moving in a circular motion, but additional equations may be needed to account for the direction of the velocity and acceleration. For example, the equation for velocity in circular motion is v(t) = ωr, where ω is the angular velocity and r is the radius of the circular path. The equations for position and acceleration may also differ depending on the specific scenario of the circular motion.
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http://mathhelpforum.com/calculus/188310-show-has-glb.html
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# Thread: show that A has a glb
1. ## show that A has a glb
If A is a nonempty subset of R that is bounded below, show that A has a greatest lower bound. That is, show that there is a number $\displaystyle m \in R$satisfying: 1) m is a lower bound for A; and 2) if x is a lower bound for A, then x<=m. [Hint: Consider the set -A={$\displaystyle -a: a \in A$} and show that m = -sup(-A) works.]
The hint is confusing me, I'm really not good at proving, any help is appreaciated!
2. ## Re: show that A has a glb
so should I just show that -m = sup(-A), then we have m= - sup(-A) by the least upper bound axiom?
3. ## Re: show that A has a glb
Originally Posted by wopashui
If A is a nonempty subset of R that is bounded below, show that A has a greatest lower bound. That is, show that there is a number $\displaystyle m \in R$satisfying: 1) m is a lower bound for A; and 2) if x is a lower bound for A, then x<=m. [Hint: Consider the set -A={$\displaystyle -a: a \in A$} and show that m = -sup(-A) works.]
The hint is confusing me, I'm really not good at proving, any help is appreaciated!
Hi wopashui,
This problem is incorrect. There are sets for which there exist lower bounds but not a greatest lower bound. For example let,
$\displaystyle \mathbb{Q}$ be the set of Rational numbers and $\displaystyle A=\left\{a\in\mathbb{Q}~|~a\in\left(\sqrt{2},5 \right) \right\}.$ Clearly there are lower bounds for the set A, such as, 1,0,-1 etc. But a greatest lower bound does not exist. Since if you take any lower bound (say $\displaystyle l\in\mathbb{Q}$) there exist infinity many rational elements in between $\displaystyle l$ and $\displaystyle \sqrt{2}$.
4. ## Re: show that A has a glb
Originally Posted by Sudharaka
Hi wopashui,
This problem is incorrect. There are sets for which there exist lower bounds but not a greatest lower bound. For example let,
$\displaystyle \mathbb{Q}$ be the set of Rational numbers and $\displaystyle A=\left\{a\in\mathbb{Q}~|~a\in\left(\sqrt{2},5 \right) \right\}.$ Clearly there are lower bounds for the set A, such as, 1,0,-1 etc. But a greatest lower bound does not exist. Since if you take any lower bound (say $\displaystyle l\in\mathbb{Q}$) there exist infinity many rational elements in between $\displaystyle l$ and $\displaystyle \sqrt{2}$.
The set has to be bounded below, the one you have given is not.
5. ## Re: show that A has a glb
actually, the glb for your example is root of 2, isn't it? Oh because the problem I have is in R, the example you gave is in Q, yes R and Q appear quite different when we examine the existence of lub or glb fir biunded sets.
6. ## Re: show that A has a glb
Originally Posted by wopashui
The set has to be bounded below, the one you have given is not.
Of course it is. As I have mentioned in my previous post examples of lower bounds for A include, 1,0,-1 etc.
Originally Posted by wopashui
actually, the glb for your example is root of 2, isn't it?
The greatest lower bound should be inside the superset. In my example, $\displaystyle \sqrt{2}\notin\mathbb{Q}$. I think you have to carefully go though the definitions Lower and Upper bound of a set, Supremum and Infimum.
Oh because the problem I have is in R, the example you gave is in Q, yes R and Q appear quite different when we examine the existence of lub or glb fir biunded sets.
Did you denote the the set of Real numbers by R? I thought it was a general set not specifically $\displaystyle \Re$.
7. ## Re: show that A has a glb
Originally Posted by wopashui
If A is a nonempty subset of R that is bounded below, show that A has a greatest lower bound. That is, show that there is a number $\displaystyle m \in R$satisfying: 1) m is a lower bound for A; and 2) if x is a lower bound for A, then x<=m.
I do not like that hint ether.
Let $\displaystyle B = \left\{ {t:\left( {\forall x \in A} \right)\left[ {t < x} \right]} \right\}$
We know that $\displaystyle B\ne\emptyset$ because $\displaystyle A$ has a lower bound.
Because $\displaystyle \left( {\exists s \in A} \right)$ the set $\displaystyle B$ is bounded above.
Now prove the $\displaystyle \sup(B)=\inf(A)$.
### bounded above and biunded below
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http://mathhelpforum.com/statistics/204208-probability-batting-orders.html
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|
text/html
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crawl-data/CC-MAIN-2016-50/segments/1480698541696.67/warc/CC-MAIN-20161202170901-00384-ip-10-31-129-80.ec2.internal.warc.gz
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# Thread: Probability with batting orders
1. ## Probability with batting orders
How many batting orders (the order in which baseball players face an opposing pitcher in a game) does a manager have available for the nine baseball players of a team, if the center fielder must bat 4th and the pitcher must bat 9th?
1. 5040
2. 362,880
3. 181,440
4. 40,320
2. ## Re: Probability with batting orders
Hey jthomp18.
For this problem you are fixing two out of the nine which means you are left to re-order the rest how you please.
If there is no other constraints for this problem then think about how possibilities you get for each position.
The first position has 7 choices. You fix one of these choices and the second position has 6 (since you already used up 1 for the first). Then you have 5,4,3,2 and 1 choice(s) for the rest of them. These are all independent choices (once you choose the first choice, the second choice only depends on the number of people that are left) so you can multiply these choices.
What does this process give you as an answer?
3. ## Re: Probability with batting orders
This is permutations. 7! =5040
4. ## Re: Probability with batting orders
Originally Posted by Tim28
This is permutations. 7! =5040
chiro had explained HOW to get the answer. Do you think it is at all helpful to simply give the answer without any explanation?
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en
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Name: Quiz 2-4: Dividing Numbers
Multiple Choice
Identify the letter of the choice that best completes the statement or answers the question.
1.
Find:
a. c. b. d.
2.
Due to melting snow and heavy rains, a river’s flow reached a high of 40,500 cubic feet per second before decreasing. Five months later, the river flowed at 5700 cubic feet per second. What was the average change in river flow for each month?
a. –6960 cubic feet per second c. –9240 cubic feet per second b. 6960 cubic feet per second d. –34,800 cubic feet per second
3.
The table shows the amount of merchandise sold by a store.What was the average change in the number of digital cameras sold in each of the five years from 1995 to 2000?
Type of Merchandise 1995 2000 35 mm cameras 1534 1753 digital cameras 152 1197 camcorders 439 515
a. 269.8 digital cameras c. –209 digital cameras b. 239.4 digital cameras d. 209 digital cameras
4.
Assume that the table shows the number of acres in a rural county planted in orchards. What is the average change in acres of pear orchards for the eight years from 1992 to 2000?
Type of Orchard 1992 2000 Apple 5258 4890 Pear 6865 5921 Cherry 2823 3035
a. –118 acres c. –46 acres b. –944 acres d. –1598.25 acres
5.
The stock of a certain company was valued at 54 points. Twelve months later, the same stock was valued at 30 points. What was the average change in the value of the stock for each of the last twelve months?
a. -2 points c. -4.5 points b. -7 points d. -2.5 points
6.
The strip of land between a cliff and a coastal highway is shrinking due to erosion. Suppose that the land between the cliff and the highway in one section shrank from a width of 66 feet to a width of 48 feet over a period of six years. What was the average change in width of the land between the cliff and the highway for each year?
a. -19 ft c. 3 ft b. -3 ft d. 19 ft
7.
Find the quotient.
a. 0.13 c. 1.3 b. d. 1.13
8.
Find the quotient.
a. c. b. d.
9.
Find the quotient.
a. c. b. d.
10.
Simplify.
a. 1 + 7x c. 1 + 14x b. 8x d. 2 + 7x
11.
Find 18 (-).
a. -27 c. -12 b. 27 d. 12
12.
Simplify .
a. 3x + 2 c. -x + 2 b. -3x + 2 d. x - 2
13.
Divide:
14.
Due to melting snow and heavy rains, a river’s flow reached a high of 35,600 cubic feet per second before decreasing. Six months later, the river flowed at 4700 cubic feet per second. What was the average change in river flow for each month?
15.
The table shows the amount of merchandise sold by a store.What was the average change in the number of digital cameras sold in each of the five years from 1995 to 2000?
Type of Merchandise 1995 2000 35 mm cameras 1273 1435 digital cameras 212 1082 camcorders 388 527
16.
Assume that the table shows the number of acres in a rural county planted in orchards. What is the average change in acres of apple orchards for the eight years from 1992 to 2000?
Type of Orchard 1992 2000 Apple 6228 5796 Pear 6524 5988 Cherry 3060 3478
17.
The stock of a certain company was valued at 34 points. Twelve months later, the same stock was valued at 22 points. What was the average change in the value of the stock for each of the last twelve months?
18.
The strip of land between a cliff and a coastal highway is shrinking due to erosion. Suppose that the land between the cliff and the highway in one section shrank from a width of 59 feet to a width of 31 feet over a period of seven years. What was the average change in width of the land between the cliff and the highway for each year?
19.
Find the quotient.
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https://math.stackexchange.com/questions/3735010/need-help-in-understanding-that-any-permutation-can-be-written-as-a-product-of-t
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# Need help in understanding that any permutation can be written as a product of two involutions.
I have seen this proof Is any permutation the product of two involutions? but that is unclear to me
This proves it for only the cycle $$(1,2,3...n)$$
How does proving it only for the cycle $$(1, 2, 3...n)$$ helps proving for any kind of permutation?
I am new to group theory and I do not know anything other than the following
1. Linear algaebra
2. Any permutation can be written as a product of transpositions.
3. Any permutation can be written as a product of disjoint cycles.
4. A permutation is an involution precisely if it can be written as a product of one or more non-overlapping transpositions.
• By 3, Any permutation can be written as the product of disjoint cycles. So, if you can prove that a cycle is the product of 2 involutions, then you can show that the permutation is the product of disjoint "product of 2 involutions", which hopefully you can then show is simply the product of 2 involutions. This turns out to be true because the product of several involutions which have disjoint terms is still an involution. Putting it all together, $P = \prod C_i = \prod A_i B_i = (\prod A_i) ( \prod B_i)$ gives the permutation as the product of 2 involutions. – Calvin Lin Jun 26 '20 at 7:24
It might help to consider an example. Suppose we have the following disjoint cycle decomposition of a permutation: $$\sigma = (1 \ 7\ 4\ 10)(2\ 9\ 8)(3\ 5\ 6).$$ To begin, separately decompose each cycle: $$(1 \ 7\ 4\ 10) = \tau_{1,1}\tau_{1,2}, \quad (2\ 9\ 8) = \tau_{2,1}\tau_{2,2} \quad (3\ 5\ 6) = \tau_{3,1} \tau_{3,2}.$$ Note that $$\tau_{1,1},\tau_{1,2}$$ are permutations that only affect the elements $$1,4,7,10$$. Similarly, $$\tau_{2,1},\tau_{2,2}$$ only affect $$2,9,8$$. In other words, for any $$i \neq j$$, the elements $$\tau_{i,p}, \tau_{j,q}$$ are disjoint permutations, which means that $$\tau_{ip}\tau_{jq} = \tau_{jq}\tau_{ip}$$.
With that established, we can use this commutativity property to "move" the transpositions $$\tau_{i,1}$$ to the left. That is, we can write $$\sigma = \tau_{11}(\tau_{12}\color{red}{\tau_{21}})\tau_{22}\tau_{31}\tau_{32}\\ = \tau_{11} (\color{red}{\tau_{21}}\tau_{12}) \tau_{22}\color{red}{\tau_{31}} \tau_{32}\\ = \tau_{11}\tau_{21}\color{red}{\tau_{31}}\tau_{12}\tau_{22}\tau_{32}\\ = (\tau_{11}\tau_{21}\tau_{31})(\tau_{12}\tau_{22}\tau_{32}).$$ Now, we see that $$\sigma$$ is a product of the involutions $$\tau_{11}\tau_{21}\tau_{31}$$ and $$\tau_{12}\tau_{22}\tau_{32}$$.
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# How many minimum number of flowers he needs to carry such that he offers equal number of flowers in all temples?
618 views
All the rivers are magical, moment a worshiper crosses a river with any number of flowers, it becomes double. (For eg 1 flower becomes 2 flowers, 2 become 4 and so on).
How many minimum number of flowers a worshiper needs to carry from beginning such that he offers equal number of flowers in all the three temples, and he is left with zero flowers at the end of fourth river.
posted Nov 6, 2014
## 1 Solution
+1 vote
He started with 7 Flowers, put them into river so it becomes 14, then he offers 8 flower to first temple, left with 6. then it will become 12 into second river. Offers 8 to second 1.and left with 4. Then put them into third 1 and they became 8 and he offers all to third 1. Left with 0.
solution Nov 6, 2014
Similar Puzzles
Rohan is on his way to visit his girlfriend, who lives at the end of the state.It's her birthday, and he want to give her the cakes that he has made.Between his place and her girlfriend's house, he need to cross 7 toll bridges.
Before you can cross the toll bridge, you need to give them half of the cakes you are carrying, but as they are kind trolls, they each give you back a single cake.
How many cakes do Rohan have to carry with him so he can reach his girlfriend's home with exactly 2 cakes?
+1 vote
There are (n+1) people in a party, they might or might not know each others names.
There is one celebrity in the group(total n +1 people), celebrity does not know any of n peoples by name and all n people know celebrity by name.
You are given the list of people's names(n+1), You can ask only one question from the people. DO YOU KNOW THIS NAME ?
HOW MANY MINIMUM NUMBER OF QUESTIONS YOU NEED TO ASK TO KNOW THE CELEBRITY NAME?
NOTE: assume all names are unique.
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### Home > AC > Chapter 4 > Lesson 4.1.4 > Problem4-32
4-32.
For each equation below, solve for $x$. Check your solution, if possible, and show all work.
1. $3x − 6 + 1 = − 2x − 5 + 5x$
1. $−2x − 5 = 2 − 4x − \left(x − 1\right)$
Combine like terms.
$3x − 5 = 3x − 5$
Add $5$ to both sides.
$3x − 5 = 3x − 5$
$+5 +5$
$3x = 3x$
Since $3x$ is always equal to $3x,$ any number makes this statement true.
Follow the steps in part (a).
$x=\frac{8}{3}$
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https://www.shaalaa.com/question-bank-solutions/in-the-figure-qrqs-qtpr-and-1-2-show-that-pqs-tqr-criteria-for-similarity-of-triangles_267730
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# In the figure QRQS=QTPR and ∠1 = ∠2. Show that ∆PQS ~ ∆TQR. - Mathematics
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Sum
In the figure (QR)/(QS) = (QT)/(PR) and ∠1 = ∠2. Show that ∆PQS ~ ∆TQR.
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#### Solution
In ΔPQR,
∠PQR = ∠PRQ
∴ PQ = PR .......(i)
Given,
(QR)/(QS) = (QT)/(PR) ........Using equation (i), we get
(QR)/(QS) = (QT)/(QP) ........(ii)
In ΔPQS and ΔTQR, by equation (ii)
(QR)/(QS) = (QT)/(QP)
∠Q = ∠Q
∴ ΔPQS ~ ΔTQR .......[By SAS similarity criterion]
Concept: Criteria for Similarity of Triangles
Is there an error in this question or solution?
#### APPEARS IN
NCERT Mathematics Exemplar Class 10
Chapter 6 Triangles
Exercise 6.3 | Q 4 | Page 140
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# Study Notes for CAT 2021: Time & Calendar
Updated : March 26th, 2021
Share via |
Reasoning is one of the most scoring sections in SSC Exams. But some topics are considered to be confusing by many of the aspirants. One such topic is “Time & Calendar”. In order to make the topic easy for all of you, here we are providing some Tricks to solve Time and Calendar Related Questions in Reasoning Section easily, accurately and in minimum time. We hope these tricks prove to be useful for you all.
Reasoning is one of the most scoring sections in SSC Exams. But some topics are considered to be confusing by many of the aspirants. One such topic is “Time & Calendar”. In order to make the topic easy for all of you, here we are providing some Tricks to solve Time and Calendar Related Questions in Reasoning Section easily, accurately and in minimum time. We hope these tricks prove to be useful for you all.
## Short Tricks to solve Time and Calendar related questions
Trick for solving questions where year is 2000 or more
1. Consider the last 3 digit of the year. If it is less than 100, add 100 to it. For example- In year 2012, the last three digit is 012 which is less than 100 so add 100 to add and make it 112 (100+012). Once done, divide it by 4 and keep the result.
2. Then use the code of the month from calender shown above. If the year is a leap year, consider the code for leap year.
3. Then write the date.
4. At the end, add all these data and find the result.
5. To find the day, divide the result by 7. You will get some remainder.
6. Match the code of remainder with above given code table and find the answer.
Trick for solving Questions where year is 1999 or less
1. First of all, take the last two digit of year and divide it by 4. For example- In case of year 1985, divide 85 by 4 and keep the result for future use.
2. Then use the code of the month from calendar shown above. If the year is a leap year, consider the code for leap year.
3. Then write the date.
4. At the end, add all these data and find the result.
5. To find the day, divide the result by 7. You will get some remainder.
6. Match the code of remainder with above given code table and find the answer.
Examples on above Tricks
Q1. What was the day on 31st Oct 1984?
(a) Friday
(b) Sunday
(c) Wednesday
(d) Monday
Sol- In 1984, divide 84/4 = 21
Code for Oct = 0 (Refer the above shown calendar for codes)
Mentioned Date= 31
Result = 84+21+0+31 =136
To find the day of week, divide the result by 7 and write the remainder = 136/7 = 3 (remainder)
3 is the code for Wednesday so the answer will be Wednesday.
Q2. What was the day on 27th Dec 1985?
(a) Friday
(b) Monday
(c) Tuesday
(d) Sunday
Sol- Here the year is 1985 so divide 85/4= 21
Code for Dec = 5
Mentioned Date= 27
Result = 85+21+5+27 =138
To find the day of week, divide the result by 7 and write the remainder = 138/7 = 5(remainder)
5 is the code for Friday so the answer will be Friday.
Q3. Find the day of the week on 26th Jan 2012?
(a) Tuesday
(b) Thursday
(c) Friday
(d) Sunday
SolHere the year is 2012 so follow the second rule and consider the last 3 digit i.e 012. It is less than hundred so add 100 to it. After adding 100, it becomes 112. Now, divide it by 4. You will get 28. Also note that it is a leap year.
Code of Jan = 6 (due to leap year)
Mentioned Date= 26
Result = 112+28+6+26 = 172
To find the day of week, divide the result by 7 and write the remainder = 172/7= 4
4 is the code for Thursday so the answer will be Thursday.
We hope these tricks would have cleared all your doubts related to the topic.
=======================
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Blog
# How to Calculate Difference in Time in Excel?
Time is a precious commodity, and managing it can be a challenge. Knowing how to calculate differences in time can be a useful skill when it comes to managing your schedule and staying on top of deadlines. In this guide, we’ll walk you through how to calculate the difference between two times in Excel. We’ll also explain how to use the TIMEVALUE function to easily determine the amount of time between two dates. With this knowledge, you can easily calculate the differences between two points in time and keep your tasks on track.
## Calculating Time Difference in Excel
Calculating the difference between two times in Excel is a simple task. It can be done in a few steps and can be used for a variety of purposes. Whether it’s to track time spent on a project or to calculate the amount of time in between two events, using Excel to calculate the time difference is a quick and easy way to get the information you need.
### Difference Between Two Times
The first step in calculating the difference between two times in Excel is to enter the starting and ending times into two different cells. Once the times are entered, you will then need to create a formula that subtracts the start time from the end time. This formula will calculate the difference between the two times and will give you the amount of time in between them.
The formula that is used to calculate the difference between two times is simple. It is simply the end time minus the start time. For example, if you are calculating the difference between a starting time of 8:00 a.m. and an ending time of 9:30 a.m., the formula would be “=9:30-8:00”. This formula will give you the result of 1.5 hours.
### Formatting the Time Difference
Once the formula has been used to calculate the difference between two times, you may want to format the time difference in a certain way. Excel has a variety of different formats that can be used to display the time difference. Depending on what you are using the time difference for, you may want to display it in hours, minutes, or seconds.
For example, if you are calculating the difference between two times and want the result to be displayed in minutes, you can use the formula “=TEXT(D2-D1,”m”)”. This formula will display the result in minutes. Similarly, if you want the result to be displayed in seconds, you can use the formula “=TEXT(D2-D1,”s”)”. This formula will display the result in seconds.
### Calculating Time Difference in Hours, Minutes, and Seconds
If you need to calculate the time difference in hours, minutes, and seconds, you can use the formula “=HOUR(D2-D1)*60+MINUTE(D2-D1)+SECOND(D2-D1)/60”. This formula will calculate the time difference in hours, minutes, and seconds. For example, if you are calculating the difference between a starting time of 8:00 a.m. and an ending time of 9:30 a.m., the formula would be “=HOUR(9:30-8:00)*60+MINUTE(9:30-8:00)+SECOND(9:30-8:00)/60”. This formula will give you the result of 1 hour, 30 minutes.
### Calculating Elapsed Time
In some cases, you may want to calculate the elapsed time between two times. This can be done by using the formula “=TEXT(D2-D1,”h:mm:ss”)”. This formula will calculate the elapsed time between two times and will display it in hours, minutes, and seconds.
### Conclusion
Calculating the difference between two times in Excel is a simple task that can be done in a few easy steps. Using formulas, you can quickly and easily calculate the time difference or elapsed time between two times. This can be useful for tracking time spent on projects or for calculating the amount of time in between two events.
### What Information Do I Need to Calculate the Difference in Time in Excel?
To calculate the difference in time in Excel, you will need to know the starting time and the ending time. You will also need to know the time format that you are using, as this will affect the formula used to calculate the difference. For example, if you are using a 24-hour time format, you will need to use a different formula than if you are using a 12-hour AM/PM time format.
### What Formula Do I Need to Use to Calculate the Difference in Time in Excel?
The formula you need to use to calculate the difference in time in Excel depends on the time format you are using. If you are using a 24-hour format, the formula is =ENDTIME – STARTTIME. If you are using a 12-hour AM/PM format, the formula is =TEXT(ENDTIME-STARTTIME,”hh:mm:ss”). Both of these formulas will give you the difference in time in hours, minutes, and seconds.
### How Do I Format My Times for Calculating the Difference in Time in Excel?
When formatting your times for calculating the difference in time in Excel, you will need to make sure that the times are in the same format. If you are using a 24-hour format, you will need to make sure that the times are in the format hh:mm:ss (for example, 12:45:00). If you are using a 12-hour AM/PM format, you will need to make sure that the times are in the format hh:mm:ss AM/PM (for example, 12:45:00 PM).
### What Does the Result of Calculating the Difference in Time in Excel Look Like?
The result of calculating the difference in time in Excel will be a numerical value representing the difference in hours, minutes, and seconds between the two times. For example, if the starting time is 10:00 AM and the ending time is 11:45 AM, the result will be 1:45:00. This means that the difference between the two times is 1 hour, 45 minutes, and 0 seconds.
### Can I Calculate the Difference in Time in Excel for Dates That Span Multiple Days?
Yes, you can calculate the difference in time in Excel for dates that span multiple days. For example, if the starting time is 10:00 PM on one day and the ending time is 12:00 AM the next day, the result will be 2:00:00. This means that the difference between the two times is 2 hours, 0 minutes, and 0 seconds.
### What Happens if I Input Negative Values for Calculating the Difference in Time in Excel?
If you input negative values for calculating the difference in time in Excel, the result will be a negative value. For example, if the starting time is 11:00 PM and the ending time is 9:00 PM, the result will be -2:00:00. This means that the difference between the two times is 2 hours, 0 minutes, and 0 seconds, but in a negative direction.
### Calculate Time Difference in Excel
Calculating the difference between two times in Excel is a quick and easy task. By following the step-by-step instructions outlined in this article, you can easily calculate the difference between two times in Excel and save yourself time and effort. With the help of the formula provided, you can accurately calculate the difference between two times, no matter how big the difference is. So, why not give it a try and make use of this handy tool?
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## Pages
Showing posts with label free study guide. Show all posts
Showing posts with label free study guide. Show all posts
## Monday, December 10, 2018
### Free Algebra Study Guide with Videos
This Algebra study guide is designed to supplement your current textbook. Your textbook is well written in a patient style and the appropriate sections should be read before each class meeting.
Chapter 1 - Real Numbers and Their Operations 1.1 Real Numbers and The Number Line 1.2 Adding and Subtracting Integers 1.3 Multiplying and Dividing Integers 1.4 Fractions 1.5 Review of Decimals and Percents 1.6 Exponents and Square Roots 1.7 Order of Operations 1.8 Sample Exam Questions Chapter 2 - Linear Equations and Inequalities 2.1 Introduction to Algebra 2.2 Simplifying Algebraic Expressions 2.3 Linear Equations: Part I 2.4 Linear Equations: Part II 2.5 Applications of Linear Equations 2.6 Ratio and Proportion Applications 2.7 Introduction to Inequalities and Interval Notation 2.8 Linear Inequalities (one variable) 2.9 Review Exercises and Sample Exam Chapter 3 - Graphing Lines 3.1 Rectangular Coordinate System 3.2 Graph by Plotting Points 3.3 Graph using Intercepts 3.4 Graph using the y-intercept and Slope 3.5 Finding Linear Equations 3.6 Parallel and Perpendicular Lines 3.7 Introduction to Functions 3.8 Linear Inequalities (Two Variables) 3.9 Review Exercises and Sample Exam Chapter 4 - Solving Linear Systems 4.1 Solving Linear Systems by Graphing 4.2 Solving Linear Systems by Substitution 4.3 Solving Linear Systems by Elimination 4.4 Applications of Linear Systems 4.5 Solving Systems of Linear Inequalities (Two Variables) 4.6 Review Exercises and Sample Exam [ Elementary Algebra Exam #1 Solutions ] [ Elementary Algebra Exam #2 Solutions ] [ Elementary Algebra Exam #3 Solutions ] [ Elementary Algebra Exam #4 Solutions ] --- EA Sample Final Exam Ch. 1-7 --- Chapter 5 - Polynomials and Their Operations 5.1 Rules of Exponents (Integer Exponents) 5.2 Introduction to Polynomials and Evaluating 5.3 Adding and Subtracting Polynomials 5.4 Multiplying Polynomials and Special Products 5.5 Dividing Polynomials (Synthetic Division) 5.6 Negative Exponents and Scientific Notation 5.7 Review Exercises and Sample Exam Chapter 6 - Factoring and Solving by Factoring 6.1 Introduction to Factoring 6.2 Factoring Trinomials 6.3 Factoring Trinomials ax^2 + bx + c 6.4 Factoring Binomials 6.5 General Guidelines for Factoring Polynomials 6.6 Solving Equations by Factoring 6.7 Applications involving Quadratic Equations 6.8 Review Exercises and Sample Exam Chapter 7 - Rational Expressions and Equations 7.1 Simplifying Rational Expressions 7.2 Multiplying and Dividing Rational Expressions 7.3 Adding and Subtracting Rational Expressions 7.4 Complex Fractions 7.5 Solving Rational Equations 7.6 Applications of Rational Equations 7.7 Variation 7.8 Review Exercises and Sample Exam Do you find this site helpful? If so please share the link and click the +1 button. Also, feel free to copy-and-paste anything you find useful here. All we ask is that you link back to this site: OpenAlgebra.com
Intermediate Algebra (IA) (Algebra 2) topics below.
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http://math.oregonstate.edu/home/programs/undergrad/CalculusQuestStudyGuides/SandS/SeriesTests/geometric.html
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|
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crawl-data/CC-MAIN-2018-17/segments/1524125945448.38/warc/CC-MAIN-20180421203546-20180421223546-00220.warc.gz
| 202,277,480
| 1,594
|
# Geometric Series
A series is called geometric if each term in the series is obtained from the preceding one by multiplying it by a common ratio. For example, the series
is geometric, since each term is obtained by multiplying the preceding term by 1/2. In general, a geometric series is of the form
.
Geometric series are useful because of the following result:
The geometric series
is convergent if |r| < 1, and its sum is
Otherwise, the geometric series is divergent.
So, for our example above, a=1, and r=1/2, and the sum of the series is
.
| 135
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https://www.hackmath.net/en/example/5547
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crawl-data/CC-MAIN-2018-26/segments/1529267866984.71/warc/CC-MAIN-20180624160817-20180624180817-00606.warc.gz
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| 7,069
|
Calculate the surface area and volume of a regular quadrangular pyramid:
sides of bases (bottom, top): a1 = 18 cm, a2 = 6cm
angle α = 60 ° (Angle α is the angle between the side wall and the plane of the base.)
S =? , V =?
Result
h = 10.392 cm
S = 936 cm2
V = 1621.2 cm3
#### Solution:
Leave us a comment of example and its solution (i.e. if it is still somewhat unclear...):
Be the first to comment!
#### To solve this example are needed these knowledge from mathematics:
See also our right triangle calculator. Tip: Our volume units converter will help you with converion of volume units. Do you want to convert area units?
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# How to Read a Ruler
Need to measure something but getting hung up on all those lines on a ruler? You’re in the right place. We’re here to explain what the ruler markings mean so taking measurements will be a breeze. Whether you need to know how to read an inch ruler or how to read a metric ruler (cm ruler), this easy guide to ruler measurements has got you covered.
Method 1
1. 1Get an inch ruler. You’ll know it’s an inch ruler because it will have 12 lines that denote inches on the ruler. 12 inches equals 1 foot (0.305 m). Each foot is broken down into inches. Each inch is broken down into 15 smaller marks, equaling 16 marks in total for each inch on the ruler.[1]
• The longer the line on the surface of the ruler, the bigger the measurement is. Ranging from 1 inch to 1/16 of an inch, the lines decrease in size as the unit of measurement does.
• Make sure you read the ruler from left to right. If you are measuring something, align it with the left side of the zero mark on the ruler. The left side of the line where the object ends will be its measurement in inches.
1. 2Learn the inch marks. A ruler is made up of 12 inch marks. These are typically the numbered marks on the ruler and are denoted by the longest lines on the ruler. For example, if you need to measure a nail, place one end directly on the left side of the ruler. If it ends directly above the long line next to the large number 5, then the nail is 5 inches long.
• Some rulers will also denote 1/2 inches with numbers, so make sure you are using the largest numbers with the longest lines as your inch markers.
1. 3Learn the 1/2 inch marks. The 1/2 inch marks will be the second longest lines on the ruler, half as long as the inch marks. Each 1/2 inch mark will come midway between each inch number because it is half of an inch. This means that marks directly between the 0 and 1 inch, 1 and 2 inches, 2 and 3 inches, and so on across the ruler, are the 1/2 inch marks. In total, there are 24 of these marks on a 12 inch ruler.[2]
• For example, place the ruler against a pencil with the eraser at the far left of the ruler. Mark where the tip of the pencil lead ends on the ruler. If the pencil point ends at the shorter line halfway between the 4 and 5 inches marks, then your pencil is 4 and 1/2 inches long.
1. 4Learn the 1/4 of an inch marks. Halfway in between each 1/2 inch line, there will be a smaller line that denotes a 1/4 of an inch. In the first inch, these marks will mark 1/4, 1/2, 3/4, and 1 inch. Although the 1/2 inch and 1 inch marks have their own lines, they are still part of the 1/4 of an inch measurements because 2/4 of an inch equals half an inch and 4/4 of an inch equals 1 inch. There are a total of 48 of these marks on a 12 inch ruler.[3]
• For example, if you measure a carrot and the tip falls on the line halfway between the 6 1/2 and 7 inch lines, the carrot is 6 and 3/4 inches long.
1. 5Learn the 1/8 of an inch marks. The 1/8 of an inch marks are the smaller marks found directly in between the 1/4 of an inch marks on the ruler. Between 0 and 1 inch, there are marks that denote 1/8, 1/4 (or 2/8), 3/8, 1/2 (or 4/8), 5/8, 6/8 (or 3/4), 7/8, and 1 (or 8/8) of an inch. In total, there are 96 of these marks on a 12 inch ruler.[4]
• For example, you measure a piece of fabric and the edge falls on the 6th line after the 4 inch mark, which is directly in between the 1/4 of an inch mark and the 1/2 inch mark. This means that your fabric is 4 and 3/8 inches long.
1. 6Learn the 1/16 of an inch marks. The small lines halfway between each 1/8 of an inch denote 1/16 of an inch. These are also the smallest lines on the ruler. The very first line on the left hand side of the ruler is the 1/16 of an inch mark. Between 0 and 1 inch, there are marks that denote 1/16, 2/16 (or 1/8), 3/16, 4/16 (or 1/4), 5/16, 6/16 (or 3/8), 7/16, 8/16 (or 1/2), 9/16, 10/16 (or 5/8), 11/16, 12/16 (3/4), 13/16, 14/16 (or 7/8), 15/16, 16/16 (or 1) of an inch. There are a total of 192 of these lines on the ruler.[5]
• For example, you measure a flower stem and the end of the stem falls on the 11th line after the 5 inch mark. The flower stem is 5 and 11/16 inches long.
• Not every ruler will have the 1/16 inch mark. If you plan on measuring things that are small or you need to be extremely accurate, make sure the ruler you use has these marks.
Method 2
1. 1Get a metric ruler. A metric ruler is based on the International System of Units (SI), sometimes called the metric system, and is divided into either millimeters or centimeters instead of inches. Rulers are often 30 centimeters long, which are designated by large numbers on the ruler. Between each centimeter (cm) mark, there should be 10 smaller marks called millimeters (mm).
• Make sure you read the ruler from left to right. If you are measuring an object, align it with the left side of the zero mark on the ruler. The left side of the line where the object ends will be its measurement in centimeters. This way the line thickness will not affect the measurement.
• Unlike with the English ruler, the measurements for the metric ruler are written in decimals instead of fractions. For example, 1/2 a centimeter is written as 0.5 cm. [6]
1. 2Learn the centimeter marks. The large numbers next to the longest lines on the ruler denote the centimeter marks. A metric ruler has 30 of these marks. For example, place the bottom of a crayon on the far left side of the ruler to measure it. Note where the tip falls. If the crayon ends directly on the long line next to the large number 14, your crayon is exactly 14 cm long.[7]
1. 3Learn the 1/2 of a centimeter marks. Halfway between each centimeter, there is a slightly shorter line that denotes 1/2 of a centimeter, or 0.5 cm. There are a total of 60 of these marks on a 30 cm ruler.[8]
• For example, you measure a button and the edge ends on the fifth line right between the 1 and 2 centimeter marks. Your button is 1.5 cm long.
• For example, to measure 0.6 cm, count one thick line (5 mm) and one thin line (1 mm).
1. 4Learn the millimeter marks. Between each 0.5 cm line, there are four additional lines that denote the millimeter marks. There are a total of 10 lines per centimeter, with the 0.5 cm line acting as the 5 millimeter mark, making each centimeter 10 mm long. There are 300 millimeter marks on a 30 cm ruler.[9]
• For example, if you measure a piece of paper and it ends on the 7th mark between the 24 and 25 centimeter mark, it means your object is 247 mm, or 24.7 cm long.
## Community Q&A
Question: What is 55.5? Is that larger than 55 1/4?
Answer: The 55.5 is larger than 55 1/4. the .5 on the 55.5 would equal 1/2. Therefore, 55.5 is equal to 55 1/2 which is 1/4″ larger than 55 1/4.
Question: Can I learn to read a ruler in one day?
Answer: Yes, but it really depends on what type of ruler you want to learn as well as how fast you pick up new material
Question: What does it mean when mm is shown just beside the 0 in a ruler?
Answer: Each small line represents 1mm. Therefore, the first line past the big number (for instance 25) will represent 25.1cm or 251mm.
Question: Where can I find the centimeter markers on a ruler?
Answer: The centimeters side is usually the part of the ruler where the markers are shorter and closer together. It reads cm, and has more numbers.
Question: What does 0.75 cm look like?
Answer: It’s in the middle between the 7th and 8th millimeter lines in a centimeter. In other words, it ends in the middle of the second half of a centimeter.
Question: Is 7/8 larger than 1 inch?
Answer: 7/8 is smaller than 1 inch. 1 inch represents a whole, while 7/8 represents 7 parts of a whole (8 parts).
Question: Is 12 inches longer than a foot? I am feeling stumped by this.
Answer: They’re the same 12in = 1ft.
Question: Why there is a space at the beginning of a ruler?
Answer: Some lower quality rulers have spaces at the beginning to make the rulers easier to use. Higher quality rulers are often made of non-elastic materials like steel or aluminum, and their markings start without any space.
Question: Is 5.5 mm closer to a half inch or a quarter inch?
Answer: A quarter inch. 6 mm is almost a quarter inch, whereas half an inch looks closer to 12-13 mm, so 5.5 would be close to a quarter of an inch.
Question: Why are there five holes in my 12″ ruler?
Answer: So you can put the ruler in a 3- or 5-ring binder to use in school or in an office environment.
Question: How do I read centimeters on a ruler?
Answer: 1 centimeter is equivalent to .39 inches. Many rulers come with centimeters marked on them, but a little math is required if that is not the case.
Question: What does 4.5mm look like on a ruler?
Answer: This measurement would be the same as 4 1/2 mm. In other words, this measurement would be halfway in between 4 and 5 mm on a ruler.
Question: Which value is larger, .4 or .5?
Answer: .5 is larger because .4 is equal to 4/10, which reduces down to 2/5 and .5 is equal to 5/10. This reduces to 1/2, and 1/2 is larger than 2/5.
Question: Regarding the centimeters, I’m confused about the 10 lines. I see 11 lines, counting the 0.5 mm. Shouldn’t the 0.5 mm count as well?
Answer: You aren’t really counting the lines — you are counting the spaces. The “first” line and the “second” line are the start and end of the first millimeter. The “second” line and the “third” line are the start and end of the second millimeter. If you insist on “counting lines,” then start with 0 for the first line. Every line after that will give you the total number of “spaces” or millimeters that you are looking for.
Question: What is 1/8 of a ruler?
Answer: The 1/8 marks are the smaller marks found directly between the 1/4 marks. There are 96 1/8 marks on a 1-inch ruler.
Question: Is there a 30 and a half inch mark on the 30 inch ruler?
Answer: The measurement marks on a ruler go up to 30. The ruler itself (including the plastic) is 31 inches long.
Question: Where is the 16th on a ruler?
Answer: The 16th is the second small mark on a ruler, because it takes 2 mm to make a 16th.
Question: How do I convert 9.906 millimeters to inches?
Answer: Since one inch is defined as 25.4 mm, you would need to divide 9.906 by 25.4, which would give you 0.39 inches.
Question: How do I find inches on a ruler?
Answer: You should see the word “inch” printed near a particular type of mark, which would tell you which lines indicate inches. Normally, the inch lines will be the longest lines on the ruler if the measurements are on one side.
Question: How many inches are in a foot?
Answer: Twelve inches are in a foot.
Question: Where is 2/3″ on a ruler?
Answer: It would be between 1/2″ and 3/4″ (between-inch subdivisions on a ruler are in halves, quarters, eighths, etc., not thirds).
Question: How do I find quarter inches on a ruler?
Answer: It is half of the half inch. For example, if you were looking for the 4 1/4, then you would look at the mark between the 4 in and the 4 1/2 (4.5) and that would be a quarter inch.
Question: Is 1/4″ the same as 10 mm on a ruler?
Answer: No, 1/4″ is only 6.35 mm.
Question: How many inches are in 1.3 cm?
Answer: 1/2″. Mark 1.3 cm on paper then place ruler inch side up and you will see it is 1/2″. On metric ruler, (mm or cm) the smaller lines are mm and the larger lines are cm. A foot is 30.48 cm. There are 10mm in each cm.
Question: Why do you have to make even number fractions on the ruler odd?
Answer: We can only count the odd numbers since even numbers have fractions with a small denominator.
Question: How do I know if my ruler can fit in a binder?
Answer: Check the height of the binder; if it is smaller than 12 inches (30cm), then the rule it won’t fit, since all rulers are 12 inches (30cm) long.
Question: I’m pleating a face mask. Instructions say each pleat should be 2cm. How do I measure that in inches?
Answer: Half an inch is a about a centimeter, so measure an inch and you should be fine (it is a fraction more but an inch will be fine).
Question: Who created the ruler?
Answer: The first documented (1851) inventor of the folding ruler is Anton Ullrich. A flexible ruler was later invented by Frank Hunt (1902). However, people would likely have made measuring rules, sticks, etc. from the dawn of civilization.
Question: Where is .6 on a ruler? Is it a notch past the half inch mark?
Answer: It is the 3/8 notch. The .6 notch is two notches before the half inch mark on the ruler.
## Tips
• Make sure you always use the correct side of the ruler for the task at hand. You don’t want to get the centimeters and the inches mixed up or your measurements won’t be correct. Remember that there are 12 large numbers on an English ruler and 30 numbers on the metric ruler.
• Learning to read a ruler takes practice, especially converting the numbers in the measurements. Just remember to practice using your ruler and you’ll get better at it.
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763,
279,
1176,
17560,
11,
1521,
15785,
690,
1906,
220,
16,
14,
19,
11,
220,
16,
14,
17,
11,
220,
18,
14,
19,
11,
323,
220,
16,
17560,
13,
10541,
279,
220,
16,
14,
17,
17560,
323,
220,
16,
17560,
15785,
617,
872,
1866,
5238,
11,
814,
527,
2103,
961,
315,
279,
220,
16,
14,
19,
315,
459,
17560,
22323,
1606,
220,
17,
14,
19,
315,
459,
17560,
17239,
4376,
459,
17560,
323,
220,
19,
14,
19,
315,
459,
17560,
17239,
220,
16,
17560,
13,
2684,
527,
264,
2860,
315,
220,
2166,
315,
1521,
15785,
389,
264,
220,
717,
17560,
49080,
8032,
18,
933,
6806,
1789,
3187,
11,
422,
499,
6767,
264,
75294,
323,
279,
11813,
17503,
389,
279,
1584,
42436,
1990,
279,
220,
21,
220,
16,
14,
17,
323,
220,
22,
17560,
5238,
11,
279,
75294,
374,
220,
21,
323,
220,
18,
14,
19,
15271,
1317,
627,
16,
13,
220,
20,
24762,
279,
220,
16,
14,
23,
315,
459,
17560,
15785,
13,
4194,
791,
220,
16,
14,
23,
315,
459,
17560,
15785,
527,
279,
9333,
15785,
1766,
6089,
304,
1990,
279,
220,
16,
14,
19,
315,
459,
17560,
15785,
389,
279,
49080,
13,
28232,
220,
15,
323,
220,
16,
17560,
11,
1070,
527,
15785,
430,
79164,
220,
16,
14,
23,
11,
220,
16,
14,
19,
320,
269,
220,
17,
14,
23,
705,
220,
18,
14,
23,
11,
220,
16,
14,
17,
320,
269,
220,
19,
14,
23,
705,
220,
20,
14,
23,
11,
220,
21,
14,
23,
320,
269,
220,
18,
14,
19,
705,
220,
22,
14,
23,
11,
323,
220,
16,
320,
269,
220,
23,
14,
23,
8,
315,
459,
17560,
13,
763,
2860,
11,
1070,
527,
220,
4161,
315,
1521,
15785,
389,
264,
220,
717,
17560,
49080,
8032,
19,
933,
6806,
1789,
3187,
11,
499,
6767,
264,
6710,
315,
13354,
323,
279,
6964,
17503,
389,
279,
220,
21,
339,
1584,
1306,
279,
220,
19,
17560,
1906,
11,
902,
374,
6089,
304,
1990,
279,
220,
16,
14,
19,
315,
459,
17560,
1906,
323,
279,
220,
16,
14,
17,
17560,
1906,
13,
1115,
3445,
430,
701,
13354,
374,
220,
19,
323,
220,
18,
14,
23,
15271,
1317,
627,
16,
13,
220,
21,
24762,
279,
220,
16,
14,
845,
315,
459,
17560,
15785,
13,
4194,
791,
2678,
5238,
42436,
1990,
1855,
220,
16,
14,
23,
315,
459,
17560,
79164,
220,
16,
14,
845,
315,
459,
17560,
13,
4314,
527,
1101,
279,
25655,
5238,
389,
279,
49080,
13,
578,
1633,
1176,
1584,
389,
279,
2163,
1450,
3185,
315,
279,
49080,
374,
279,
220,
16,
14,
845,
315,
459,
17560,
1906,
13,
28232,
220,
15,
323,
220,
16,
17560,
11,
1070,
527,
15785,
430,
79164,
220,
16,
14,
845,
11,
220,
17,
14,
845,
320,
269,
220,
16,
14,
23,
705,
220,
18,
14,
845,
11,
220,
19,
14,
845,
320,
269,
220,
16,
14,
19,
705,
220,
20,
14,
845,
11,
220,
21,
14,
845,
320,
269,
220,
18,
14,
23,
705,
220,
22,
14,
845,
11,
220,
23,
14,
845,
320,
269,
220,
16,
14,
17,
705,
220,
24,
14,
845,
11,
220,
605,
14,
845,
320,
269,
220,
20,
14,
23,
705,
220,
806,
14,
845,
11,
220,
717,
14,
845,
320,
18,
14,
19,
705,
220,
1032,
14,
845,
11,
220,
975,
14,
845,
320,
269,
220,
22,
14,
23,
705,
220,
868,
14,
845,
11,
220,
845,
14,
845,
320,
269,
220,
16,
8,
315,
459,
17560,
13,
2684,
527,
264,
2860,
315,
220,
5926,
315,
1521,
5238,
389,
279,
49080,
8032,
20,
933,
6806,
1789,
3187,
11,
499,
6767,
264,
23153,
19646,
323,
279,
842,
315,
279,
19646,
17503,
389,
279,
220,
806,
339,
1584,
1306,
279,
220,
20,
17560,
1906,
13,
578,
23153,
19646,
374,
220,
20,
323,
220,
806,
14,
845,
15271,
1317,
627,
6806,
2876,
1475,
49080,
690,
617,
279,
220,
16,
14,
845,
17560,
1906,
13,
1442,
499,
3197,
389,
30090,
2574,
430,
527,
2678,
477,
499,
1205,
311,
387,
9193,
13687,
11,
1304,
2771,
279,
4194,
81,
8646,
499,
1005,
4194,
4752,
1521,
15785,
382,
3607,
220,
17,
271,
16,
13,
220,
16,
1991,
264,
18767,
49080,
13,
118586,
18767,
49080,
374,
3196,
389,
279,
7327,
744,
315,
36281,
320,
14137,
705,
7170,
2663,
279,
4194,
16282,
1887,
11,
323,
374,
18255,
1139,
3060,
2606,
55336,
477,
4194,
1189,
55336,
4619,
315,
15271,
13,
432,
27898,
527,
3629,
220,
966,
2960,
55336,
1317,
11,
902,
527,
24073,
555,
3544,
5219,
389,
279,
49080,
13,
28232,
1855,
2960,
26402,
320,
6358,
8,
1906,
11,
1070,
1288,
387,
220,
605,
9333,
15785,
2663,
2606,
55336,
320,
3906,
4390,
6806,
7557,
2771,
499,
1373,
279,
49080,
505,
2163,
311,
1314,
13,
1442,
499,
527,
30090,
459,
1665,
11,
5398,
433,
449,
279,
2163,
3185,
315,
279,
7315,
1906,
389,
279,
49080,
13,
578,
2163,
3185,
315,
279,
1584,
1405,
279,
1665,
10548,
690,
387,
1202,
19179,
304,
2960,
55336,
13,
1115,
1648,
279,
1584,
26839,
690,
539,
7958,
279,
19179,
627,
6806,
27140,
449,
279,
6498,
49080,
11,
279,
22323,
369,
279,
18767,
49080,
527,
5439,
304,
4194,
8332,
25137,
4619,
315,
65995,
13,
1789,
3187,
11,
220,
16,
14,
17,
264,
2960,
26402,
374,
5439,
439,
220,
15,
13,
20,
10166,
13,
4194,
58,
21,
933,
16,
13,
220,
17,
24762,
279,
2960,
26402,
15785,
13,
4194,
791,
3544,
5219,
1828,
311,
279,
22807,
5238,
389,
279,
49080,
79164,
279,
2960,
26402,
15785,
13,
362,
18767,
49080,
706,
220,
966,
315,
1521,
15785,
13,
1789,
3187,
11,
2035,
279,
5740,
315,
264,
90016,
263,
389,
279,
3117,
2163,
3185,
315,
279,
49080,
311,
6767,
433,
13,
7181,
1405,
279,
11813,
17503,
13,
1442,
279,
90016,
263,
10548,
6089,
389,
279,
1317,
1584,
1828,
311,
279,
3544,
1396,
220,
975,
11,
701,
90016,
263,
374,
7041,
220,
975,
10166,
1317,
8032,
22,
933,
16,
13,
220,
18,
24762,
279,
220,
16,
14,
17,
315,
264,
2960,
26402,
15785,
13,
4194,
43727,
3195,
1990,
1855,
2960,
26402,
11,
1070,
374,
264,
10284,
24210,
1584,
430,
72214,
220,
16,
14,
17,
315,
264,
2960,
26402,
11,
477,
220,
15,
13,
20,
10166,
13,
2684,
527,
264,
2860,
315,
220,
1399,
315,
1521,
15785,
389,
264,
220,
966,
10166,
49080,
8032,
23,
933,
6806,
1789,
3187,
11,
499,
6767,
264,
3215,
323,
279,
6964,
10548,
389,
279,
18172,
1584,
1314,
1990,
279,
220,
16,
323,
220,
17,
2960,
26402,
15785,
13,
4718,
3215,
374,
220,
16,
13,
20,
10166,
1317,
627,
6806,
1789,
3187,
11,
311,
6767,
220,
15,
13,
21,
10166,
11,
1797,
832,
12314,
1584,
320,
20,
9653,
8,
323,
832,
15792,
1584,
320,
16,
9653,
4390,
16,
13,
220,
19,
24762,
279,
2606,
26402,
15785,
13,
4194,
26556,
1855,
220,
15,
13,
20,
10166,
1584,
11,
1070,
527,
3116,
5217,
5238,
430,
79164,
279,
2606,
26402,
15785,
13,
2684,
527,
264,
2860,
315,
220,
605,
5238,
824,
2960,
26402,
11,
449,
279,
220,
15,
13,
20,
10166,
1584,
15718,
439,
279,
220,
20,
2606,
26402,
1906,
11,
3339,
1855,
2960,
26402,
220,
605,
9653,
1317,
13,
2684,
527,
220,
3101,
2606,
26402,
15785,
389,
264,
220,
966,
10166,
49080,
8032,
24,
933,
6806,
1789,
3187,
11,
422,
499,
6767,
264,
6710,
315,
5684,
323,
433,
10548,
389,
279,
220,
22,
339,
1906,
1990,
279,
220,
1187,
323,
220,
914,
2960,
26402,
1906,
11,
433,
3445,
701,
1665,
374,
220,
14125,
9653,
11,
477,
220,
1187,
13,
22,
10166,
1317,
382,
567,
12332,
1229,
36121,
271,
14924,
25,
3639,
374,
220,
2131,
13,
20,
30,
2209,
430,
8294,
1109,
220,
2131,
220,
16,
14,
19,
1980,
16533,
25,
578,
220,
2131,
13,
20,
374,
8294,
1109,
220,
2131,
220,
16,
14,
19,
13,
279,
662,
20,
389,
279,
220,
2131,
13,
20,
1053,
6273,
220,
16,
14,
17,
13,
15636,
11,
220,
2131,
13,
20,
374,
6273,
311,
220,
2131,
220,
16,
14,
17,
902,
374,
220,
16,
14,
19,
22308,
8294,
1109,
220,
2131,
220,
16,
14,
19,
382,
14924,
25,
3053,
358,
4048,
311,
1373,
264,
49080,
304,
832,
1938,
1980,
16533,
25,
7566,
11,
719,
433,
2216,
14117,
389,
1148,
955,
315,
49080,
499,
1390,
311,
4048,
439,
1664,
439,
1268,
5043,
499,
3820,
709,
502,
3769,
271,
14924,
25,
3639,
1587,
433,
3152,
994,
9653,
374,
6982,
1120,
30488,
279,
220,
15,
304,
264,
49080,
1980,
16533,
25,
9062,
2678,
1584,
11105,
220,
16,
3906,
13,
15636,
11,
279,
1176,
1584,
3347,
279,
2466,
1396,
320,
2000,
2937,
220,
914,
8,
690,
4097,
220,
914,
13,
16,
6358,
477,
220,
13860,
3906,
382,
14924,
25,
11208,
649,
358,
1505,
279,
2960,
26402,
24915,
389,
264,
49080,
1980,
16533,
25,
578,
2960,
55336,
3185,
374,
6118,
279,
961,
315,
279,
49080,
1405,
279,
24915,
527,
24210,
323,
12401,
3871,
13,
1102,
16181,
10166,
11,
323,
706,
810,
5219,
382,
14924,
25,
3639,
1587,
220,
15,
13,
2075,
10166,
1427,
1093,
1980,
16533,
25,
1102,
753,
304,
279,
6278,
1990,
279,
220,
22,
339,
323,
220,
23,
339,
2606,
26402,
5238,
304,
264,
2960,
26402,
13,
763,
1023,
4339,
11,
433,
10548,
304,
279,
6278,
315,
279,
2132,
4376,
315,
264,
2960,
26402,
382,
14924,
25,
2209,
220,
22,
14,
23,
8294,
1109,
220,
16,
17560,
1980,
16533,
25,
220,
22,
14,
23,
374,
9333,
1109,
220,
16,
17560,
13,
220,
16,
17560,
11105,
264,
4459,
11,
1418,
220,
22,
14,
23,
11105,
220,
22,
5596,
315,
264,
4459,
320,
23,
5596,
3677,
14924,
25,
2209,
220,
717,
15271,
5129,
1109,
264,
4579,
30,
358,
1097,
8430,
357,
61292,
555,
420,
382,
16533,
25,
2435,
3207,
279,
1890,
220,
717,
258,
284,
220,
16,
728,
382,
14924,
25,
8595,
1070,
374,
264,
3634,
520,
279,
7314,
315,
264,
49080,
1980,
16533,
25,
4427,
4827,
4367,
60996,
617,
12908,
520,
279,
7314,
311,
1304,
279,
60996,
8831,
311,
1005,
13,
35321,
4367,
60996,
527,
3629,
1903,
315,
2536,
12,
63064,
7384,
1093,
9699,
477,
25674,
11,
323,
872,
65172,
1212,
2085,
904,
3634,
382,
14924,
25,
2209,
220,
20,
13,
20,
9653,
12401,
311,
264,
4376,
17560,
477,
264,
8502,
17560,
1980,
16533,
25,
362,
8502,
17560,
13,
220,
21,
9653,
374,
4661,
264,
8502,
17560,
11,
20444,
4376,
459,
17560,
5992,
12401,
311,
220,
717,
12,
1032,
9653,
11,
779,
220,
20,
13,
20,
1053,
387,
3345,
311,
264,
8502,
315,
459,
17560,
382,
14924,
25,
8595,
527,
1070,
4330,
20349,
304,
856,
220,
717,
22308,
49080,
1980,
16533,
25,
2100,
499,
649,
2231,
279,
49080,
304,
264,
220,
18,
12,
477,
220,
20,
77029,
55114,
311,
1005,
304,
2978,
477,
304,
459,
5274,
4676,
382,
14924,
25,
2650,
656,
358,
1373,
2960,
55336,
389,
264,
49080,
1980,
16533,
25,
220,
16,
2960,
26402,
374,
13890,
311,
662,
2137,
15271,
13,
9176,
60996,
2586,
449,
2960,
55336,
13160,
389,
1124,
11,
719,
264,
2697,
7033,
374,
2631,
422,
430,
374,
539,
279,
1162,
382,
14924,
25,
3639,
1587,
220,
19,
13,
20,
3906,
1427,
1093,
389,
264,
49080,
1980,
16533,
25,
1115,
19179,
1053,
387,
279,
1890,
439,
220,
19,
220,
16,
14,
17,
9653,
13,
763,
1023,
4339,
11,
420,
19179,
1053,
387,
42436,
304,
1990,
220,
19,
323,
220,
20,
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https://www.tutorialspoint.com/program-to-find-sum-of-longest-sum-path-from-root-to-leaf-of-a-binary-tree-in-python
| 1,679,814,756,000,000,000
|
text/html
|
crawl-data/CC-MAIN-2023-14/segments/1679296945433.92/warc/CC-MAIN-20230326044821-20230326074821-00504.warc.gz
| 1,163,841,358
| 9,819
|
# Program to find sum of longest sum path from root to leaf of a binary tree in Python
Suppose we have a binary tree, we have to find the sum of the longest path from the root to a leaf node. If there are two same long paths, return the path with larger sum.
So, if the input is like
then the output will be 20.
To solve this, we will follow these steps −
• Define a function rec() . This will take curr
• if curr is null, then
• return(0, 0)
• bigger := maximum of rec(left of curr) , rec(right of curr)
• return a pair (bigger[0] + 1, bigger[1] + value of curr)
• From the main method do the following −
• ret := rec(root)
• return the 1th index of ret
Let us see the following implementation to get better understanding −
## Example
Live Demo
class TreeNode:
def __init__(self, val, left=None, right=None):
self.val = val
self.left = left
self.right = right
class Solution:
def solve(self, root):
def rec(curr):
if not curr:
return (0, 0)
bigger = max(rec(curr.left), rec(curr.right))
return (bigger[0] + 1, bigger[1] + curr.val)
return rec(root)[1]
ob = Solution()
root = TreeNode(2)
root.left = TreeNode(10)
root.right = TreeNode(4)
root.right.left = TreeNode(8)
root.right.right = TreeNode(2)
root.right.left.left = TreeNode(6)
print(ob.solve(root))
## Input
root = TreeNode(2)
root.left = TreeNode(10)
root.right = TreeNode(4)
root.right.left = TreeNode(8)
root.right.right = TreeNode(2)
root.right.left.left = TreeNode(6)
## Output
20
| 399
| 1,463
|
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| 3.640625
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CC-MAIN-2023-14
|
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|
en
| 0.642449
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https://thefloatingschoolid.org/and-pdf/3254-square-and-cube-of-1-to-30-pdf-931-686.php
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# Square And Cube Of 1 To 30 Pdf
File Name: square and cube of 1 to 30 .zip
Size: 22653Kb
Published: 13.05.2021
In arithmetic and algebra , the cube of a number n is its third power , that is, the result of multiplying three instances of n together. The cube is also the number multiplied by its square :. It is an odd function , as. The volume of a geometric cube is the cube of its side length, giving rise to the name.
## Square Root Table 1 1000 Pdf 21
All questions and answers from the Mathematics Solutions Book of Class 8 Math Chapter 2 are provided here for you for free. Hence, the perfect squares between 1 and are 1, 4, 9, 16, 25, 36, 49, 64, 81, , , , , , , , , , , , and Write 3-digit numbers ending with 0, 1, 4, 5, 6, 9, one for each digit, but none of them is a perfect square. Find the th and th triangular numbers, and find their sum. Verify the Statement 8 for this sum. Therefore, when a perfect number is divided by 5, then the remainder will be 0, 1, 4, 0, 1 or 4.
## Cubes and Cube Roots
To cube a number, just use it in a multiplication 3 times Note: we write "3 Cubed" as 3 3 the little 3 means the number appears three times in multiplying. The cube root of a number is The cube root of 27 is This is the special symbol that means "cube root", it is the "radical" symbol used for square roots with a little three to mean cube root. You can use it like this: we say "the cube root of 27 equals 3".
A square root 74 of a number is a number that when multiplied by itself yields the original number. Every positive real number has two square roots, one positive and one negative. If the radicand 77 , the number inside the radical sign, is nonzero and can be factored as the square of another nonzero number, then the square root of the number is apparent. In this case, we have the following property:. The radicand may not always be a perfect square. If a positive integer is not a perfect square, then its square root will be irrational.
Add standard and customized parametric components - like flange beams, lumbers, piping, stairs and more - to your Sketchup model with the Engineering ToolBox - SketchUp Extension - enabled for use with the amazing, fun and free SketchUp Make and SketchUp Pro. We don't collect information from our users. Only emails and answers are saved in our archive. Cookies are only used in the browser to improve user experience. Some of our calculators and applications let you save application data to your local computer.
Squares and cubes form an important part of Quantitative Aptitude section. Earlier we discussed an article on how to increase your calculation speed during this lockdown. Such questions involving squares and cubes can always be asked in PO, Clerk and also in SSC exams so in this article we are explaining some quick tips and tricks to use squares and cubes along with letting the aspirants know the squares and cubes of numbers from 1 to Knowing squares and cubes is an important mathematical operation by which aspirants can actually make their calculations fast and moreover, there will be less probability of error. It is essential for the candidates to maintain speed and accuracy in the bank exams or for that matter in any such competitive exam.
Use this table to find the squares and square roots of numbers from 1 to You can also use this table to estimate the square roots of larger numbers. For instance, if you want to find the square root of , look in the middle column until you find the number that is closest to You don't need to use prime factorization or other methods every time you find the square roots.
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Сьюзан спустилась по лестнице на несколько ступенек. Горячий воздух снизу задувал под юбку. Ступеньки оказались очень скользкими, влажными из-за конденсации пара. Она присела на решетчатой площадке. - Коммандер. Стратмор даже не повернулся. Он по-прежнему смотрел вниз, словно впав в транс и не отдавая себе отчета в происходящем.
Я не могу, - повторила. - Я не могу выйти за тебя замуж. - Она отвернулась.
Дэвид Беккер умрет. Халохот поднимался вверх с пистолетом в руке, прижимаясь вплотную к стене на тот случай, если Беккер попытается напасть на него сверху. Железные подсвечники, установленные на каждой площадке, стали бы хорошим оружием, если бы Беккер решил ими воспользоваться. Но если держать дистанцию, можно заметить его вовремя. У пистолета куда большая дальность действия, чем у полутораметрового подсвечника.
number system notes for ssc cgl pdf, number system questions pdf, number system tricks for ssc, number system in 1 Squares of numbers from 51 to Square values up to number 30 and cube values up to number 16 are given below.
#### Practice Mock Tests
- Откроем пачку тофу. - Нет, спасибо. - Сьюзан шумно выдохнула и повернулась к. - Я думаю, - начала она, -что я только… -но слова застряли у нее в горле. Она побледнела. - Что с тобой? - удивленно спросил Хейл. Сьюзан встретилась с ним взглядом и прикусила губу.
Между шифровалкой и стоянкой для машин не менее дюжины вооруженных охранников. - Я не такой дурак, как вы думаете, - бросил Хейл. - Я воспользуюсь вашим лифтом. Сьюзан пойдет со. А вы останетесь. - Мне неприятно тебе это говорить, - сказал Стратмор, - но лифт без электричества - это не лифт.
По сути, это был самый настоящий шантаж. Он предоставил АНБ выбор: либо рассказать миру о ТРАНСТЕКСТЕ, либо лишиться главного банка данных. Сьюзан в ужасе смотрела на экран. Внизу угрожающе мигала команда: ВВЕДИТЕ КЛЮЧ Вглядываясь в пульсирующую надпись, она поняла. Вирус, ключ, кольцо Танкадо, изощренный шантаж… Этот ключ не имеет к алгоритму никакого отношения, это противоядие. Ключ блокирует вирус.
Вижу, - сказал Бринкерхофф, стараясь сосредоточиться на документе. - Это данные о сегодняшней производительности. Взгляни на число дешифровок.
Уже на середине комнаты она основательно разогналась. За полтора метра до стеклянной двери Сьюзан отпрянула в сторону и зажмурилась. Раздался страшный треск, и стеклянная панель обдала ее дождем осколков. Звуки шифровалки впервые за всю историю этого здания ворвались в помещение Третьего узла.
Ему было не привыкать работать допоздна даже по уикэндам; именно эти сравнительно спокойные часы в АНБ, как правило, были единственным временем, когда он мог заниматься обслуживанием компьютерной техники. Просунув раскаленный паяльник сквозь проволочный лабиринт у себя над головой, он действовал с величайшей осмотрительностью: опалить защитную оболочку провода значило вывести аппарат из строя. Еще несколько сантиметров, подумал Джабба. Работа заняла намного больше времени, чем он рассчитывал. Когда он поднес раскаленный конец паяльника к последнему контакту, раздался резкий звонок мобильного телефона.
Но глаза… твои глаза, - сказал Беккер, чувствуя себя круглым дураком.
Это очень и очень плохо. - Спокойствие, - потребовал Фонтейн. - На какие же параметры нацелен этот червь. На военную информацию. Тайные операции.
Соши заливалась слезами. - Джабба, - спросил Фонтейн, - много они похитили. - Совсем мало, - сказал Джабба, посмотрев на монитор.
Не важно, сколько посетителей стоят в очереди, - секретарь всегда бросит все дела и поспешит поднять трубку. Беккер отбил шестизначный номер. Еще пара секунд, и его соединили с больничным офисом. Наверняка сегодня к ним поступил только один канадец со сломанным запястьем и сотрясением мозга, и его карточку нетрудно будет найти.
В подавленном настроении Сьюзан приняла ванну.
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1. Nieriapayda1960
Professional guide to diseases pdf computer and information security handbook 3rd edition pdf
2. Geoffrey C.
cube n3 square root number n square n2 cube n3 square root. 1. 1. 1. 41
3. Eudoxia S.
Hello Friends, Today we are sharing Square table from 1 to PDF. This is very helpful for various competitive exams and improve solve.
4. Arridano P.
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https://www.mathscinotes.com/2016/01/worked-calculation-example-of-lithium-battery-capacity-versus-load/
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crawl-data/CC-MAIN-2020-24/segments/1590347407001.36/warc/CC-MAIN-20200530005804-20200530035804-00479.warc.gz
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# Worked Calculation Example of Lithium Battery Capacity Versus Load
Quote of the Day
Successful ... politicians are insecure and intimidated men. They advance politically only as they placate, appease, bribe, seduce, bamboozle or otherwise manage to manipulate the demanding and threatening elements in their constituencies.
— Walter Lippmann
## Introduction
Figure 1: 18650 Lithium -Ion
Cylindrical Battery (right)
versus a standard AA battery
(left). (Source)
I frequently am asked to comment on data that other engineers send me. This morning I received some test data obtained from an engineer measuring the backup time of an Uninterruptible Power Source (UPS) containing multiple lithium-ion (Li-Ion) batteries. The engineer was disappointed with the backup time provided by this UPS and wanted to know if his test results were reasonable considering the battery capacity of the UPS. While there were numerous circuit parameters measured during this testing, the critical information was the battery voltage versus time.
The engineer's routine calculations showed that the UPS should provide 4 hours of backup time, but he measured 2.7 hours. While batteries are listed with a nominal charge capacity, their actual capacity varies strongly with the load presented to the battery. His testing was performed at room temperature and with a load of 0.8 A.
In this post, I will show that his test results are reasonable when when the effects of the load current are taken into account. I will perform the calculations three ways:
• Method 2: Using a graphical curve of capacity versus load.
• Method 3: Using numerical methods for interpolating digitized capacity graphs.
## Background
### Definitions
C-Rate
C-rate is the charge/discharge current normalized to the battery capacity. A charge/discharge rate of one C for one hour draws a charge equal to the battery capacity. For example, the 1C discharge rate for a 2.2 A-hour battery is 2.2 A.
Cutoff Voltage (VCutoff)
The battery voltage at which the UPS terminates the discharge of the battery.
Capacity
The available charge in the battery, which is a function of the load current. A battery's charge capacity is generally specified at some low current draw. For example, the capacity of lead-acid batteries is specified at a 20 hour (C/20 or 0.05C) rate.
### UPS Characteristics
• It contains 4 series-connected, Li-Ion, 18650 cylindrical batteries (Figure 1).
• These batteries have a nominal cell voltage of 3.7 V.
• I assume that each 18650 battery is rated to have a nominal 2800 mA-hour charge capacity. The actual rating is 2700 mA-hour (minimum) and 2900 mA-hour (typical). I will average the two values for this analysis.
• The UPS load is a device that requires an input voltage between 10 V and 16 V.
• The voltage range requirement is met by connecting the batteries in series.
• The engineer modeled the load using 800 mA of constant current draw.
• Most UPS hardware stops discharging the battery at VCutoff, which for Li-Ion batteries is typically near 3.3. V. In this case, the UPS cutoff was ~2.9 V.
## Analysis
### Measured Battery Voltage Versus Time for Constant Load Current
Figure 2 shows the data that was emailed to me. I have literally seen hundreds of these test plots. This UPS I include a calculation on Figure 3 that shows that the charged cell voltage for the batteries in this 4 cell-string pack is 3.92 V. This is below the manufacturers 4.2 V charge voltage in their specification. The cutoff voltage is 2.87 V.
Figure 2: Raw Battery Data That Was Emailed To Me.
This data was measured by instrumenting the battery series inside the UPS. We will need the initial cell voltage and the final cell voltage in order to estimate the charge drawn from the battery.
Method 1 will not use this data, but Method 2 and 3 will.
### Method 1: Simple Calculation Ignoring Load Current Impact on Capacity
Figure 3 shows how to estimate the backup time provided by this UPS assuming nominal battery characteristics.
Figure 3: Nominal Calculation Example.
The key problem with this analysis is that it assumes that the battery capacity does not depend on the load. I model the effect of load current on battery capacity any time the load currents exceed 0.1C.
### Method 2: Graphical Analysis
Figure 4 shows how to compute the expected backup time using the battery's capacity versus load chart. All the calculations are shown on the graph and I obtain a backup time estimate of 2.8 hours.
The calculations shown on the graph can be described as follows:
• Us the initial cell voltage to determine how much charge is lost because the UPS did not fully charge the battery.
• Use the final cell voltage to determine how much charge is available from a fully charged battery.
• Determine the difference between the final and initial charge, which reflects the charge available for backup energy.
• Divide the available charge (in mA-hours) by the load current, which give the backup time.
Figure 4: Computing Run Time Using Capacity Chart.
### Method 3 : Numerical Analysis
#### Manufacturer's Capacity Rating Versus Load
Figure 5 shows the typical discharge specification for Li-Ion battery from Panasonic with a nominal rating of 2900 mA-hour (2700 mA-hour minimum). As you can see in Figure 4, the typical capacity is measured when the battery has a minimal load (0.2C). One unusual aspect of this chart is that you also get full capacity with a high load current (2C). I only rarely see this characteristic on a capacity chart.
Figure 5: Discharge Capacity Versus Load Current.
I digitized this data using Dagra and pasted it into Mathcad. Figure 6 shows the digitized the data and a routine to interpolate the data. I will assume that batteries actual capacity is 2800 mA-hour, the mean of the minimum and maximum.
Figure 6: Digitized and Interpolated Charge Data.
#### Mathcad Model of Battery Capacity Data
Given the discharge data shown in Figure 7, I can determine the effective capacity of the UPS battery at a 0.8 A load. Figure 6 shows the my interpolated results for the manufacturer's data and my interpolation for a 0.8 A load. The graph shows that the effective battery capacity is ~2779 mA-hour with an initial charge voltage of 4.2 V. I also show that 554 mA of charge is missing if the battery is initially charged to only 3.92 V, which is this case. In the next section, I will show how to algebraically obtain these results.
Figure 7: Plot of Battery Capacity at 0.8 A Load (0.29C).
#### Estimated Discharge Time Assuming that Capacity is Load-Dependent
In Figure 8, I use 2-dimensional interpolation to compute the battery capacity assuming VCutoff = 3.3 V and ILoad = 0.8 A.
Figure 8: Calculation of Discharge Time (2.8 hours) Assuming a Load-Dependent Charge.
My calculated discharge time of 2.8 hours roughly agrees with the measured discharge time of 2.7 hours.
## Conclusion
My estimate for the battery operating time is 2.8 hours with an 800 mA load. We actually measured 2.7 hours, so the estimate is ~4%. This is error is within reason for batteries – they are subject to individual variation.
This entry was posted in Batteries. Bookmark the permalink.
### 21 Responses to Worked Calculation Example of Lithium Battery Capacity Versus Load
1. Filip De Somer says:
Hi Marc, what exactly does the FillVec function you are using in Fig 5?
2. mathscinotes says:
Hi Filip,
Sorry about missing that utility function. FillVec is a function I copied from the Mathcad master Tom Gutman. It fills a vector with incrementing numbers. I have updated Figure 5 to include that function. Thank you for finding this error.
mathscinotes
3. Greg says:
Thank you for this excellent analysis!
4. Greg says:
Possible typos: In the calculation block titled "Backup time for an 18650 Battery String", should 3.5V read 3.7V, and if so, the result will be different, because you do appear to have used 3.5V. If you did intend to use 3.5V, you don't explain why.
In the graph title "Figure 4 shows the typical discharge specification for 2200 mA-hour Li-Ion battery from Panasonic.", should "2200" read "2800"?
• mathscinotes says:
Hi Greg,
I should have used 3.7 V instead of 3.5 V. I have made the correction. You will see me use numbers between 3.5 and 3.7 because different vendors have different nominal voltage ratings. I just hadn't updated this section when we switched to Panasonic.
The battery capacity rating is actually 2700 (min), 2900 typical. I have updated the text to reflect that. I have also included a link in that paragraph to the Panasonic specification. Again, I use dozens of these batteries, each with different ratings. The text did not get updated when we switched batteries.
Thank you for the help on keeping my web site accurate.
mark
5. Greg says:
Also, how exactly was the raw discharge data obtained? Since you didn't use the actual cutoff voltage from that supplied graph, I assume that it was taken by measuring the battery string disconnected from the UPS - yes?
• mathscinotes says:
This blog post describes a worksheet used by my staff and me to interpolate battery data. The interesting part of the exercise is that the interpolation must be done in 2-dimensions because the effective capacity of the battery is a function of (1) the rate of discharge (load current); and (2) the cutoff voltage. The raw discharge data is obtained from the Panasonic specification (Figure 4). The specification only gives the discharge characteristics for 4 different battery loads (0.2C, 0.5C, 1C, and 2C). Because my load was different than they measured (i.e. 0.8C). I needed to interpolate the Panasonic data to obtain the battery capacity at my load and cutoff voltage. My intent here was to show my staff how to use Mathcad to interpolate 2D data. I can do something similar in Excel (more work), and I will put that on my list of candidate posts.
I ended up measuring the effective capacity in the UPS and it agreed within experimental error (I calculated 2.8 hours and measured 2.7 hours).
mark
• Greg says:
Thanks Mark - yes - I understand why you do the 2-dimensional interpolation.
Regarding the raw data, I am referring to your figure 3. Your figure 4 is the raw data from the specs - I realise that. I am asking about the raw data that the engineer supplied to you.
Also, an 0.8A load, assuming a battery capacity of 2800mAh, is 0.29C. Looking at the raw discharge curve for the rate of 0.5C (the red trace), the available capacity for a cutoff voltage of 3.3V appears to be about 2500mAh. 2500/800 = 3.1 hours, which doesn't tally - surely the run time at 0.29C should be *at least* 3.1 hours - yes?
In the supplied raw data, the starting voltage is 3.92V per cell - not 4.2V. If that's really true, that means the batteries were not fully charged to begin with - yes?
• Greg says:
In my previous reply, I used the discharge curve from the link to the Panasonic specs you provided. However, I notice that your figure 4 doesn't quite match that graph. (have they updated the specs?) When I use your figure 4, I get a slightly lower capacity: ~2400mAh @0.5C @3.3V, which lowers the run time slightly to 3.0 hours. That still doesn't tally though.
• mathscinotes says:
The Panasonic specification I added to the post is the one I used – in my analysis, I have to digitize the spec and sometimes there are small errors in that process.
As I look at the analysis, I forgot to normalize my current draw – I have corrected that omission. I had ignored the fact that the battery was not charged as fully as the manufacturer specifies. This is fairly common with UPS hardware. They do not want to stress the batteries at all. To model that lower initial charge, I simply subtracted the missing charge from my operating time calculation. Now I get a operating time estimate of 2.5 hours, which within 7% of the measured time of 2.7 hours. I consider this reasonable accuracy for this type of calculation.
I have updated the post. Thank you for your questions -- I want to make these examples as accurate as I can.
mark
• Greg says:
Thanks Mark - much appreciated. Your figure 4 looks *very* similar to the graph in the Panasonic spec - are you sure you re-generated that by digitisation? To me, it looks like an original Panasonic graph, albeit from a slightly different version of the spec (perhaps).
Coming back to an earlier question - how exactly was the raw data (supplied to you by the engineer) obtained? The reason I ask is that the discharge cutoff voltage for the pack is ~11.5V = ~2.9V/cell, however you have used a cutoff voltage of 3.3V. Again - was the pack measured *outside* of the UPS, or in-situ? If in-situ, why didn't you simply use the supplied cutoff voltage?
I know that none of this detracts from your general approach - I'm just trying to understand what you have done to the best of my ability, because I am currently working with these types of cells - that's how/why I found your post in the first place.
• mathscinotes says:
Here is the plot that is actually in my digitizer (raw screen capture). You can see the axis that I imposed on it in red. It sure looks like what is in the specification.
The battery voltage was measured at the battery terminals with an 800 mA programmable current source connected to the UPS output. The actual cutoff voltage is set within the UPS and has a weak dependence on the load current because of resistive losses within the UPS. Some folks also adjust the cutoff voltage for temperature. As I mentioned, I do not know what they intended to have for a cutoff. I certainly can use the measured cutoff –I have modified my post to use the measured cutoff. Remember, I was just trying to respond to an engineer who was wondering why his measured result was less than he expected. I literally spent no more than a few minutes on this calculation. I was looking for a rough result.
mark
• Greg says:
Thanks again. If you were just after a rough result, why did you bother with the interpolation? I was able to get the same result just by looking at the graph. 🙂
• mathscinotes says:
Actually, I did do exactly what you did – read the data off the graph to estimate the result. I then grabbed the interpolation template so that the guys would have a worked example of how to use it. One of the purposes of this blog is to provide worked examples of everyday engineering problems. I like when folks like yourself ask questions. It shows me where things are not clear or even wrong, which are hard to see when you are in the rush of getting product out.
mark
• Greg says:
Another little nit - you say that the typical capacity is measured when the load is minimal (0.2C), however, interestingly, the capacity at the maximum published discharge rate (2C) coincides, but only for the very lowest cutoff voltage. (2.5V)
• mathscinotes says:
The high current discharge characteristic of this battery is rather unusual. Normally, you do not see the high discharge current deliver full capacity. Most of the batteries I deal with have their capacities specified at the 20 hour discharge rate, i.e. C/20 = 0.05C.
mark
• Greg says:
Just by the way, I don't understand the data in the digitised tables. Column 1 appears to be voltage, but what is column 0 - why are there negative values in column 0? (sorry for asking so many questions and I'll understand if you need a bit of a break!). My maths is pretty lousy so I hope it's not a silly question.
• mathscinotes says:
Column 0 are the x axis values from the battery load graph. This means its units are mA-hr. The negative numbers are an anomaly of how the axis was setup. I was probably off a pixel where I put the axis zero. This reflects some of the issues you see with digitizing paper graphs. Another issue is that graphs are almost always slightly distorted as they go on the page. This means that the digitization process ALWAYS has some errors.
Keep asking questions. I use this blog as a tool for both me and my staff. As people ask questions, I work to improve the exposition. For example, I have made significant changes to this post in response to your questions – including adding a graphical analysis section. There are nearly 650 posts in the blog and I get questions all the time. My readers have helped me make the posts better. Understand that I am usually writing the first draft of every post in a hurry. I generally polish them up over time and as people ask questions.
mark
• Greg says:
Thanks again. 🙂
Regarding estimating the loss of capacity due to the lower than maximum starting voltage, a small error may result from assuming a certain initial discharge rate. I.e - it may not accurately model a battery that has been at rest at that lower voltage (unloaded) for a substantial amount of time.
Have you ever researched battery modelling? I.e - a complete mathematical model, which could predict behaviour for arbitrary load patterns, taking into account recovery effects etc etc.
• mathscinotes says:
Yup, all that is true. I always tell my engineers that batteries are a nonlinear function of everything. Another problem is that they age – their performance declines with time. The manufacturers tell you that they are good for, say 500 discharges. However, this means 500 full discharges. If you don't fully discharge the battery, they can provide good service for many more cycles than their rating.
I have read every paper I could find on battery modeling over the last ten years. There has been an explosion in the amount of modeling research because of batteries being used as the prime energy source for cars. I have not applied any of these models because I am not trying to run batteries for maximum performance, which is the case in cars. However, I find this research important and if a decent software package comes available that includes a good model I will try it out.
mark
6. good information thanks sir
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# NCERT Solutions for Class 7 Maths Chapter 9 Rational Numbers
Rational numbers are a fraction with a non zero denominator. It is used in our daily lives because several measures of a quantity cannot be described by an integer or natural numbers alone. It is said to be one of the most critical chapters in Maths. This chapter deals with the number system, properties, operation and application of rational numbers. To make this chapter a little easier some tricks and easy methods of doing it are also shown in the solution. In Ex 9.1 The topics are rational numbers, positive and negative rational numbers, rational numbers on the number line, rational numbers in standard form, comparison of rational numbers and rational numbers between two rational numbers.
A rational number in standard form can also have a negative number but only on its numerator. In Ex 9.2, students will learn about addition, additive inverse, subtraction, multiplication, and rational numbers division. The additive inverse of a rational number is the same number but with the opposite sign. When the rational number and an additive inverse is added, the result will be zero. The topics and subtopics in chapter Rational number are:
• Need for rational numbers
• What are Rational numbers
• Positive and negative Rational numbers
• Rational numbers on a line
• Comparison of Rational Numbers
• Rational Number between two national number
• Subtraction of rational numbers
• Multiplication of rational numbers
• Division of rational numbers.
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Simply, put temperature value in Fahrenheit and get Kelvin value in one go. Helmenstine, Anne Marie, Ph.D. "How to Convert Fahrenheit to Kelvin." Kelvin to Fahrenheit Conversion Table. You can use the conversion equation to perform the calculation. You may want to learn to convert between Celsius, Fahrenheit, and Kelvin in any combination. Multiply the Kelvin temperature by 9/5 and subtract 459.67. The formula to convert from kelvin to Fahrenheit is: Fahrenheit = ((kelvin - 273.15) x 1.8) + 32 Guess again! 68 Degree Fahrenheit is equal to 293.15 Kelvin, so use this simple formula to convert: Kelvin = ( ( [°F] - 32 ) × 5 / 9 ) + 273.15. How do I convert Fahrenheit to Celsius in Excel 2013/2016. Plus learn how to convert °F to K Multiply this number by 5. Fahrenheit to Kelvin conversion table. Not quite! Fahrenheit to Kelvin Method #1 Subtract 32 from the Fahrenheit temperature. Conversion Formulae. Kelvin (K) = (Fahrenheit - 32) / 1.8 + 273.15 If required, there are worked examples below which use this formula to show how to convert a temperature in Fahrenheit to a temperature in Kelvin. In order to read temperatures over a wide variety of disciplines, you must learn to convert Fahrenheit to Celsius and Celsius to Kelvin. T (K) = 283.15 K. 50 °F = 283.15 K. We conclude that fifty Fahrenheit is equivalent to two hundred eighty-three point one five Kelvin: 50 Fahrenheit is equal to 283.15 Kelvin. This number is a little too high. You can also convert the temperature from Fahrenheit to Celsius and then to Kelvin, if you prefer. See if going back and changing that gives you a different answer! In both Kelvin and Celsius the difference between the cold of water and its boiling point is found to be about 100 units. Kelvin Conversion Formulas. Inverse Conversion 0 Kelvin is -273.15° Celsius. In the example of 90 °F, the answer to the second step for formula 2 is 58 ÷ 1.8 = 32.22 °C, where the 2 is a repeating decimal. The temperature T in degrees Fahrenheit (°F) is equal to the temperature T in Kelvin (K) times 9/5, minus 459.67: T (°F) = T (K) × 9/5 - 459.67. Subtract 32 from the Fahrenheit temperature. Try another answer... To convert Fahrenheit to Kelvin, use the formula K = (y °F + 459.67) x 5/9, where K equals Kelvin and y equals the temperature in Fahrenheit. How to convert Fahrenheit to Celsius. Program for Fahrenheit to Kelvin conversion. By signing up you are agreeing to receive emails according to our privacy policy. Logic to convert temperature from Fahrenheit to Kelvin … Kelvin to Fahrenheit (Swap Units) Format Accuracy Note: Fractional results are rounded to the nearest 1/64. The first step to finding the Farenheit equivalent of 427 K is to multiply it by 9/5, or 1.8 (427 x 1.8), which equals 768.6. The answer will be the temperature in Kelvin. Fahrenheit to Kelvin, K = (5/9)(F+459.67) Fahrenheit to Rankin, R = F + 459.67; Rankin to Kelvin, K = (5/9)R; Conversion Between Celsius And Kelvin. wikiHow's. Remember, Celsius will always be 32 degrees lower than Fahrenheit. T (°C) = (T (°F) - 32) / 1.8. It is similar to Celsius except that it starts at absolute zero, the lowest possible temperature. We can now use the following functions from the UliEngineering.Physics.Temperature package to convert … It is similar to Celsius except that it starts at absolute zero, the lowest possible temperature. The online Fahrenheit to Kelvin Converter is used to convert temperature from Fahrenheit (℉) to Kelvin (K). Fahrenheit is often used for surface temperatures in the United States, and Kelvin is often used for scientific equations and calculations. Definition: The Fahrenheit (symbol: °F) is a unit of … Fahrenheit or Kelvin The SI base unit for temperature is the kelvin. How to Convert Fahrenheit to Kelvin. https://www.wikihow.com/Convert-Between-Fahrenheit,-Celsius,-and-Kelvin T (°C) = (T (°F) - 32) / (9/5). So you could say 1K = 1 degree Celsius. Include your email address to get a message when this question is answered. To convert Degree Fahrenheit to Kelvin, multiply the Temperature by the conversion ratio. 1 Fahrenheit is equal to 0.55555555555556 kelvin. Definitely not! 1 … Kelvin to Fahrenheit conversion table What Temperature Does Fahrenheit Equal Celsius? Convert 300 Kelvin to degrees Fahrenheit: T (°F) = 300K × 9/5 - 459.67 = 80.33 °F. Kelvin to Fahrenheit ► How to convert Fahrenheit to Kelvin The temperature T in Kelvin (K) … Type in your own numbers in the form to convert the units! Helmenstine, Anne Marie, Ph.D. (2020, August 28). Divide this number by 9. While you might think this conversion wouldn't occur much, it turns out there is a lot of scientific and engineering equipment that uses the Fahrenheit scale! By using our site, you agree to our. Unfortunately it doesn't get any simpler than that. % of people told us that this article helped them. For a more accurate answer please select 'decimal' from the options above the result. Note that while Fahrenheit has degrees, Kelvin does not. Formulas for Fahrenheit and Celsius Conversions, Convert Temperature from Kelvin to Celsius and Back, Ph.D., Biomedical Sciences, University of Tennessee at Knoxville, B.A., Physics and Mathematics, Hastings College. Multiply by 5: 2798.35. Fahrenheit is a thermodynamic temperature scale, where the freezing point of water is 32 degrees Fahrenheit (°F) and the boiling point 212°F (at standard atmospheric pressure). Convert 68 degrees Fahrenheit to degrees Celsius: How to convert degrees Fahrenheit to degrees Celsius in Excel. Fortunately, it is easy to convert Fahrenheit to Kelvin. This answer is most likely miscalculated due to a simple calculator error, where instead of multiplying by 5/9, the calculator thought you wanted to multiply by 5 and then divide by 9. It uses zero as absolute zero, unlike Fahrenheit or Celsius, where there are negative numbers. Then, you want to multiply that number by 5/9, or 0.55556 (545.67 x 0.55556). Fahrenheit to Celsius formula. The formula is: ºF = 1.8 x (K - 273) + 32. To learn how to convert a temperature from Fahrenheit to Celisus to Kelvin, scroll down! Converting to Celsius Then Kelvin Learn the formulas. To the Fahrenheit temperature, add 459.67°. For example, say you want to convert human body temperature, 98.6° F, into its Kelvin equivalent. Kelvin to Fahrenheit converter. Conversion (temperature difference or interval) When converting a temperature interval between °F and °C, only the ratio is used, without any constant (in this case, the interval has the same numeric value in Kelvin as in degrees Celsius): f °Fahrenheit to c °Celsius or Kelvin: f °F × 5°C / 9°F = f / 1.8 °C = c °C = c K wikiHow's Content Management Team carefully monitors the work from our editorial staff to ensure that each article is backed by trusted research and meets our high quality standards. The conversion formula for calculating kelvin (K) from degrees celsius (°F) is as follows: K = (°F + 459.67) x 5/9 The Kelvin temperature scale was created in response to the need for an absolute thermometric scale, it has its zero point at absolute zero and progresses from that point. First, you will have wanted to add your starting temperature to the positive form of Farenheit's Absolute Zero (86+459.67) to get 545.67. Fahrenheit to Kelvin, K = (5/9)(F+459.67) Fahrenheit to Rankin, R = F + 459.67; Rankin to Kelvin, K = (5/9)R; Conversion Between Celsius And Kelvin. Read on for another quiz question. To get the Celsius temperature from the Fahrenheit temperature, all you have to do is subtract 32 (53-32) which will equal 11.667 when rounded to the third decimal. Also, explore tools to convert Fahrenheit or kelvin to other temperature units or learn more about temperature conversions. Absolute zero is -459.67 °F. When converted directly to Kelvin conversion example Kelvin scale is an absolute temperature,. Start by multiplying the Celsius temperature by the National Institute of Standards and Technology down. The math using simple weather conversion equations when it 's converted back degrees. May want to learn how to convert 68 degrees Fahrenheit and Kelvin read the same temperature at 574.25 of. Be familiar with subtracting, and see what you get 308.93 degrees Fahrenheit when... 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https://math.answers.com/math-and-arithmetic/What_is_230_times_46
| 1,721,276,530,000,000,000
|
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crawl-data/CC-MAIN-2024-30/segments/1720763514822.16/warc/CC-MAIN-20240718034151-20240718064151-00892.warc.gz
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0
# What is 230 times 46?
Updated: 9/24/2023
Wiki User
8y ago
230 multiplied by 46 is 10,580
Wiki User
8y ago
Earn +20 pts
Q: What is 230 times 46?
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Related questions
230
46 times.
It goes 46 times
23 x 10 = 230
230
LCM(10, 46) 230
### What is 230 by 5?
230 x 5 = 1150 230 ÷ 5 = 46
### What is the 20 percent of 230?
20% of 230 = 0.2 x 230 = 46
### Can you simplify 230 over 5?
230 / 5 is equal to 46.
230
46
### What is 5x plus 10 equals 240?
240-10=5x 5x=230 230/5=46 x=46
| 216
| 533
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Top-Rated Free Essay
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Crank and Connecting Rod
Introduction-
The motion of assemblies is determined by the configuration of links and joints. Using the configurations the operation of rotational and sliding joints are examined and observed. This kind of mechanism is very commonplace in machines. Machines are made up of a number of parts and relative motion between the various parts permits the working of the machine. As the crank is rotated the rod starts moving but the velocity is not uniform. It is greater towards one direction than the other. This principle is utilized extensively in some machines.
Aim
Understand the relative motion of the rotational and sliding joint.
Understand the movement of the rotational and sliding joint and graph the peculiar behaviour.
To investigate the reason for such movement.
Procedure
The assembly is set up such that one end of the rod moves up and down when the crank (to which the other end of the rod is attached) is rotated. So, now the movement depends on the rotation of the crank. It is made sure that there is space for the rod to move on the complete rotation of the crank. Once the setup is done, readings are taken for every 10 degrees movement in the crank. Once done, the appropriate readings are marked on the graph. Using this graph another graph for velocity with time and acceleration with time are made. Looking at the velocity graph helps find out if the movement of the slider is faster than the other.
Result
It is observed that the movement of the slider is faster towards one direction than the other while the crank is rotated. This is because the rotation for one direction is shorter than the other. This principle is widely used in various machines. One commonly used function is called the quick return mechanism. In this the faster stroke is used to do the work whereas the slower stroke is used to get it back to its position.
Calculation and Discussion
The following data is obtained upon performing the experiment.
Angle Distance (mm)
0
119
10
125
20
128
30
129
40
129
50
130
60
126
70
122
80
117
90
110
100
104
110
97
120
89
130
82
140
74
150
67
160
60
170
54
180
48
190
42
200
38
210
34
220
31
230
29
240
29
250
30
260
33
270
39
280
45
290
55
300
65
310
77
320
87
330
97
340
106
350
114
360
119
On observing the results it can be seen that the connecting rod initially moves forward then moves back and then again moves forward during the cranks entire rotation of 360 degree. If the configuration is changed different results are obtained. A graph can be plotted based on the above data as follows.
Figure -Graph of The movement of slider and the angle of rotation.
If we assume that it takes 2 sec for the crank to complete one full rotation then another graph can be plotted with time on the X axis and the movement of the slider on the Y axis. This graph shows the time dependent movement of the slider.
Figure 2- Graph of the movement of slider with respect to Time.
This graph is used to derive determine the velocity of the slider. The derivative of the movement of the slider with respect to time ie- ΔX/Δt gives the velocity of the slider when the crank is rotated.
Figure 3- Graph of velocity for the slider
It can be seen that initially the velocity is high and then as the crank is rotated the velocity falls getting to zero and then continues in the negative direction until it starts increasing again. The acceleration of the movement of slider is again found by differentiating velocity with respect to time.
Figure 4- Graph of acceleration for the slider
It is observed that the slider moves faster towards the return stroke compared to the forward stroke. This is observed when the crank is rotated in an anti clockwise manner and the results vary when the rotation is clockwise.
Conclusion
From the performed experiment it is observed that the slider moves when the crank is rotated. Depending upon the configuration of the mechanism and the rotation of the slider, it shows peculiar movement. In this case the slider moved faster towards the return stroke than the forward stroke. This conclusion is exploited in various machines and is termed as quick return mechanism. This is due to the shorter angle of movement for the slider to move one direction than the other.
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https://math.stackexchange.com/questions/1808808/fractional-anti-derivatives-and-derivatives-of-the-logarithm
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crawl-data/CC-MAIN-2020-10/segments/1581875144027.33/warc/CC-MAIN-20200219030731-20200219060731-00461.warc.gz
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# Fractional anti-derivatives and derivatives of the logarithm
Anti-derivatives and derivatives of the natural logarithm are well defined until we attempt to evaluate the fractional derivative and anti-derivatives.
The background to this problem was that I was trying to evaluate fractional derivatives of $\frac1x$, which is usually given as $D^n\frac1x=\frac{\Gamma(0)}{\Gamma(-n)}x^{-1-n}$, but that is defined only for $n=0$ and nowhere else. You can define it for $n\in\mathbb N$, but this is not the best definition one could have. ($D^n$ is the $n$th derivative with respect to $x$)
I have the general $n$th anti-derivative ($I^n$) of the natural logarithm as
$$I^n\ln(x)=\frac{x^n\left(\ln(x)-\int_0^1\frac{t^n-1}{t-1}dt\right)}{\Gamma(n+1)}$$
Proved by induction: $\frac d{dx}I^n\ln(x)=I^{n-1}\ln(x)$, and holds true for $n=1$.
I have this graphed on Desmos. (I really like that it has an integral feature. And if it is too slow, click the little circles on the right to turn off those functions.)
While this is a great formula, it doesn't really work for derivatives ($D^n=I^{-n}$), or at least, Desmos stops graphing at $n=-0.99$, but before that, it appears as though $\lim_{n\to-1}I^n\ln(x)=\frac1x$
I attempted to evaluate it for $n=-1$
$$\frac1x=D^1\ln(x)=I^{-1}\ln(x)=\lim_{n\to-1}\frac{x^n\left(\ln(x)-\int_0^1\frac{t^n-1}{t-1}dt\right)}{\Gamma(n+1)}$$
If you try to directly substitute, you get an indefinite form, so I attempted to do a limit method instead. I would like to apply L'Hostpital's rule, but I don't quite know how to deal with either the numerator nor the denominator.
I have found that
$$I^{1/2}\ln(x)=\frac{2\sqrt x\left(\ln(x)-2+\ln(4)\right)}{\sqrt\pi}$$
The $2-\ln(4)$ is wolframalpha's evaluation of $\int_0^1\frac{t^{1/2}-1}{t-1}dt$
From here, you can differentiate like normal to get $D^{(2n-1)/2}\ln(x)$, $n\in\mathbb N$.
More importantly, if the limit from above is correct, then my formula works for $I^{-n}\ln(x)$ with $n\in\mathbb N$ and I can finally define what $D^n\frac1x$ is equal to!
So the question:
How can we evaluate the limit: $\lim_{n\to-1}I^n\ln(x)$? What does it equal?
Is the fractional derivative of the natural logarithm and $\frac1x$ already known? I've made posts on this before, and it doesn't appear many people have tackled this problem. Here and Here. Both have received little attention. Attempting to use regular fractional derivative formulas are extremely messy, so I could not actually use them. If someone more experienced can, that'd be great.
• Ah, so $D^\mu\ln(x)=\frac{x^{-\mu}\left(\ln(x)-\gamma-\psi(1-\mu)\right)}{\Gamma(1-\mu)}$... ok, thanks. This was helpful. Can different methods to computing fractional differintegrals result in different results? – Simply Beautiful Art Jun 1 '16 at 20:45
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https://answerprime.com/geometry-which-figure-represents-the-image-of-trapezoid-lmnp-after-a-reflection-across-the-x-axis-2/
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Geometry which figure represents the image of trapezoid lmnp after a reflection across the x-axis?
A Step-by-step explanation: Ed2020
Consider trapezoid LMNP with vertices at points (-2,3), (-2,6), (-1,6) and (-1,4), respectively. The reflection across the x-axis has a rule: (x,y)→(x,-y). Then L(-2,3)→L'(-2,-3)M(-2,6)→M'(-2,-6)N(-1,6)→N'(-1,-6)P(-1,4)→P'(-1,-4) As you can see points L’, M’, N’ and P’ form trapezoid L’M’N’P’ and this is exactly figure A. correct choice is A.
A Step-by-step explanation: Imagine folding the graph in half on the x-axis and tracing the shape on the other side. Ps. x-axis is horizontal (left-right), y-axis is vertical (up-down)
Answer is C hope this helps
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https://math.stackexchange.com/questions/2606284/i-dont-know-how-to-solve-sum-of-series-n-cdot-frac12n
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# i dont know how to solve sum of series $n \cdot \frac{1}{2^n}$ [duplicate]
i need some ideas to solve $$\sum_{n=1}^\infty n\cdot\left(\frac12\right)^n$$ I prove that the series converges to using ratio method, but i dont know how to find the sum.
## marked as duplicate by Guy Fsone, Namaste, Jack, Hans Lundmark, user223391 Jan 15 '18 at 18:53
$$S=\sum_{n=1}^\infty n\cdot\left(\frac12\right)^n$$ Then \begin{align}S&=\frac12+2\cdot\frac14+3\cdot\frac18+4\cdot\frac1{16}+\cdots\\ &=\left(\frac12+\frac14+\frac18+\cdots\right)+\left(\frac14+\frac18+\cdots\right)+\left(\frac18+\cdots\right)+\cdots\\ &=\left(\frac12+\frac14+\frac18+\cdots\right)\cdot\left(1+\frac12+\frac14+\cdots\right)\\ &=1\cdot 2\\ &=2\end{align}
$$\sum_{n=0}^\infty x^{n} = \frac{1}{1-x}.$$
Then take the derivative of both sides
$$\sum_{n=1}^\infty nx^{n-1} = \frac{1}{(1-x)^2}.$$
Multiply by $x$
$$\sum_{n=1}^\infty nx^{n} = \frac{x}{(1-x)^2}.$$
Then plug in $x=1/2.$
use that for the finite sum is $$\sum_{i=1}^n\frac{i}{2^i}=-2\, \left( 1/2 \right) ^{n+1} \left( n+1 \right) -2\, \left( 1/2 \right) ^{n+1}+2$$ compute the Limit for $n$ tends to infinity
$$\sum_{n=1}^{\infty}\frac{n}{2^{n}}=\frac{1}{2}\sum_{n=1}^{\infty}\frac{n}{2^{n-1}}=\frac{1}{2}\left(1+x+x^2+...\right)'_{x=\frac{1}{2}}=\frac{1}{2}\left(\frac{1}{1-x}\right)'_{x=\frac{1}{2}}=$$ $$=\frac{1}{2}\cdot\frac{1}{\left(1-\frac{1}{2}\right)^2}=2$$
Are you familiar with the usual method for summing a finite geometric series?
Let $\displaystyle S_n = \sum_{k=1}^n k\cdot\left(\frac12\right)^k= \left(1 \cdot \dfrac 12 + 2 \cdot \dfrac {1}{2^2}+ 3 \cdot \dfrac {1}{2^3} + \cdots + n \cdot \dfrac {1}{2^n} \right)$
Then $\displaystyle \dfrac 12 S_n= \sum_{k=1}^n k\cdot\left(\frac12\right)^{k+1} = \left(1 \cdot \dfrac {1}{2^2} + 2 \cdot \dfrac {1}{2^3}+ 3 \cdot \dfrac {1}{2^4} + \cdots + (n-1) \cdot \dfrac {1}{2^n} + n \cdot \dfrac {1}{2^{n+1}} \right)$
So
$$S_n - \dfrac 12 S_n= \dfrac 12 + \dfrac {1}{2^2} + \dfrac {1}{2^3} + \cdots + \dfrac {1}{2^n} + n \cdot \dfrac {1}{2^{n+1}}$$
$$=\dfrac {1 - (\frac {1}{2})^{n+1}}{1 - \frac 12}-1 + n \cdot \dfrac {1}{2^{n+1}}$$
Thus
$$\dfrac 12 S_n= 2-\left( \dfrac{1}{2^n} \right)-1 - n \cdot \dfrac {1}{2^{n+1}}$$
$$S_n=4-\left( \dfrac{1}{2^{n-1}} \right)-2 - n \cdot \dfrac {1}{2^{n}}$$
So
$$\sum_{k=1}^\infty k\cdot\left(\frac12\right)^k = \displaystyle \lim_{n \to \infty} S_n =2$$
Other technique different than the showed in other answers make use of summation by parts together with particular algebraic rules named generally finite calculus.
In this fashion we want to consider the indefinite sum $\sum k x^k\,\delta k$. Using the techniques described in the last linked PDF we find that
$$\sum k x^k\,\delta k=\frac{x^k}{x-1}\cdot k-\frac1{x-1}\sum x^{k+1}\,\delta k\\\implies\sum_{k=0}^\infty kx^k=\left[\frac{x^k}{x-1}\cdot k\right]_{k=0}^{k\to\infty}-\frac1{x-1}\sum_{k=0}^\infty x^{k+1}$$
Choosing $x=1/2$ we find that
$$\sum_{k=0}^\infty k\left(\frac12\right)^k=\lim_{k\to\infty}\frac{(1/2)^k\cdot k}{-1/2}+2\sum_{k=1}^\infty (1/2)^k=0+2=2$$
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https://ghilbert-app.appspot.com/wiki/general/Orthogonality_2
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{{interfaces | imports = Interface:Orthogonality 1 | exports = Interface:Orthogonality 2 }}
This is part of a series of modules which prove a variety of geometrical theorems starting with Tarski's axioms for geometry. We follow the formalization of Julien Narboux<ref>The formal proofs are at http://www.lix.polytechnique.fr/Labo/Julien.Narboux/tarski.html Formalization of Tarski's geometry in the Coq proof assistant and are described in Julien Narboux (2007), "http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.158.8614 Mechanical Theorem Proving in Tarski’s Geometry", F. Botana and T. Recio (Eds.): ADG 2006, LNAI 4869, pp. 139–156</ref> which itself closely follows a treatise by Schwabhäuser, Szmielew, and Tarski.<ref>W. Schwabhäuser, W Szmielew, and A. Tarski (1983), ''Metamathematische Methoden in der Geometrie'', ISBN 0387129588</ref>
This page is one of several involving perpendicular lines. We prove some additional theorems about is-right-angle, and define a predicate saying that lines are perpendicular at a point. A future page will enable us to prove the existence of the midpoint of a line segment.
We import the theorems of propositional logic and predicate logic, and the geometry results so far and define some variables:
## Right angles
We've proved a number of the results relating to is-right-angle in Orthogonality definitions. Here we pick up a few more (particularly ones for which the automatic expansion of definitions in Orthogonality definitions is inconvenient, and ones which follow from those).
### Proving is-right-angle from an object
To prove is-right-angle from RightAngle requires that we come up with a point which satisfies the conditions of RightAngle. Here's a theorem which handles the logic involved in going from that point to an expression containing .<ref>not in Narboux, as coq handles this sort of thing</ref>
• (→ (∧ (is-midpoint-of B C Z) (≡ A C A Z)) (is-right-angle A B C)) (RightAngleObject)
### Swapping the vertex with one of the legs
Another degenerate case is is-right-angle A B C ∧ is-right-angle A C B → B = C.<ref>l8_7 in Narboux</ref>
Let A′ be the symmetric point of A through the point C.
First assume B ≠ C (if not, we are done). Then expand is-right-angle A B C by the definition: ∃ c′ (B is-midpoint-of C c′ ∧ A C ≡ A c′). We also flip is-right-angle A C B to is-right-angle B C A and expand it according to the definition: ∃ a′ (C is-midpoint-of A a′ ∧ B A ≡ B a′).
• (→ (∧ (¬ (= B C)) (∧ (is-right-angle A B C) (is-right-angle A C B))) (∃ c′ (∧ (∧ (¬ (= B C)) (∧ (is-right-angle A B C) (is-right-angle A C B))) (∧ (is-midpoint-of B C c′) (≡ A C A c′))) )) (RightAngleVertexLeg-cprime)
• ( → (∧ (∧ (¬ (= B C)) (∧ (is-right-angle A B C) (is-right-angle A C B))) (∧ (is-midpoint-of B C C′) (≡ A C A C′))) (∃ a′ (∧ (∧ (∧ (¬ (= B C)) (∧ (is-right-angle A B C) (is-right-angle A C B))) (∧ (is-midpoint-of B C C′) (≡ A C A C′))) (∧ (is-midpoint-of C A a′) (≡ B A B a′))) )) (RightAngleVertexLeg-aprime)
Now we apply RightAngleLeg to get is-right-angle c′ C A.
• ( → (∧ (∧ (∧ (¬ (= B C)) (∧ (is-right-angle A B C) (is-right-angle A C B))) (∧ (is-midpoint-of B C C′) (≡ A C A C′))) (∧ (is-midpoint-of C A A′) (≡ B A B A′))) (is-right-angle C′ C A) ) (RightAngleVertexLeg-cprime-c-a)
We can paraphrase the definition of is-right-angle C′ C A as "the symmetric point of A through C is the same distance from C′ as A is". By symmetric point uniqueness, said symmetric point is just A′. We express this via the "uniqueness lemma", is-right-angle C′ C A ∧ C is-midpoint-of A A′ → C′ A ≡ C′ A′, which we prove after a few lemmas which reflect parts of its proof.
Our next step is A′ C ≡ A′ C′.
• ( → (∧ (∧ (∧ (¬ (= B C)) (∧ (is-right-angle A B C) (is-right-angle A C B))) (∧ (is-midpoint-of B C C′) (≡ A C A C′))) (∧ (is-midpoint-of C A A′) (≡ B A B A′))) (≡ A′ C A′ C′) ) (RightAngleVertexLeg-aprime-c-aprime-cprime)
Next is is-right-angle A′ B C.
• ( → (∧ (∧ (∧ (¬ (= B C)) (∧ (is-right-angle A B C) (is-right-angle A C B))) (∧ (is-midpoint-of B C C′) (≡ A C A C′))) (∧ (is-midpoint-of C A A′) (≡ B A B A′))) (is-right-angle A′ B C) ) (RightAngleVertexLeg-aprime-b-c)
• ( → (∧ (∧ (∧ (¬ (= B C)) (∧ (is-right-angle A B C) (is-right-angle A C B))) (∧ (is-midpoint-of B C C′) (≡ A C A C′))) (∧ (is-midpoint-of C A A′) (≡ B A B A′))) (= B C) ) (RightAngleVertexLeg-b-not-c-1)
• (→ (∧ (¬ (= B C)) (∧ (is-right-angle A B C) (is-right-angle A C B))) (= B C)) (RightAngleVertexLeg-b-not-c)
• (→ (∧ (is-right-angle A B C) (is-right-angle A C B)) (= B C)) (RightAngleVertexLeg)
### A leg which is perpendicular to itself
Another degenerate case is is-right-angle A B A → A = B.<ref>l8_8 in Narboux</ref> The proof is that RightAngleVertexLeg gives us is-right-angle A B A ∧ is-right-angle A A B → A = B, but is-right-angle A A B is a theorem, so we are done.
### Three points which are both perpendicular and collinear
If three points are both perpendicular and collinear, then one of the legs must be an empty line segment.<ref>l8_9 in Narboux</ref>
• (→ (∧ (is-right-angle A B C) (collinear A B C)) (∨ (= A B) (= C B))) (RightAngleCollinear)
### Slight variant of RightAngleABB
We'll want this straightforward consequence of RightAngleABB in a moment.
### A congruence theorem
The only reason we rederive this theorem, rather than importing it, is to avoid editing all the interfaces between Betweenness of points and here.
### An angle congruent to a right angle is a right angle
That is, is-right-angle A B C ∧ A B C ≅ A′ B′ C′ → is-right-angle A′ B′ C′.<ref>l8_10 in Narboux</ref>
We start with the B = C case, where we first conclude B′ = C′ and so the conclusion follows from RightAngleABB.
• (→ (∧ (= B C) (∧ (is-right-angle A B C) (≅ A B C A′ B′ C′))) (is-right-angle A′ B′ C′)) (RightAngleCongruence-b-c)
Here's the sketch of the B ≠ C case. Let D be the point we get by expanding the definition of is-right-angle A B C (that is, B is-midpoint-of C D ∧ A C ≡ A D). Let D′ be the symmetric point of C′ through B′. Now we just need A′ C′ ≡ A′ D′. We apply outer five segment with baselines C B D{{{ and {{{C′ B′ D′ and points A and A′, which gives us D A ≡ D′ A′. We have A C ≡ A D from the construction of D and A′ C′ ≡ A C from A B C ≅ A′ B′ C′. So by transitivity, we have A′ C′ ≡ A′ D′ which is what we needed.
It will be most convenient to start with the construction of D and D′. First we construct D by expanding the definition of is-right-angle and moving terms inside the quantifier.
• (→ (∧ (¬ (= B C)) (∧ (is-right-angle A B C) (≅ A B C A′ B′ C′))) (∃ d (∧ (¬ (= B C)) (∧ (∧ (is-midpoint-of B C d) (≡ A C A d)) (≅ A B C A′ B′ C′))))) (RightAngleCongruence-d)
Next we construct D′ as the symmetric point of C′ through B′.
• (→ (∧ (¬ (= B C)) (∧ (∧ (is-midpoint-of B C D) (≡ A C A D)) (≅ A B C A′ B′ C′))) (∃ d′ (∧ (∧ (¬ (= B C)) (∧ (∧ (is-midpoint-of B C D) (≡ A C A D)) (≅ A B C A′ B′ C′))) (is-midpoint-of B′ C′ d′)) ) ) (RightAngleCongruence-dprime)
Having constructed our points, we can get going, starting with B′ ≠ C′, which follows from B ≠ C and A B C ≅ A′ B′ C′.
• (→ (∧ (∧ (¬ (= B C)) (∧ (∧ (is-midpoint-of B C D) (≡ A C A D)) (≅ A B C A′ B′ C′))) (is-midpoint-of B′ C′ D′)) (¬ (= B′ C′))) (RightAngleCongruence-bprime-not-cprime)
Next is outer five segment with baselines C B D{{{ and {{{C′ B′ D′ and points A and A′, to get D A ≡ D′ A′.
• (→ (∧ (∧ (¬ (= B C)) (∧ (∧ (is-midpoint-of B C D) (≡ A C A D)) (≅ A B C A′ B′ C′))) (is-midpoint-of B′ C′ D′)) (≡ D A D′ A′)) (RightAngleCongruence-d-a-dprime-aprime)
Now we apply transitivity on some line segment congruences which we already have to get A′ C′ ≡ A′ D′.
That's all the major pieces of is-right-angle A′ B′ C′. We just need to do a bit of assembly.
• (→ (∧ (∧ (¬ (= B C)) (∧ (∧ (is-midpoint-of B C D) (≡ A C A D)) (≅ A B C A′ B′ C′))) (is-midpoint-of B′ C′ D′)) (is-right-angle A′ B′ C′)) (RightAngleCongruence-aprime-bprime-cprime)
• (→ (∧ (¬ (= B C)) (∧ (is-right-angle A B C) (≅ A B C A′ B′ C′))) (is-right-angle A′ B′ C′)) (RightAngleCongruence-b-not-c)
• (→ (∧ (is-right-angle A B C) (≅ A B C A′ B′ C′)) (is-right-angle A′ B′ C′)) (RightAngleCongruence)
## Perpendicular lines at a point
The line A B is perpendicular to the line C D at the point X if that point lies on both lines and if choosing one point from each line plus the vertex X always produces a right angle. In symbols, A B C D ⟂at X is defined as A ≠ B ∧ C ≠ D ∧ collinear X A B ∧ collinear X C D ∧ ∀ u ∀ v (collinear u A B ∧ collinear v C D → is-right-angle u X v).
### Definition as a theorem
As usual, we'll need a theorem form of the definition.
• (↔ (⟂at A B C D X) (∧ (∧ (∧ (∧ (¬ (= A B)) (¬ (= C D))) (collinear X A B)) (collinear X C D)) (∀ u (∀ v (→ (∧ (collinear u A B) (collinear v C D)) (is-right-angle u X v)))))) (PerpendicularAt)
### Symmetry
Here we prove A B C D ⟂at X ↔ C D A B ⟂at X.<ref>l8_12 and perp_in_symmetry in Narboux</ref> The concept is pretty simple: expand the definition and apply symmetry to each piece. The only thing which makes this proof a bit long is the number of pieces.
### Only one point on each line is needed
The definition of ⟂at might seem a bit odd, in that it would intuitively appear that one point on each line which forms a right angle would suffice, rather than needing to make an assertion about all points on those lines. In fact, this intuition is correct subject to the condition that the points being chosen on the line do not equal the vertex. In symbols, A ≠ B ∧ C ≠ D ∧ collinear X A B ∧ collinear X C D ∧ ∃ u ∃ v (collinear u A B ∧ collinear v C D ∧ u ≠ X ∧ v ≠ X ∧ is-right-angle u X v) → A B C D ⟂at X.<ref>l8_13_2 in Narboux</ref>
The proof is based on the idea that A, B, u are on a line and we'll also consider an arbitrary point, which we'll call U0, on that line. We'll use RightAngleLeg to turn is-right-angle u X v to is-right-angle U0 X v (and some collinearity transitivity to set up the hypotheses for RightAngleLeg). Then we'll do much the same for C, D, v, and an arbitrary point V0 on the line C D{{{, which will turn {{{is-right-angle v X U0 to is-right-angle V0 X U0. Fortunately, the first half and the similar half are similar enough that we can break it off into a lemma which we'll be able to apply twice.
The lemma is A ≠ B ∧ collinear A B U0 ∧ collinear A B X ∧ collinear A B U ∧ is-right-angle U X V ∧ U ≠ X → is-right-angle U0 X V. In this case, RightAngleLeg is is-right-angle U X V ∧ U ≠ X ∧ collinear X U U0 → is-right-angle U0 X V. The first two hypotheses we have, so the first part of our proof is headed towards collinear X U U0. We'll start by applying collinearity transitivity twice.<ref>based on Narboux's proof of l8_13_2 but streamlined, as Narboux also asserts collinear A X U0 and collinear A U X, which don't seem to be used.</ref>
The first collinearity is collinear B U U0 by transitivity from B ≠ A, collinear B A U and collinear B A U0.
• (→ (∧ (∧ (∧ (∧ (∧ (¬ (= A B)) (collinear A B U0)) (collinear A B X)) (collinear A B U)) (is-right-angle U X V)) (¬ (= U X))) (collinear B U U0)) (PerpendicularAtThereExists-b-u-u0)
The other collinearity is collinear B U X by transitivity from A ≠ B, collinear A B U and collinear A B X.
• (→ (∧ (∧ (∧ (∧ (∧ (¬ (= A B)) (collinear A B U0)) (collinear A B X)) (collinear A B U)) (is-right-angle U X V)) (¬ (= U X))) (collinear B U X)) (PerpendicularAtThereExists-b-u-x)
At this point we prove collinear X U U0 by considering B = U and B ≠ U cases.
For the B = U case, we first apply transitivity to give collinear B X U0 (from A ≠ B, A B X and A B U0).
• (→ (∧ (∧ (∧ (∧ (∧ (¬ (= A B)) (collinear A B U0)) (collinear A B X)) (collinear A B U)) (is-right-angle U X V)) (¬ (= U X))) (collinear B X U0)) (PerpendicularAtThereExists-b-x-u0) The rest of the B = U case is just substituting U for B in collinear B X U0 to get collinear X U U0.
• (→ (∧ (= B U) (∧ (∧ (∧ (∧ (∧ (¬ (= A B)) (collinear A B U0)) (collinear A B X)) (collinear A B U)) (is-right-angle U X V)) (¬ (= U X)))) (collinear X U U0)) (PerpendicularAtThereExists-b-u)
The B ≠ U case applies transitivity to B ≠ U, collinear B U U0 and collinear B U X, to give collinear U U0 X.
• (→ (∧ (¬ (= B U)) (∧ (∧ (∧ (∧ (∧ (¬ (= A B)) (collinear A B U0)) (collinear A B X)) (collinear A B U)) (is-right-angle U X V)) (¬ (= U X)))) (collinear X U U0)) (PerpendicularAtThereExists-b-not-u) Combining the two cases gives collinear X U U0.
• (→ (∧ (∧ (∧ (∧ (∧ (¬ (= A B)) (collinear A B U0)) (collinear A B X)) (collinear A B U)) (is-right-angle U X V)) (¬ (= U X))) (collinear X U U0)) (PerpendicularAtThereExists-x-u-u0)
We're now ready to prove the lemma A ≠ B ∧ collinear A B U0 ∧ collinear A B X ∧ collinear A B U ∧ is-right-angle U X V ∧ U ≠ X → is-right-angle U0 X V. We just need to apply RightAngleLeg which is is-right-angle U X V ∧ U ≠ X ∧ collinear X U U0 → is-right-angle U0 X V.
• (→ (∧ (∧ (∧ (∧ (∧ (¬ (= A B)) (collinear A B U0)) (collinear A B X)) (collinear A B U)) (is-right-angle U X V)) (¬ (= U X))) (is-right-angle U0 X V)) (PerpendicularAtThereExists-half)
Now we apply this lemma twice to give A ≠ B ∧ C ≠ D ∧ collinear X A B ∧ collinear X C D ∧ (collinear U A B ∧ collinear V C D ∧ U ≠ X ∧ V ≠ X ∧ is-right-angle U X V) ∧ (collinear U0 A B ∧ collinear V0 C D) → is-right-angle U0 X V0.
The first application of the lemma is A ≠ B ∧ collinear A B U0 ∧ collinear A B X ∧ collinear A B U ∧ is-right-angle U X V ∧ U ≠ X → is-right-angle U0 X V.
• (→ (∧ (∧ (∧ (∧ (∧ (¬ (= A B)) (¬ (= C D))) (collinear X A B)) (collinear X C D)) (∧ (∧ (∧ (∧ (collinear U A B) (collinear V C D)) (¬ (= U X))) (¬ (= V X))) (is-right-angle U X V))) (∧ (collinear U0 A B) (collinear V0 C D))) (is-right-angle U0 X V)) (PerpendicularAtThereExists-u0-x-v)
Applying the lemma a second time is only slightly more complicated.
• (→ (∧ (∧ (∧ (∧ (∧ (¬ (= A B)) (¬ (= C D))) (collinear X A B)) (collinear X C D)) (∧ (∧ (∧ (∧ (collinear U A B) (collinear V C D)) (¬ (= U X))) (¬ (= V X))) (is-right-angle U X V))) (∧ (collinear U0 A B) (collinear V0 C D))) (is-right-angle U0 X V0)) (PerpendicularAtThereExists-u0-x-v0)
Now we need to handle the logic to turn that into our desired theorem.
• ( → (∧ (∧ (∧ (∧ (¬ (= A B)) (¬ (= C D))) (collinear X A B)) (collinear X C D)) (∃ u (∃ v (∧ (∧ (∧ (∧ (collinear u A B) (collinear v C D)) (¬ (= u X))) (¬ (= v X))) (is-right-angle u X v))))) (→ (∧ (collinear U0 A B) (collinear V0 C D)) (is-right-angle U0 X V0))) (PerpendicularAtThereExists-1)
• ( → (∧ (∧ (∧ (∧ (¬ (= A B)) (¬ (= C D))) (collinear X A B)) (collinear X C D)) (∃ u (∃ v (∧ (∧ (∧ (∧ (collinear u A B) (collinear v C D)) (¬ (= u X))) (¬ (= v X))) (is-right-angle u X v))))) (⟂at A B C D X) ) (PerpendicularAtThereExists)
### Perpendicular lines meet at right angles
Narboux doesn't explicitly state the following lemma (apparently his coq tactics cover it), but it states that if lines are perpendicular at a point, they form a right angle there. The proof may seem a bit long, but the idea is simple: expand A B C D ⟂at X according to the definition to get ∀ u ∀ v (collinear u A B ∧ collinear v C D → is-right-angle u X v). Substitute A for u and C for v to get collinear A A B ∧ collinear C C D → is-right-angle A X C, and then detach collinear A A B and collinear C C D as they are theorems.
### Builder
Equals can be substituted for equals, in the context of ⟂at. Because the definition of ⟂at is long, this proof is kind of long, but it is just a straightforward application of the builders for everything making up the definition of ⟂at.
• (→ (∧ (∧ (∧ (∧ (= A A′) (= B B′)) (= C C′)) (= D D′)) (= X X′)) (↔ (⟂at A B C D X) (⟂at A′ B′ C′ D′ X′))) (PerpendicularAtBuilder)
### Commutativity
As with line segment congruence, we use the word ''commutativity'' to refer to exchanging the points within each line (that is, A B C D ⟂at X ↔ B A D C ⟂at X), and the word ''symmetry'' to exchanging the two lines (A B C D ⟂at X ↔ C D A B ⟂at X).
We already proved symmetry, so next is commutativity on the left side.<ref>perp_in_left_commutativity in Narboux</ref> The proof is a straightforward exercise in expanding the definition and then commuting the relevant pieces.
Right commutativity follows from left commutativity and symmetry.<ref>perp_in_right_commutativity in Narboux</ref>
Commutativity follows from left and right commutativity.<ref>perp_in_commutativity in Narboux</ref>
### No line is perpendicular to itself
Here we show ¬ A B A B ⟂at X. The proof is by contradiction: A ≠ B from the definition, but A = B will follow from two applications of RightAngleLegItself.<ref>l8_14_1 in Narboux</ref>
## Export
We now export to Interface:Orthogonality 2.
Also, since this is currently the last proof module for geometry, we export to Interface:Basic geometry.
## References
<references/>
• Tarski, Alfred; Givant, Steven (1999), "Tarski's system of geometry", The Bulletin of Symbolic Logic 5 (2): 175–214, doi:10.2307/421089, MR1791303, ISSN 1079-8986
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28. The Relationship between the 0.729166667" digit and the Egyptian Royal Cubit of 20.625"
Updated: Feb 23, 2022
The 0.729166667" digit is an important unit. It goes 1,575,000,000 times into the circumference of the earth (the polar circumference of 24,857.95454545 miles to be exact), and it fits well with the remen of 14.583333", this remen consisting of 20 of these digits. But the 20.625" Egyptian Royal cubit seems not to fit, unless of course by means of the square root of 2. And this itself has to be taken as 99/70. So we have 20.625 x 7 / (2 x 99) = 0.729166667. The relationship between the digit and the Egyptian Royal Cubit, as 0.729166667" and 20.625" respectively, is of course as the side of a square is to it's diagonal.
However, the relationship between the 0.729166667" digit and the Egyptian Royal Cubit of 20.625" could also be though of as this: the digit goes in 1,400,000 x 4,374 / 4,375 times into a hypothetical 25,920 ancient metres unit of length (using the 39.375" conversion to metres). And the 20.625" cubit goes into this 25,920 mm unit 49,500 x 4374 / 4375 x 9,801 / 9,800 times. Royal Egyptian cubit and digit, as 20.625" and 0.729166667", are linked like this: cubit x 49,500 / 1,400,000 x 9,800/9,801 = digit.
This also has the advantage of linking up with the Neal / Michell value for the Egyptian Royal Cubit of 20.618181818", as 25,920 x 39.375 / 49,500 = 20.6181818181. This is because the link between these two cubits, the 20.6181818" one and the 20.625" one, is 9,801/9,800 x 4,375/4,374.
So we could define the digit of 0.7291666667" as simply 25,920 ancient metres divided by 49,500, and multiplied by 4,375/4,374 (ragisma) and 9,801/9,800.
This offers an intriguing connection between the digit, the inch and the ancient metre.
A hypothetical unit of 25,920 ancient metres in fact lends itself to many other possible connections. For example, 25,920 / 49,500 = 0.523636363..., and this is the value in metres of the Neal / Michell Royal Egyptian Cubit, otherwise known as 1,134/55 = 20.618181818 inches. The 1,134 part of this fraction is in fact 25,920 x 4,375 / 100,000. The 55 part of the fraction is 49,500 / 9, and also links up with an approximation of Phi or Phi squares using the Fibonacci numbers. For example 55/34 = 1.617647, and 144/55 = 2.618181818.
There is good reason to think of these values in metres, of the ancient kind at least, using the 39.375" conversion. For example, 25,920 / 49,500 = 0.523636363... is 2.618181818 + 3.141818181818, these last two numbers being values for Phi squared (144/55) and pi (144/55 x 6/5).
To convert from the ancient metre to the modern one, the 8,001/8,000 ratio works perfectly.
And so the digit of 0.7291666667" is 25,920 / 140 x 4,375 / 4,374 ancient mm, or 25,920 / 140 x 4,375 / 4,374 x 39.375 / 10,000 ancient mm, multiplied by 8,000/8,001 to obtain the modern metre value. (39.375 = 10,000 / 254 x 8,001/8,000)
The 39.375 conversion between ancient metre and inch is in fact 9 x 4,375.
The 0.729166667" digit is itself already connected to the ragisma 4,375/4,374, as it can also be written 9 x 9 x 9 / 1,000 x 4,375 / 4,374 . Curiously, 25,920 / 140 x 4,375 = 81 = 9 x 9. This means then that the digit expressed in ancient centimetres, 1.851851851851, is 9 x 9 / 4,374.
And expressed in inches, that's simply 9 x 9 x 9 x 4375/ 4,374 x 1/1,000.
The remen of 14.5833333", in relation to 25,920 ancient mm, is 25920 / 1,800 x 7,875 / 7,776, or 14.4 x 7,875/7,776. The remen is an Egyptian Royal Cubit of 20.625" divided by 99/70. This approximation of root 2 is also 25,920 / (2.618181818 x 7,000).
7,875/7,776 is an interesting ratio that comes up from time to time in metrology, and here it seems to bridge the values in inches and metres between the 25,920 mm unit and the 14.583333" remen. 7,875 inches are in fact 200 ancient metres. And 14.4 / 7,776 is in fact the digit in ancient metres: 0.0185185185....
The 0.7291666667" or 1.851851851 (ancient) cm digit is 2 (ancient) metres / 1,080.
The moon's radius is 1,080 miles.
I thought I might try and see if the 0.729166667" digit might connect to other units.
I went back to a text I was looking at a few months ago by Mauss, to see how the measures he gave fitted in with a 0.72916666" digit. The first unit he mentions is the Assyrian and Persian Royal Cubit, also referred to as the "grande Hachémique", which he assigns 658.285 mm to. The conversion rate for all his measures is 39.375" to the metre. We know this because he gives the value of the English yard in millimetres as 914.2857. This article is from 1892, so before the 1897 Weights and Measures Act, and long before 1930 when the 25.4 mm inch was adopted. The conversion rate is curious however as I've not been able to find any mentrion anywhere of this rate ever having been in existence. Still, it ties in perfectly with the 0.7291666667" digit.
914.2857, which you could take as 6.4/7, multplied by 39.375 = 36, so 36 inches.
Interestingly, 6.4 m = 252", so 7 yards, is quite compatible with the 0.729166667" digit.
6.4 metres are obviously 54 x 0.72916667 x 6.4, or perhaps also 21.6 x 16 digits, 14.4 x 24 digits, 12.8 x 27 digits, etc.
This makes the yard 14.4 x 24 x 0.72916667 / 7 inches = 12 x 12 x 12 x 2 x 0.729166667 / 70", and the English foot = 115.2 / 7 x 0.729166667"
To go back to the Assyrian and Persian royal cubit that Mauss writes about, 658.285 mm, this is also a seventh division of something. 4608 / 7 = 658.2857142857 mm = 25,920 / 1,000 inches, which is 2.16 English feet, and also 248.832 / 7 digits, or 12⁵ / 7,000.
25.92" for an Assyrian cubit is an interesting number. In the article on weights and measures here, an Assyrian cubit is mentioned, "a royal cubit of 7/6 the U cubit, or 25.20, and four monuments show a cubit averaging 25.28 (...) we may take 25.24 as the nearest approach to the ancient Persian unit". (p 4 on the webpage)
The 25.2" value would work well with the 0.729166667" digit, being 2 x 12³ / 100 digits.
The same article gives the size of the "U" as 10.806 inches. If it were in fact 10.8 inches, that would also go well with the digit, as 12³ x 6/700 digits. (In fact in the little table below this sentence in the article the "U" is down as 10.80, 6 of these are a qanu, and there are more multiples of this, worth 129.6", 648", 7776" and 233,280", whose names are hard to read.)
The article then says there is a 2"U" unit of 21.6", and this ties in well with the Mauss article.
Mauss has a unit which is 5/6 of the royal Assyrian and Persian cubit, which he calls the worker's cubit, or Coudée ouvriere, worth, as he states it, 548.571 mm, which is also 3,840 / 7 mm, and which is also 21.6" exactly.
The 21.6" unit doesn't seem to work with the digit at first, but bring in the number 7 again and it does: 21.6" x 7 = 151.2" = 12⁴ x 0.729166667 / 100
Royal assyrian cubit = 4,608 / 7 mm = 12⁵ / 7,000 digits = 25.92"
Worker's cubit = 3,840 / 7 mm = 12⁴ / 700 digits = 21.6"
This worker's unit of 548.57142 mm is also 3 digits x 144/55 x 6/5 x 22/7, so the two pis, 3.142857 and 3.141818181. Three digits are perhaps a tenth of Ezekiel's cubit, as Jim Alison says in his article: "Ezekiel’s cubit of 30 digits, which is contained 72,000,000 times in the polar circumference, or 18,000,000 times in the quarter circumference, or 200,000 times in one degree of latitude"
Royal foot = 2,304 / 7 mm = `12² x 16 / 7 mm = 6 x 12⁴ / 700 digits = 12.96"
`Royal foot x 7/8 = 288 mm = 11.34" = 55/100 x 20.6181818 (Michell / Neal Egyptian Egyptian cubit)
So the 20.6181818" cubit = 6 x 12⁴ / 440 digits = 28.8 / 55 = 0.52363636 metres = 2 x Phi squared / 10 = metres (with Phi squared as 144 / 55). This would make the metre an integral part of Neal and Michell's Egyptian cubit, via Phi squared. (And of course, as pointed out earlier, this unit of 0.52363636.. = 3.1418181818... - 2.618181818... metres, or (144/55 x 6/5) - (144/55) metres.)
Other units mentioned in the Mauss article include:
• Dieulafoy's worker's cubit = 550 mm = 20.625 x 24/25 x 3/2 digits = 21.65625"
• Dieulafoy's worker's foot = 330 mm = 20.625 x 24/25 x 18/20 digits = 12.99375"
• Dieulafoy's royal cubit = 660 mm = 20.625 x 24/25 x 18/10 digits = 25.9875"
• Watering cubit ("coudée de l'arroseur") = 576 mm = 126 x 12³ / 7,000 digits = 22.68"
• Hand cubit ("coudée de la main") = 480 mm = 105 x 12³ / 7,000 = 18.9". This is also the quim cubit and divides into 24 digits of 0.7875".
• Iron cubit / Black cubit ("coudée de fer / coudée noire") = 540 mm = 105 x 12³ x 9 / (8 x 7,000) digits = 21.2625". This is also the same as the "coudée des étoffes", the fabric cubit, which divides up into 27 digits. These digits would be of 0.7875", which is 0.729166667 x 1.08 inches, same as the hand cubit digits. After all, a digit of 0.729166667" or 1.851851851 ancient cm, is 2 / 108 ancient metres.
32 of these digits of 0.7875", or 2 ancient cm, make up the Hachemi cubit of 0.64 metres = 25.2"
These 0.7875" digits connect back to the English foot as 320 / 21 digits of 0.7875". The Egyptian Royal cubit of 20.625" contains 550/21 of these digits. The Neal / Michell Egyptian cubit contains 10 x Phi squared of these cubits, or 10 x 144/55. The 14.58333" remen would contain 2000 / (3 x 36) of these 0.7875" digits. This remen is also 4/9 x 2.618181818 / 3.1418181818 metres. If you take 1,000 of these digits of 0.7875" that's also 20 metres.
It's actually surprisingly helpful to think of the digit and Egyptian / Royal Cubit in terms of metres, however much certain researchers might like to disagree. The idea of a hypothetical unit of 25,920 ancient metres is intriguing.
The 39.375" 'metre' is just another subdivision of the 1,575,000,000" meridian circumference, and a third of it is Jim Alison's Northern foot of 13.125", which is Jim Wakefield's 13.2" inch Saxon foot x 175/176, 13.2 x 100,000 x 10/3 x 360 = 1,584,000,000. The question is always going to be what was the value for the meridian circumference in the first place? the 12" English foot relates to the 1,575,000,000" circumference, as 1,575,000,000 x 9 x 176 / (360 x 12 x 250,000 x 1.1 x 175) = 12, and the 0.729166667" digit is the 12" foot multiplied by 1.1, 175/176, and divided by 18.
Are all units of measure linked? The idea has come up in the work of various researchers recently. It was also a common idea in 18th century France. Bailly, Letronne, and Gosselin found in their research that many units of measure could be traced back to an accurate survey of the size of the planet, in particular to the length of the polar meridian. Paucton, also from the same period, wrote about various types of feet being already integer parts of a degree. He also says if you're designing a new system, you should really take one particular length for a degree and stick to it, or else it will be too confusing when people travel abroad. And he says that the project of post-revolutionary France in implementing a universal system of measure based on an exact division of the earth's meridian arc is actually the very same project that was undertaken in the "remotest antiquity", when the units designed were based precisely on a meridian degree.
Quote Paucton Voilà préciſément quel étoit le ſyſtême métrique des peuples dans l'antiquité la plus reculée . Cette partie de la légiſlation leur avoit paru mériter une attention particuliere . Ils fixerent d'une maniere irrévocable leurs meſures en les rendant dépendantes de ta grandeur d'un degré du Méridien . Ils en prirent préciſément la quatre cent millieme partie qu'ils appellerent , tantôt pied & tantôt coudée (...)
This is precisely the measurement system of the peoples of the remotest antiquity. This part of their legislation seemed to warrant particular attention. They irrevocably fixed their measures making them dependant on the length of a meridian degree. They tool precisely the 400,000th part, which they sometimes called foor, sometimes cubit (...)
Jim Alison recently wrote on GHMB:
The heart of Washington DC is defined by the NS sections 3-6 and the EW sections B-D. This forms a perfect square of 3600 x 3600 meters, or 10,800 x 10,800 Northern feet, with an area of 116,640,000 square Northern feet. I wonder where L'enfant was coming from with that?
Jim has done some amazing work on the city of Washington D.C., see here. He has found that the metre, albeit in its ancient form of 39.375", comes up in surprising places, for example in the US, where the metre was never actually adopted, despite the closeness of the founders of the country with the French revolutionaries who implemented the metre in France as part of their design for a new country. L'Enfant, who was responsible for designing the city of Washington, was of course French, and both a revolutionary anda trained painter at the French Academy, and trained by his father who himself wa a court painter. Jim Alison has shown that L'Enfant used a grid defined in metres to place his design for the city in.
In the same post, Jim Alison also wrote:
If, instead of saying the proposed metric system was based on the most recent, most extensive and most accurate global survey in the history of the world, they had said their proposed metric system was the same as the oldest system of measurement in the history of the world, would the rest of the world, or even the French, have considered adopting it?
One of my webpages about the global alignment of Teotihuacan, Washington DC, Stonehenge, Troy, etc., has to do with the street plan of Washington DC that was designed by Pierre L'enfant. Although the U.S. rejected the metric system, Washington DC, despite the diagonal avenues, is based on a due NS-EW street grid, and a metric grid, with NS lengths of 900 meters, and EW lengths of 1200 meters, defines the locations of the main buildings and monuments and the slopes, angles and distances of the diagonal avenues.
See his webpage here.
I wondered if you could extend Jim's grid slightly to the east to include the hospital / prison site, which is at the apex of on of the huge triangles in the city's design. In another post, I wrote about how if you superimpose a picture of Orion's Belt onto L'Enfant's plan, it forms a nice little trio with the other two more important sites: the White House and the Capitol. In the mind of a revolutionary, perhaps the place where 'broken' citizens go to get 'fixed' should be placed in an almost equally important part of the city as the Congress and President's House, one which is directly linked to them.You can then get a nice 2:1 rectangle, with an area of 259,200 square km. Again this number: 25,920, that I was imagining as a possible unit of measure that might make sense of the relationship between the digit of 0.72916666" and the Royal cubit of 20.625".
Here is my variation on Jim Alison's grid.
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slides9_v1
This works exactly if population values have a normal
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Unformatted text preview: is about the Population Mean: Case 1 σ known Case 1: σ known We start with unrealistic case where σ is known. This works exactly if population values have a normal distribution, and approximately if not. One-Tailed Tests A left-tailed test has the H0 and H1 : H0 :µ ≥ µ0 H1 :µ < µ0 A right-tailed test has the H0 and H1 : H0 :µ ≤ µ0 H 1 :µ > µ 0 Utku Suleymanoglu (UMich) Hypothesis Testing 9 / 39 Testing Hypothesis about the Population Mean: Case 1 σ known Test statistic for tests with known σ ’s will have the test statistic: z= x − µ0 ¯ √ σ/ n Now, we need to come up with a testing criteria. There are two equivalent ways of doing this: p-value approach Critical value (rejection region) approach These are best explained with an example. We will discuss the logic of hypothesis testing with this example. Important Note: We will discuss hypothesis testing regarding µ and p in different scenarios. The first scenario is for µ where σ is known. I will spend an extra amount of time on this case to explain to you the logic of hypothesis testing. This logic carries through everything we are going to do, so I will not repeat it again. Don’t mistake me spending a lot of time on the first case for other cases not being important. Utku Suleymanoglu (UMich) Hypothesis Testing 10 / 39 Testing Hypothesis about the Population Mean: Case 1 σ known Long Running Example Suppose you think the average lifespan of energy-saving light bulbs is less than 3 years. You collect a sample of 25 newly builty bulbs and measure their lifespan. You get x = 2.5. You ¯ (somehow) know standard deviation of their lifespan is σ = 1.5. Then we have the hypotheses: H0 :µ ≥ 3 H 1 :µ < 3 This is a left-tailed test. Relevant test statistic for this test (for all Case 1 cases, right or left-tailed or two-tailed) is: z= x − µ0 ¯ 2.5 − 3 = −1.66 √= σ/ n 1.5/5 We will see why we use this. Utku Suleymanoglu (UMich) Hypothesis Testing 11 / 39 Testing Hypothesis about the...
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This note was uploaded on 03/17/2014 for the course ECON 404 taught by Professor Staff during the Spring '08 term at University of Michigan.
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https://www.physicsforums.com/threads/power-loss-ac-vs-dc.975217/page-2
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# B Power loss: AC vs DC
#### DaveE
But something i read from someone very recently tells me that current is actually pushed by the electric field that goes along the whole wire and that electrons don't actually push on each other at all like megnetic freight carts on a track. Is this true?
Really, if you want to try to understand electricity, you need to give up on analogies like water, marbles and magnetic freight carts.
#### mfb
Mentor
How though? I know power drained by a resistor is P = I*I *R so lower I means lower power drained but how can you deliver higher voltage with less current when voltage and current are proportional? How do you wind up with the same amount of power being carried?
And, isn't the load at the end of the cable, like a vacuum cleaner for instance, just another resistance in the circuit, so aren't you just lowering the power the vacuum clearer gets?
You want to deliver a specific power P to the consumer. Pconsumer=Ugrid I. A higher voltage means you need a lower current to deliver the same power.
Losses in the cable are Pcable = I2 Rcable. A lower current means lower losses in the cable.
We can also plug the first equation into the second: Pcable = Rcable P2consumer/(U2grid). We can't change the power the consumer wants. We can lower the cable resistance, and we can increase the cable voltage to reduce the power lost in the cable.
#### Nugatory
Mentor
How though? I know power drained by a resistor is P = I*I *R so lower I means lower power drained but how can you deliver higher voltage with less current when voltage and current are proportional? How do you wind up with the same amount of power being carried?
And, isn't the load at the end of the cable, like a vacuum cleaner for instance, just another resistance in the circuit, so aren't you just lowering the power the vacuum clearer gets?
You can think of the load as another resistance but you must remember that the load is not the vacuum cleaner itself, but rather the step-down transformer at the end of the transmission line. Its resistance will vary with the power demand so the voltage and the current in the transmission line are not proportional. Instead, we have V=IR with R varying and V held constant by the generator so that I and R are inversely proportional to one another.
#### anorlunda
Mentor
Gold Member
Maxwell's Equations tell us that electromagnetic effects propagate at a finite speed. That speed is c in a vacuum. In other media, such as a wire the propagation speed is in the range 0.6-0.8 c.
At those speeds, AC versus DC is irrelevant. Ordinary circuit analysis breaks down and you must treat wires as waveguides, and antennas.
If you are trying to teach students about that, then using circuits is the wrong approach. You need to consider Maxwell's Equations. Unfortunately, the math is difficult so it is not usually taught at the high school level, but rather in field theory courses to college seniors or post graduate students.
#### vanhees71
Gold Member
This math will also tell you that for dispersive media it is very important to clearly define which "speed of propagation" you are talking about... (phase velocity, group velocity, front velocity,...).
#### John3509
Really, if you want to try to understand electricity, you need to give up on analogies like water, marbles and magnetic freight carts.
Why? When I use the analogy of magnetic fright cars to describe electrons pushing each other down the wire in case someone does does not know what I mean by electrons pushing each other down the wire. I now know electrons may not literally push each other down the wire but whats wrong with using a an anology to just convey the idea of electrons pushing each other?
#### DaveE
Why? When I use the analogy of magnetic fright cars to describe electrons pushing each other down the wire in case someone does does not know what I mean by electrons pushing each other down the wire. I now know electrons may not literally push each other down the wire but whats wrong with using a an anology to just convey the idea of electrons pushing each other?
Analogies are ok in the proper context, but analogies are always wrong. That's why they're called analogies. If you really want to understand a subject, you need to get past that and study the subject for what it is, not what it is like.
#### John3509
You can think of the load as another resistance but you must remember that the load is not the vacuum cleaner itself, but rather the step-down transformer at the end of the transmission line. Its resistance will vary with the power demand so the voltage and the current in the transmission line are not proportional. Instead, we have V=IR with R varying and V held constant by the generator so that I and R are inversely proportional to one another.
But if I and R are inversely proportional does that still means V and I are proportional?
#### John3509
You want to deliver a specific power P to the consumer. Pconsumer=Ugrid I. A higher voltage means you need a lower current to deliver the same power.
Losses in the cable are Pcable = I2 Rcable. A lower current means lower losses in the cable.
We can also plug the first equation into the second: Pcable = Rcable P2consumer/(U2grid). We can't change the power the consumer wants. We can lower the cable resistance, and we can increase the cable voltage to reduce the power lost in the cable.
Here is what confuses me about this, the load of the consumer or transformer to me exact, is another resistor in the circuit, you are minimizing the poser consumed by one resistence but maximising it for the other? How can that work? Im guessing it has something to do with the fact that you have 2 different P's in your equation, just not sure how to put the pieces together in my mind.
and V = IR, how to you achieve delivering the current with high voltage but low current?
#### mfb
Mentor
The power you have to deliver to the consumer is determined by the consumer - you can't change that if you want to keep the grid healthy.
With a given power arriving at the customer you want to minimize the power loss in the grid. Reducing the resistance of the cables is an obvious way to do so, increasing the transmission voltage is another one.
If high voltage safety and insulation wouldn't be an issue we could deliver this high voltage directly to the customer, the customer would use a very large resistance (as P=V2/R), this resistance can be larger for high voltages, and it can be much larger than the resistance of the cable. You can't do that in practice, of course, so the voltage is transformed down near the customer.
#### jbriggs444
Homework Helper
But if I and R are inversely proportional does that still means V and I are proportional?
I and R are inversely proportional if you hold V constant. Which means that this inverse proportionality, by itself, gives you exactly zero information on what happens if V is allowed to vary.
V and I are directly proportional if you hold R constant. Which means that this direct proportionality, by itself, gives you exactly zero information on what happens if R is allowed to vary.
In the case at hand, R is not held constant while you vary V. It varies, for one thing, because you will use different step down transformers if you choose to run your transmission lines at 20,000 volts versus at 40,000 volts.
#### Nugatory
Mentor
But if I and R are inversely proportional does that still means V and I are proportional?
No. If I and R are inversely proportional with V constant, then V and I cannot be proportional - one of them is fixed and the other is not.
#### John3509
I and R are inversely proportional if you hold V constant. Which means that this inverse proportionality, by itself, gives you exactly zero information on what happens if V is allowed to vary.
V and I are directly proportional if you hold R constant. Which means that this direct proportionality, by itself, gives you exactly zero information on what happens if R is allowed to vary.
In the case at hand, R is not held constant while you vary V. It varies, for one thing, because you will use different step down transformers if you choose to run your transmission lines at 20,000 volts versus at 40,000 volts.
Oh right, one of them always has to be a constant of proportionality, now I remember.
#### John3509
So I had some time to think about it, but its just not making sense to me for some reason.
Here is how I see it
As electrons pass a resistor they loose potential, that is the voltage drop, this happening over time is the power the resistor drains
You cant control current directly, only indirectly by controlling the things that effect it, resistance and voltage.
What I don't get is:
How are you controlling the current in power distribution systems?
How can you achieve a high voltage but low current? Wouldn't increasing the transmission voltage increase the current?
Also the wording "power transmitted" is throwing me off, I know I used it too but now that I think about it it makes no sense to me, how I understand it is power used up by a resistor or load.
And finally, Im imagining a system with two resisters, one representing the total resistance in the wiring and one for the consumer load. If you minimize the amount of power consumed by the wire resistor by lowering current, aren't you also minimizing the power used by the consumer? Ideally you can have no power lost by setting current to 0, but then the consumer appliances will have no power to consume eighter. So the power both resistances consume has to be proportional right? isn't it in series?
#### vanhees71
Gold Member
The "power drain" in a resistor is due to dissipation. In the most simple classical picture you have conduction electrons in the wire which can move quasi freely, but there's friction. If you have a DC voltage after some short time you have a constant current density, i.e., the electrons are not accelerated anymore due to the electric field. That's the case when the friction force is as large as the electric force on that electrons.
The power transmission is, BTW, not through electron transport (that wouldn't make sense in the AC case at all, because there the electrons stay more or less where they are) but through the electromagnetic field. It is very illuminating to analyse the coaxial cable for DC as well as AC in detail. The DC case if masterfully discussed in
A. Sommerfeld, Lectures on Theoretical Physics, Vol. 3
#### Nugatory
Mentor
What I don't get is:
How are you controlling the current in power distribution systems?
You don’t. You control the voltage (holding it fixed) and the current varies with the load. For example, the power company supplies power at 240 volts to my house. When I’m running a device using 240 W the current in the supply wires to my house is .1 A; turn on a second such device and the effective resistance is halved and the current doubles to .2 A so that twice as much power is being transmitted at the same voltage.
How can you achieve a high voltage but low current? Wouldn't increasing the transmission voltage increase the current?
You are forgetting the stepdown transformer. If the power company is using a 2400 V transmission line to get power to my house (where everything runs on 240 V) they will install a 10:1 stepdown transformer at the end of line and connect my house supply line to that. If they decide that a 9600 V transmission line makes more sense, they will replace the 10:1 stepdown transformer with a 40:1 one as part of the change. Nothing will change for me, I’m still getting 240 V in my house but now the power company transmission line can deliver the same amount of power with 1/4 the current.
And finally, Im imagining a system with two resisters, one representing the total resistance in the wiring and one for the consumer load. If you minimize the amount of power consumed by the wire resistor by lowering current, aren't you also minimizing the power used by the consumer? .... So the power both resistances consume has to be proportional right? isn't it in series?
You are forgetting the stepdown transformer again. The consumer appliances are not in series with the transmission line resistance; they are in series with my household wiring which is connected to the output of the stepdown transformer. It is the stepdown transformer that is in series with the transmission line resistance. The power “consumed” by it is of course the power that’s running the appliances in my house; it’s resistance varies with the load so the current in the line also varies while the voltage is fixed.
#### sophiecentaur
Gold Member
Im imagining a system with two resisters, one representing the total resistance in the wiring and one for the consumer load. If you minimize the amount of power consumed by the wire resistor by lowering current, aren't you also minimizing the power used by the consumer?
There is a logic in the way you need to approach this. You start with a SUPPLY VOLTAGE at the generator. The appliance has a particular resistance (based on the Power it is designed for) and that determines the amount of Current that flows.
A practical point to make here is that the supply cables and generator are made with as LOW a resistance as is practical (only a few Ohms total). The current running through your appliance also flows through the cables and they will get a bit warm and waste Power. If you were using 100V mains voltage then the current for a given power (say 1kW) will be 10A. Change the system so that you are using 200V supply and a suitable 1kW appliance (you need a different one) will use 5A.
The resistance of the supply cables has very little effect on the 1kW figure (we ignore any change for simplicity to start with) so you have half the current going through the 200V system, which means 1/4 of the power dissipated. (P=I2R).
The only possible problem is that the risk of Shock has gone up a bit. But the number of accidents in Europe are not significant compared with US so a higher operating voltage can be very good value.
#### mfb
Mentor
But the number of accidents in Europe are not significant compared with US so a higher operating voltage can be very good value.
This could also be a result of the plug types. Europlugs have no metal you can touch when there is an electric connection as parts of their pins are isolated. The US plugs don't have that isolation - you can easily touch a live metal part there.
#### sophiecentaur
Gold Member
This could also be a result of the plug types. Europlugs have no metal you can touch when there is an electric connection as parts of their pins are isolated. The US plugs don't have that isolation - you can easily touch a live metal part there.
Undoubtedly. It's something that is easily dealt with.
#### John3509
You don’t. You control the voltage (holding it fixed) and the current varies with the load. For example, the power company supplies power at 240 volts to my house. When I’m running a device using 240 W the current in the supply wires to my house is .1 A; turn on a second such device and the effective resistance is halved and the current doubles to .2 A so that twice as much power is being transmitted at the same voltage.
You are forgetting the stepdown transformer. If the power company is using a 2400 V transmission line to get power to my house (where everything runs on 240 V) they will install a 10:1 stepdown transformer at the end of line and connect my house supply line to that. If they decide that a 9600 V transmission line makes more sense, they will replace the 10:1 stepdown transformer with a 40:1 one as part of the change. Nothing will change for me, I’m still getting 240 V in my house but now the power company transmission line can deliver the same amount of power with 1/4 the current.
You are forgetting the stepdown transformer again. The consumer appliances are not in series with the transmission line resistance; they are in series with my household wiring which is connected to the output of the stepdown transformer. It is the stepdown transformer that is in series with the transmission line resistance. The power “consumed” by it is of course the power that’s running the appliances in my house; it’s resistance varies with the load so the current in the line also varies while the voltage is fixed.
So when you increase the voltage but also use a steeper step down transformer you can lower the current? I did not know transformers could do that, in that case that answers my question spot on. Thanks.
Back to the original topic, about the propagation of voltage through a long wire, someone at the beginning of the thread mentioned synchronization of the voltage is a problem on a long wire, can someone elaborate on what that means?
#### sophiecentaur
Gold Member
use a steeper step down transformer you can lower the current?
This is not the way it goes; there is a right way and a wrong way to think of a problem like this. Your domestic (250V) supply will provide a 1kW appliance with the current will pass (4A) because its resistance will be designed to be about 65Ω. The transformer will only need to take 1/100 of that current from its 25,000V supply to give 1kW (VI is the same). So the 25000 V supply will 'see' a load of 650,000Ω, when the load has been 'transformed' by the transformer.
Best to get used to that before trying to 'understand' how the currents flowing in the windings of a transformer and the magnetic fields in the Iron are related. Transformers are seldom totally understood by most of the people who use them and who design circuits with transformers in them. So no need to worry. Something to take up at your leisure.
"Power loss: AC vs DC"
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# (0, 0) : order 1; (0, 1) : order 4; (0, 2) : order 2; (0, 3) : order 4; (1, 0) : order 2; (1, 1) : order 4; (1, 2) : order 2; (1, 3) : order 4.
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## Transcription
1 11.01 List the elements of Z 2 Z 4. Find the order of each of the elements is this group cyclic? Solution: The elements of Z 2 Z 4 are: (0, 0) : order 1; (0, 1) : order 4; (0, 2) : order 2; (0, 3) : order 4; (1, 0) : order 2; (1, 1) : order 4; (1, 2) : order 2; (1, 3) : order 4. This group is not cyclic since no element can generate the whole group List the elements of Z 3 Z 4. Find the order of each of the elements is this group cyclic? Solution: The elements of Z 3 Z 4 are: (0, 0) : order 1; (0, 1) : order 4; (0, 2) : order 2; (0, 3) : order 4; (1, 0) : order 3; (1, 1) : order 12; (1, 2) : order 6; (1, 3) : order 12; (2, 0) : order 3; (2, 1) : order 12; (2, 2) : order 6; (2, 3) : order 12. This group is cyclic since it can be generated by either of the elements (1, 1), (1, 3), (2, 1), and (2, 3). 18
2 11.13 Disregarding the order of the factors, write direct products of two or more groups of the form Z n so that the resulting product is isomorphic to Z 60 in as many ways as possible. Solution: There are 4 different ways: Z 60 = Z 2 2 Z 3 Z 5 = Z 4 Z 3 Z 5, Z 60 = Z Z 5 = Z 12 Z 5, Z 60 = Z Z 3 = Z 20 Z 3, Z 60 = Z 2 2 Z 3 5 = Z 4 Z a. The cyclic subgroup of Z 24 generated by 18 has order 4. b. Z 3 Z 4 is of order 12. c. The element (4, 2) of Z 12 Z 8 has order 12. d. The Klein 4-group is isomorphic to Z 2 Z 2. e. Z 2 Z Z 4 has 8 elements of finite order Find the maximum possible order for some element of Z 4 Z 6. Solution: (1, 1) in Z 4 Z 6 has the maximum order lcm(4, 6) = Are the groups Z 8 Z 10 Z 24 and Z 4 Z 12 Z 40 isomorphic? Why or why not? Solution: We decompose both groups into indecomposible ones: Z 8 Z 10 Z 24 Z 8 (Z 2 Z 5 ) (Z 8 Z 3 ) = Z 2 (Z 8 ) 2 Z 3 Z 5, Z 4 Z 12 Z 40 Z 4 (Z 4 Z 3 ) (Z 8 Z 5 ) = (Z 4 ) 2 Z 8 Z 3 Z 5. So they are not isomorphic How many abelian groups (up to isomorphism) are there of order 24? of order 25? of order (24)(25)? Solution: 24 = = = So there are 3 abelian groups of order 24: Z 2 3 Z 3, Z 2 Z 2 2 Z 3, Z 2 Z 2 Z 2 Z = 5 2 = 5 5. So there are 2 abelian groups of order 25: Z 5 2, Z 5 Z 5. 19
3 Because gcd(24, 25) = 1, there are 3 2 = 6 abelian groups of order (24)(25): Z 2 3 Z 3 Z 5 2, Z 2 3 Z 3 Z 5 Z 5, Z 2 Z 2 2 Z 3 Z 5 2, Z 2 Z 2 2 Z 3 Z 5 Z 5, Z 2 Z 2 Z 2 Z 3 Z 5 2, Z 2 Z 2 Z 2 Z 3 Z 5 Z Mark each of the following true or false: a. (T) If G 1 and G 2 are any groups, then G 1 G 2 is always isomorphic to G 2 G 1. b. (T) Computation in an external direct product of groups is easy if you know how to compute in each component group. c. (F) Groups of finite order must be used to form an external direct product. d. (T) A group of prime order could not be the internal direct product of two proper nontrivial subgroups. e. (F) Z 2 Z 4 is isomorphic to Z 8. f. (F) Z 2 Z 4 is isomorphic to 8. g. (F) Z 3 Z 8 is isomorphic to 4. h. (F) Every element in Z 4 Z 8 has order 8. i. (F) The order of Z 12 Z 15 is 60. j. (T) Z m Z n has mn elements whether m and n are relatively prime or not a. How many subgroups of Z 5 Z 6 are isomorphic to Z 5 Z 6? Solution: No subgroup of Z 5 Z 6 is isomorphic to Z 5 Z 6. b. How many subgroups of Z Z are isomorphic to Z Z? Solution: There are infinite many subgroups of Z Z that are isomorphic to Z Z. They are of the form mz nz for positive integers m and n with m 1 or n Mark each of the following true or false: 20
4 a. (T) Every abelian group of prime order is cyclic. b. (F) Every abeliang roup of prime power order is cyclic. c. (F) Z 8 is generated by {4, 6}. d. (T) Z 8 is generated by {4, 5, 6}. e. (T) All finite abelian groups are classified up to isomorphism by Theorem f. (F) Any two finitely generated abelian gruops witht he same Betti number are isomorphic. g. (T) Every abelian group of order divisible by 5 contains a cyclic subgroup of order 5. h. (F) Every abelian group of order divisible by 4 contains a cyclic subgroup of order 4. i. (T) Every abelian group of order divisible by 6 contains a cyclic subgroup of order 6. j. (T) Every finite abelian group has a Betti number of Prove that a direct product of abelian groups is abelian. Solution: Suppose G i are abelian groups. We prove that n i=1 G i is an abelian group. Let (a 1,, a n ) and (b 1,, b n ) be elements of n i=1 G i. Then (a 1,, a n )(b 1,, b n ) = (a 1 b 1,, a n b n ) = (b 1 a 1,, b n a n ) = (b 1,, b n )(a 1,, a n ). This shows that the binary operation on n i=1 G i is commutative. So n i=1 G i is an abelian group Let G be an abelian group. Let H be the subset of G consisiting of the identity e together with all elements of G of order 2. Show that H is a subgroup of G. Solution: We show that H meets the criteria of a subgroup of G: 1. (Closed) If a, b H, then a 2 = b 2 = e. So (ab) 2 = a 2 b 2 = e. This implies that ab H. 2. (Identity) The identity is in H by definition. 21
5 3. (Inverse) If a H, then a 2 = e and so a 1 = a H. Therefore, H is a subgroup of G Prove that if a finite abelian group has order a power of a prime p, then the order of every element in the group is a power of p. Can the hypothesis of commutativity be dropped? Why, or why not? Solution: Suppose a group G has the order p k for some prime p and some positive integer k. Then the order m of every element a of G divides the order p k of G. So m = p r for some integer 0 r k. That is, the order of a is a power of p. The hypothesis of commutativity can be dropped, since we do not use the commutativity in the above argument. 22
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https://seehuhn.github.io/MATH3714/I03-lm-output.html
| 1,712,919,817,000,000,000
|
text/html
|
crawl-data/CC-MAIN-2024-18/segments/1712296815919.75/warc/CC-MAIN-20240412101354-20240412131354-00191.warc.gz
| 458,601,674
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# Interlude: Understanding the lm() Output
We have seen that the lm() function returns an object which contains a lot of information about the fitted model. We can inspect this information by using the summary() function. The aim of this section is to help you understand how the output of summary() relates to the mathematical expressions we have considered in the previous sections.
We will use the stackloss dataset as an example:
m <- lm(stack.loss ~ ., data = stackloss)
summary(m)
Here I have marked different parts of the output using red shaded boxes and using the letters A to I. We discuss each of the marked sections in turn:
• The first section, part A, contains summary statistics for the fitted residuals $$\hat\varepsilon_1, \ldots, \hat\varepsilon_n$$. The values shown are the minimum, first quartile, median, third quartile, and maximum of the residuals. This is the same information we can get using the command summary(resid(m)). The mean is omitted from A, since if always equals zero.
• Column B shows the estimated coefficient vector $$\hat\beta$$. This is computed using the formula from lemma 2.1.
• Column C shows the standard error of the estimated coefficients. The $$i$$th entry is the (estimated) standard deviation of $$\hat\beta_i$$, computed as $$\sqrt{\hat\sigma^2 C_{ii}}$$, where $$C = (X^\top X)^{-1}$$. This corresponds to the variance shown in equation (4.2), where the true variance is $$\sigma^2$$ replaced with the estimate $$\hat\sigma^2$$. These quantities are used to compute the t-test statistic in equation (5.2).
• Column D shows the t-test statistic for the coefficients. The values are computed using equation (5.2).
• Column E replicates the information from column D in different form, showing $$p$$-values instead of the test statistics.
• The field F shows the estimated standard deviation $$\hat\sigma$$. This is computed as the square root of $$\hat\sigma^2$$ from equation (4.6).
• Field G shows the value $$n - p - 1$$. This is the number of degrees of freedom in lemma 5.2. The value is needed when performing hypothesis tests and computing confidence intervals for individual coefficients.
• Field H shows the $$R^2$$ value and adjusted $$R^2$$ value. These are computed using definitions 11.1 and 11.2.
• Field I shows the $$F$$-test statistic for testing the hypothesis $$H_i\colon \beta_1 = \cdots = \beta_p = 0$$ (omitting the coeficient $$\beta_0$$ for the intercept). This value can be used to test the hypothesis that the inputs have no effect on the output. The $$F$$-test statistic is computed using equation (6.1), the degrees of freedom shown are $$k = p$$ and $$n - p - 1$$ from lemma 6.1.
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| 2,674
|
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https://toolslick.com/math/calculator/decimal-to-percent/0.001
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|
text/html
|
crawl-data/CC-MAIN-2024-26/segments/1718198861342.11/warc/CC-MAIN-20240613025523-20240613055523-00741.warc.gz
| 533,550,532
| 13,255
|
# 0.001 as a percent
Nov 5, 2021
This tool takes a decimal number and converts it to a percentage.
Percentage:
..
Decimal value in percentage
Percentage is the representation of a ratio as a fraction of 100. The symbol % is suffixed to values to clearly indicate that they are percentages. The earliest usage of percentages was to levy taxes in Ancient rome.
Decimal is the numeric system most used by us humans. It consists of the Hindu-Arabic numerals, a set of 10 digits from 0 to 9.
Formula
#### Formula
Expressing a number as a percentage is very simple. All you have to do is multiply it by 100.
Percentage = d * 100
Where:-
• d = decimal number
Alternative Option
The other option is to just shift the decimal point two places to the right.
• 0.001 → 0.01
• 0.01 → 0.1
• Add a percent sign at the end: 0.1%
Example
Suppose we want to convert 0.001 to it's percent form. Then, according to the formula:-
• d * 100
• 0.001 * 100
• Percentage = 0.1
DecimalPercent
0.0010.1%
0.0020.2%
0.0030.3%
0.0040.4%
0.0050.5%
0.011%
0.022%
0.033%
0.044%
0.055%
0.110%
0.220%
0.330%
0.440%
0.550%
1100%
2200%
3300%
4400%
5500%
| 359
| 1,134
|
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https://testbook.com/question-answer/the-bar-graph-given-below-presents-the-production--5ea9451ff60d5d33d955442e
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# The bar graph given below presents the production of wheat (in tonnes) by a big farm during the years 2011 – 2018. In how many of the given years was the production of wheat greater than the average production of the period?
Free Practice With Testbook Mock Tests
## Options:
1. 4
2. 3
3. 2
4. 5
### Correct Answer: Option 1 (Solution Below)
This question was previously asked in
SSC MTS Previous Paper 27 (Held On: 16 August 2019 Shift 3)
## Solution:
Total production during the years = 2500 + 2000 + 6000 + 4500 + 6500 + 5000 + 7500 + 7000 = 41,000
Average production during the years = 41000/8 = 5,125
There are 4 years 2013, 2015, 2017 and 2018.
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# 加载中...
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## Chapter 13-1: Graphs-unweighted graphs
(2009-09-09 09:12:00)
### 杂谈
(1)
图是一种与树有些相像的数据结构。实际上，从数学意义上说，树是图的一种。然而，在计算机程序设计中，图的应用方式与树不同。
The data structures examined previously in this book have an architecture dictated by the algorithms used on them. For example, a binary tree is shaped the way it is because that shape makes it easy to search for data and insert new data. The edges in a tree represent quick ways to get from node to node.
Graphs, on the other hand, often have a shape dictated by a physical problem. For example, nodes in a graph may represent cities, while edges may represent airline flight routes between the cities. Another more abstract example is a graph representing the individual tasks necessary to complete a project. In the graph, nodes may represent tasks, while directed edges (with an arrow at one end) indicate which task must be completed before another. In both cases, the shape of the graph arises from the specific real-world situation.
Before going further, we must mention that, when discussing graphs, nodes are called vertices (the singular is vertex)(顶点). This is probably because the nomenclature for graphs is older than that for trees, having arisen in mathematics centuries ago. Trees are more closely associated with computer science.
（Two vertices are said to be adjacent to one another if they are connected by a single edge.）
Paths
A path is a sequence of edges. Figure 13.1 shows a path from vertex B to vertex J that passes through vertices A and E. We can call this path BAEJ. There can be more than one path between two vertices; another path from B to J is BCDJ.
Connected Graphs
A graph is said to be connected if there is at least one path from every vertex to every other vertex.
在非常抽象的图的问题中，只是简单地把顶点编号，从0到N-1（这里N是顶点数）。不需要任何变量类型存储顶点，因为它们的用处来自于它们之间的相互关系。
然而在大多数情况下，顶点表示某个真实世界的对象，这个对象必须用数据项来描述。例如，如果在一个飞机航线模拟程序中，顶点代表城市，那么它需要存储城市名字、海拔高度、地理位置和其他相关信息。因此，通常用一个顶点类的对象来表示一个顶点。
Our example programs store only a letter (like A), used as a label for identifying the vertex, and a flag for use in search algorithms, as we'll see later. Here's how the Vertex class looks:
class Vertex
{
public char label; // label (e.g. 'A')
public boolean wasVisited;
public Vertex(char lab) // constructor
{
label = lab;
wasVisited = false;
}
} // end class Vertex
A graph, however, doesn't usually have the same kind of fixed organization as a tree. In a binary tree, each node has a maximum of two children, but in a graph each vertex may be connected to an arbitrary number of other vertices.
To model this sort of free-form organization, a different approach to representing edges is preferable to that used for trees. Two methods are commonly used for graphs:the adjacency matrix and the adjacency list.
图不像树，拥有几种固定的结构。二叉树中，每个节点最多有两个子节点，但图的每个顶点可以与任意多个顶点连接。为了模拟这种自由形式的组织结构，需要用一种不同的方法表示边，比树的表示方法更合适些。一般用两个方法表示图：即邻接矩阵和邻接表。
(2)
邻接矩阵是一个二维数组，数据项表示两点间是否存在边。如果图有N个顶点，邻接矩阵就是N*N的数组。
主对角线上的实体不代表任何真实世界的信息，所以为了方便，也可以把主对角线的值设为1.
注意，这个矩形的上三角是下三角的镜像；两个三角包含了同样的信息。这个冗余信息看似低效，但在大多数计算机语言中，创造一个三角形数组比较困难，所以只好求其次接受这个冗余。这也要求当增加一条边时，必须更新邻接矩阵的两部分，而不是一部分。
(3)
表示边的另一种方法是邻接表。邻接表中的表指的是第5章“链表”中讨论的那种链表。实际上，邻接表是一个链表数组（或者是链表的链表）。每个单独的链表表示了有哪些顶点与当前顶点领接。
(4)
为了向图中添加顶点，必须用new保留字生成一个新的顶点对象，然后插入到顶点数组vertexList中。
邻接矩阵（或者邻接表）提供了关于当前顶点的位置信息。特别是，当前顶点通过边与哪些顶点相连。为了回答关于顶点序列的更一般问题，就必须求助于其他的算法。
(5)
在图中实现的最基本的操作之一就是搜索从一个指定顶点可以到达哪些顶点。
还有另外一种情形可能需要找到所有当前顶点可到达的顶点。
假设已经创建了这么一个图。现在需要一种算法来提供系统的方法，从某个特定的顶点开始，沿着边移动到其他顶点。移动完毕后，要保证访问了和起始点相连的每一个顶点。
There are two common approaches to searching a graph: depth-first search (DFS) and breadth-first search (BFS). Both will eventually reach all connected vertices. The difference is that the depth-first search is implemented with a stack, whereas the breadthfirst search is implemented with a queue. These mechanisms result, as we'll see, in the graph being searched in different ways.
有两种常用的方法可以用来搜索图：即深度优先搜索（DFS）和广度优先搜索（BFS）。它们最终都会到达所有连通的顶点。深度优先搜索通过栈来实现，而广度优先搜索通过队列实现。
(6)
在搜索到尽头的时候，深度优先搜索用栈记住下一步的走向。
为了实现深度优先搜索，找一个起始点——本例为顶点A。需要做三件事：首先访问该顶点，然后把该点放入栈中，以便记住它，最后标记该节点，这样就不会再访问它。
下面可以访问任何与顶点A相连的顶点，只要还没有访问过它。假设顶点按字母顺序访问，所以下面访问顶点B。然后标记它，并放入栈中。
深度优先搜索与迷宫问题类似。迷宫在英国很流行，可以由一方给另一方设置障碍，由另一方想办法通过。迷宫由狭窄的过道（认为是边）和过道的交汇点（顶点）组成。
Suppose that someone is lost in the maze. She knows there's an exit and plans to traverse the maze systematically to find it. Fortunately, she has a ball of string and a marker pen. She starts at some intersection and goes down a randomly chosen passage, unreeling the string. At the next intersection, she goes down another randomly chosen passage, and so on, until finally she reaches a dead end.
假设有个人在迷宫中迷路。她知道有一个出口，并且计划系统地搜索迷宫找到出口。幸运的是，她有一团线和一支笔。她从某个交汇点开始，任意选择一个通路，从线团上退下一些线。在下一个交汇点，她继续随机选择一条通路，再退下一些线，直到最后她到达死胡同。
At the dead end she retraces her path, reeling in the string, until she reaches the previous intersection. Here she marks the path she's been down so she won't take it again, and tries another path. When she's marked all the paths leading from that intersection, she returns to the previous intersection and repeats the process.
到达死胡同时，她按原路返回，在把线绕上，直到到达前一个交汇点。她标记了以前走过的路径，所以不会重复走那些通路，而选择未选择的通路。当她标记了这个交汇点的所有通路，就会再回到上一个交汇点，并且重复这个过程。
The string represents the stack: It "remembers" the path taken to reach a certain point.
线代表栈：它“记住”了走向某个特定点的路径。
// returns an unvisited vertex adj to v
{
for(int j=0; j<nVerts; j++)
return j;
return -1;
现在开始考察Graph类中的dfs()方法，这个方法实际执行了深度优先搜索。下面会看到这段代码如何包含了前面提出的三条规则。它循环执行，知道栈为空。每次循环中，它做四件事：
1. 用peek()方法检查栈顶的顶点.
2. 试图找到这个顶点还未访问的邻接点.
3. 如果没有找到,出栈。
4. 如果找到这样的顶点，访问这个顶点，并把它放入栈。
在dfs()方法的最后，重置了所有wasVisited标记位，这样就可以在稍后继续使用dfs()方法。栈此时已为空，所以不需要重置。
////////////////////////////////////////////////////////////////
class Vertex
{
public char label; // label (e.g. 'A')
public boolean wasVisited;
// ------------------------------------------------------------
public Vertex(char lab) // constructor
{
label = lab;
wasVisited = false;
}
// ------------------------------------------------------------
// end class Vertex
////////////////////////////////////////////////////////////////
class Graph
{
private final int MAX_VERTS = 20;
private Vertex vertexList[]; // list of vertices
private int nVerts; // current number of vertices
private StackX theStack;
// ------------------------------------------------------------
public Graph() // constructor
{
vertexList = new Vertex[MAX_VERTS];
nVerts = 0;
for(int y=0; y<MAX_VERTS; y++) // set adjacency
for(int x=0; x<MAX_VERTS; x++) // matrix to 0
theStack = new StackX();
// end constructor
// ------------------------------------------------------------
{
vertexList[nVerts++] = new Vertex(lab);
}
// ------------------------------------------------------------
public void addEdge(int start, int end)
{
}
// ------------------------------------------------------------
public void displayVertex(int v)
{
System.out.print(vertexList[v].label);
}
// ------------------------------------------------------------
public void dfs() // depth-first search
// begin at vertex 0
vertexList[0].wasVisited = true; // mark it
displayVertex(0); // display it
theStack.push(0); // push it
while( !theStack.isEmpty() ) // until stack empty,
{
// get an unvisited vertex adjacent to stack top
int v = getAdjUnvisitedVertex( theStack.peek() ); //peek()只是返回，不是弹出栈顶元素
if(v == -1) // if no such vertex,
theStack.pop();
else // if it exists,
{
vertexList[v].wasVisited = true; // mark it
displayVertex(v); // display it，标记只标记一次，显示也只一次
theStack.push(v); // push it
}
// end while
// stack is empty, so we're done
for(int j=0; j<nVerts; j++) // reset flags
vertexList[j].wasVisited = false;
// end dfs
// ------------------------------------------------------------
// returns an unvisited vertex adj to v
{
for(int j=0; j<nVerts; j++)
return j;
return -1;
// ------------------------------------------------------------
// end class Graph
(7)
正如深度优先搜索中看到的，算法表现得好像要尽快地远离起始点似的。相反，在广度优先搜索中，算法好像要尽可能地靠近起始点。它首先访问起始顶点的所有邻接点，然后再访问较远的区域。这种搜索不能用栈，而要用队列来实现。
A是起始点，所以访问它，并标记为当前顶点。然后应用下面几条规则：
At each moment, the queue contains the vertices that have been visited but whose neighbors have not yet been fully explored. (Contrast this with the depth-first search, where the contents of the stack is the route you took from the starting point to the current vertex.)
在每一时刻，队列所包含的顶点时那些本身已经被访问，而它的邻居还有未访问的顶点。（对比深度优先搜索，栈的内容是起始点到当前顶点经过的所有顶点。）
Graph类的bfs()方法和dfs()方法类似，只是用队列代替了栈，嵌套的循环代替了单层循环。外层循环等待队列为空，而内层循环依次寻找当前顶点的未访问邻接点。
广度优先搜索有一个有趣的属性：它首先找到与起始点相距一条边的所有顶点，然后是与起始点相距两条边的顶点，依次类推。如果要寻找起始顶点到指定顶点的最短距离，那么这个属性非常有用。首先执行BFS，当找到指定顶点时，就可以说这条路径是到这个顶点的最短路径。如果有更短的路径，BFS算法就应该已经找到过它了。
class Queue
{
private final int SIZE = 20;
private int[] queArray;
private int front;
private int rear;
// -------------------------------------------------------------
public Queue() // constructor
{
queArray = new int[SIZE];
front = 0;
rear = -1;
}
// -------------------------------------------------------------
public void insert(int j) // put item at rear of queue
{
if(rear == SIZE-1)
rear = -1;
queArray[++rear] = j;
}
// -------------------------------------------------------------
public int remove() // take item from front of queue
{
int temp = queArray[front++];
if(front == SIZE)
front = 0;
return temp;
}
// -------------------------------------------------------------
public boolean isEmpty() // true if queue is empty
{
return ( rear+1==front || (front+SIZE-1==rear) );
}
// -------------------------------------------------------------
// end class Queue
////////////////////////////////////////////////////////////////
class Vertex
{
public char label; // label (e.g. 'A')
public boolean wasVisited;
// -------------------------------------------------------------
public Vertex(char lab) // constructor
{
label = lab;
wasVisited = false;
}
// -------------------------------------------------------------
// end class Vertex
////////////////////////////////////////////////////////////////
class Graph
{
private final int MAX_VERTS = 20;
private Vertex vertexList[]; // list of vertices
private int nVerts; // current number of vertices
private Queue theQueue;
// ------------------------------------------------------------
public Graph() // constructor
{
vertexList = new Vertex[MAX_VERTS];
nVerts = 0;
for(int j=0; j<MAX_VERTS; j++) // set adjacency
for(int k=0; k<MAX_VERTS; k++) // matrix to 0
theQueue = new Queue();
// end constructor
// -------------------------------------------------------------
{
vertexList[nVerts++] = new Vertex(lab);
}
// -------------------------------------------------------------
public void addEdge(int start, int end)
{
}
// -------------------------------------------------------------
public void displayVertex(int v)
{
System.out.print(vertexList[v].label);
}
// -------------------------------------------------------------
public void bfs() // breadth-first search
// begin at vertex 0
vertexList[0].wasVisited = true; // mark it
displayVertex(0); // display it
theQueue.insert(0); // insert at tail
int v2;
while( !theQueue.isEmpty() ) // until queue empty,
{
int v1 = theQueue.remove(); // remove vertex at head
// until it has no unvisited neighbors
// get one,
vertexList[v2].wasVisited = true; // mark it
displayVertex(v2); // display it
theQueue.insert(v2); // insert it
// end while
// end while(queue not empty)
// queue is empty, so we're done
for(int j=0; j<nVerts; j++) // reset flags
vertexList[j].wasVisited = false;
// end bfs()
// -------------------------------------------------------------
// returns an unvisited vertex adj to v
{
for(int j=0; j<nVerts; j++)
return j;
return -1;
// -------------------------------------------------------------
// end class Graph
////////////////////////////////////////////////////////////////
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# Search by Topic
#### Resources tagged with Combinatorics similar to Beans Eating Beans:
Filter by: Content type:
Stage:
Challenge level:
### There are 40 results
Broad Topics > Decision Mathematics and Combinatorics > Combinatorics
### Tri-colour
##### Stage: 3 Challenge Level:
Six points are arranged in space so that no three are collinear. How many line segments can be formed by joining the points in pairs?
### How Many Dice?
##### Stage: 3 Challenge Level:
A standard die has the numbers 1, 2 and 3 are opposite 6, 5 and 4 respectively so that opposite faces add to 7? If you make standard dice by writing 1, 2, 3, 4, 5, 6 on blank cubes you will find. . . .
### Plum Tree
##### Stage: 4 and 5 Challenge Level:
Label this plum tree graph to make it totally magic!
### Shuffle Shriek
##### Stage: 3 Challenge Level:
Can you find all the 4-ball shuffles?
### Greetings
##### Stage: 3 Challenge Level:
From a group of any 4 students in a class of 30, each has exchanged Christmas cards with the other three. Show that some students have exchanged cards with all the other students in the class. How. . . .
### Knight Defeated
##### Stage: 4 Challenge Level:
The knight's move on a chess board is 2 steps in one direction and one step in the other direction. Prove that a knight cannot visit every square on the board once and only (a tour) on a 2 by n board. . . .
### Postage
##### Stage: 4 Challenge Level:
The country Sixtania prints postage stamps with only three values 6 lucres, 10 lucres and 15 lucres (where the currency is in lucres).Which values cannot be made up with combinations of these postage. . . .
### Euromaths
##### Stage: 3 Challenge Level:
How many ways can you write the word EUROMATHS by starting at the top left hand corner and taking the next letter by stepping one step down or one step to the right in a 5x5 array?
### Doodles
##### Stage: 4 Challenge Level:
Draw a 'doodle' - a closed intersecting curve drawn without taking pencil from paper. What can you prove about the intersections?
### Master Minding
##### Stage: 3 Challenge Level:
Your partner chooses two beads and places them side by side behind a screen. What is the minimum number of guesses you would need to be sure of guessing the two beads and their positions?
### An Investigation Based on Score
##### Stage: 3
Class 2YP from Madras College was inspired by the problem in NRICH to work out in how many ways the number 1999 could be expressed as the sum of 3 odd numbers, and this is their solution.
### Russian Cubes
##### Stage: 4 Challenge Level:
I want some cubes painted with three blue faces and three red faces. How many different cubes can be painted like that?
### Painting Cubes
##### Stage: 3 Challenge Level:
Imagine you have six different colours of paint. You paint a cube using a different colour for each of the six faces. How many different cubes can be painted using the same set of six colours?
### Permute It
##### Stage: 3 Challenge Level:
Take the numbers 1, 2, 3, 4 and 5 and imagine them written down in every possible order to give 5 digit numbers. Find the sum of the resulting numbers.
### Cube Paths
##### Stage: 3 Challenge Level:
Given a 2 by 2 by 2 skeletal cube with one route `down' the cube. How many routes are there from A to B?
### Ways of Summing Odd Numbers
##### Stage: 3
Sanjay Joshi, age 17, The Perse Boys School, Cambridge followed up the Madrass College class 2YP article with more thoughts on the problem of the number of ways of expressing an integer as the sum. . . .
### Magic Caterpillars
##### Stage: 4 and 5 Challenge Level:
Label the joints and legs of these graph theory caterpillars so that the vertex sums are all equal.
### Bell Ringing
##### Stage: 3 Challenge Level:
Suppose you are a bellringer. Can you find the changes so that, starting and ending with a round, all the 24 possible permutations are rung once each and only once?
### N000ughty Thoughts
##### Stage: 4 Challenge Level:
How many noughts are at the end of these giant numbers?
### Symmetric Tangles
##### Stage: 4
The tangles created by the twists and turns of the Conway rope trick are surprisingly symmetrical. Here's why!
### Paving Paths
##### Stage: 3 Challenge Level:
How many different ways can I lay 10 paving slabs, each 2 foot by 1 foot, to make a path 2 foot wide and 10 foot long from my back door into my garden, without cutting any of the paving slabs?
### Magic W
##### Stage: 4 Challenge Level:
Find all the ways of placing the numbers 1 to 9 on a W shape, with 3 numbers on each leg, so that each set of 3 numbers has the same total.
### Ordered Sums
##### Stage: 4 Challenge Level:
Let a(n) be the number of ways of expressing the integer n as an ordered sum of 1's and 2's. Let b(n) be the number of ways of expressing n as an ordered sum of integers greater than 1. (i) Calculate. . . .
### Lost in Space
##### Stage: 4 Challenge Level:
How many ways are there to count 1 - 2 - 3 in the array of triangular numbers? What happens with larger arrays? Can you predict for any size array?
### Deep Roots
##### Stage: 4 Challenge Level:
Find integer solutions to: $\sqrt{a+b\sqrt{x}} + \sqrt{c+d.\sqrt{x}}=1$
### Small Change
##### Stage: 3 Challenge Level:
In how many ways can a pound (value 100 pence) be changed into some combination of 1, 2, 5, 10, 20 and 50 pence coins?
### Penta Colour
##### Stage: 4 Challenge Level:
In how many different ways can I colour the five edges of a pentagon red, blue and green so that no two adjacent edges are the same colour?
### Snowman
##### Stage: 4 Challenge Level:
All the words in the Snowman language consist of exactly seven letters formed from the letters {s, no, wm, an). How many words are there in the Snowman language?
### Olympic Magic
##### Stage: 4 Challenge Level:
in how many ways can you place the numbers 1, 2, 3 … 9 in the nine regions of the Olympic Emblem (5 overlapping circles) so that the amount in each ring is the same?
### Euler's Officers
##### Stage: 4 Challenge Level:
How many different ways can you arrange the officers in a square?
### Tangles
##### Stage: 3 and 4
A personal investigation of Conway's Rational Tangles. What were the interesting questions that needed to be asked, and where did they lead?
### Flagging
##### Stage: 3 Challenge Level:
How many tricolour flags are possible with 5 available colours such that two adjacent stripes must NOT be the same colour. What about 256 colours?
### One Basket or Group Photo
##### Stage: 2, 3, 4 and 5 Challenge Level:
Libby Jared helped to set up NRICH and this is one of her favourite problems. It's a problem suitable for a wide age range and best tackled practically.
### Links and Knots
##### Stage: 4 and 5
Some puzzles requiring no knowledge of knot theory, just a careful inspection of the patterns. A glimpse of the classification of knots, prime knots, crossing numbers and knot arithmetic.
### In a Box
##### Stage: 4 Challenge Level:
Chris and Jo put two red and four blue ribbons in a box. They each pick a ribbon from the box without looking. Jo wins if the two ribbons are the same colour. Is the game fair?
##### Stage: 3 Challenge Level:
Is it possible to use all 28 dominoes arranging them in squares of four? What patterns can you see in the solution(s)?
##### Stage: 4 Challenge Level:
A walk is made up of diagonal steps from left to right, starting at the origin and ending on the x-axis. How many paths are there for 4 steps, for 6 steps, for 8 steps?
### Counting Binary Ops
##### Stage: 4 Challenge Level:
How many ways can the terms in an ordered list be combined by repeating a single binary operation. Show that for 4 terms there are 5 cases and find the number of cases for 5 terms and 6 terms.
### Molecular Sequencer
##### Stage: 4 and 5 Challenge Level:
Investigate the molecular masses in this sequence of molecules and deduce which molecule has been analysed in the mass spectrometer.
### Scratch Cards
##### Stage: 4 Challenge Level:
To win on a scratch card you have to uncover three numbers that add up to more than fifteen. What is the probability of winning a prize?
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https://tallahasseescene.com/2019/11/02/what-the-numbers-say-about-ali-larter-11-02-2019/
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crawl-data/CC-MAIN-2019-51/segments/1575540518337.65/warc/CC-MAIN-20191209065626-20191209093626-00337.warc.gz
| 560,915,604
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|
# What The Numbers Say About Ali Larter (11/02/2019)
How will Ali Larter do on 11/02/2019 and the days ahead? Let’s use astrology to conduct a simple analysis. Note this is for entertainment purposes only – don’t get too worked up about the result. I will first find the destiny number for Ali Larter, and then something similar to the life path number, which we will calculate for today (11/02/2019). By comparing the difference of these two numbers, we may have an indication of how good their day will go, at least according to some astrology enthusiasts.
PATH NUMBER FOR 11/02/2019: We will take the month (11), the day (02) and the year (2019), turn each of these 3 numbers into 1 number, and add them together. Here’s how it works. First, for the month, we take the current month of 11 and add the digits together: 1 + 1 = 2 (super simple). Then do the day: from 02 we do 0 + 2 = 2. Now finally, the year of 2019: 2 + 0 + 1 + 9 = 12. Now we have our three numbers, which we can add together: 2 + 2 + 12 = 16. This still isn’t a single-digit number, so we will add its digits together again: 1 + 6 = 7. Now we have a single-digit number: 7 is the path number for 11/02/2019.
DESTINY NUMBER FOR Ali Larter: The destiny number will take the sum of all the letters in a name. Each letter is assigned a number per the below chart:
So for Ali Larter we have the letters A (1), l (3), i (9), L (3), a (1), r (9), t (2), e (5) and r (9). Adding all of that up (yes, this can get tedious) gives 42. This still isn’t a single-digit number, so we will add its digits together again: 4 + 2 = 6. Now we have a single-digit number: 6 is the destiny number for Ali Larter.
CONCLUSION: The difference between the path number for today (7) and destiny number for Ali Larter (6) is 1. That is lower than the average difference between path numbers and destiny numbers (2.667), indicating that THIS IS A GOOD RESULT. But don’t go jumping for joy yet! As mentioned earlier, this is not at all guaranteed. If you want to see something that we really strongly recommend, check out your cosmic energy profile here. Check it out now – what it returns may blow your mind.
### Abigale Lormen
Abigale is a Masters in Business Administration by education. After completing her post-graduation, Abigale jumped the journalism bandwagon as a freelance journalist. Soon after that she landed a job of reporter and has been climbing the news industry ladder ever since to reach the post of editor at Tallahasseescene.
#### Latest posts by Abigale Lormen (see all)
Abigale Lormen
Abigale is a Masters in Business Administration by education. After completing her post-graduation, Abigale jumped the journalism bandwagon as a freelance journalist. Soon after that she landed a job of reporter and has been climbing the news industry ladder ever since to reach the post of editor at Tallahasseescene.
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https://www.studypool.com/discuss/1012602/which-is-a-value-of-x-when-3x2-4-40?free
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crawl-data/CC-MAIN-2017-43/segments/1508187821189.10/warc/CC-MAIN-20171017125144-20171017145144-00883.warc.gz
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##### Which is a value of x, when 3x2 4 = 40?
label Algebra
account_circle Unassigned
schedule 1 Day
account_balance_wallet \$5
Which is a value of x, when 3x2 + 4 = 40?
Jun 1st, 2015
I assume its 3x^2 +4 = 40
3x^2 = 40 - 4
3x^2 = 36
x^2 = 36/3 =12
x = sqrt (12) and -sqrt(12)
= 2sqrt(3) and -2sqrt(3)
Please let me know if you have any questions and best me if you are satisfactory.
Jun 1st, 2015
what are you supposed to do after you put the square root symbol with x and the 12?
Jun 1st, 2015
sqrt (12) = sqrt(4*3)
sqrt (4) = 2
Taking 4 outside the root, we get sqrt (12) = 2 * sqrt(3)
= 2sqrt (3)
Jun 1st, 2015
...
Jun 1st, 2015
...
Jun 1st, 2015
Oct 17th, 2017
check_circle
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https://codegolf.stackexchange.com/questions/197379/proportion-of-self-avoiding-walks-on-a-square-lattice/197382#197382
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| 159,512,178
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# The challenge
Given a positive integer N, compute the proportion of N-step walks on a plane that don't intersect themselves.
Each step can have any of the 4 possible directions North, East, South, West.
A walk intersects itself if it visits a previously visited point.
# Examples
• N=1: a single-step walk obviously doesn't intersect itself. So the result is 1.
• N=2: given the first step in any direction, there are 3 possible directions that avoid intersection, and one that goes back to the origin, causing intersection. So the result is 3/4 = 0.75.
• N=3: if the second step doesn't cause intersection, which happens 3/4 of the times, the third step will not cause intersection with probability again 3/4. So the result is (3/4)^2 = 0.5625.
• N=4: things become more interesting because proper loops can be formed. A similar computation as above gives (3/4)^3 - 8/4^4 = 0.390625, where the second term accounts for the 8 proper loops out of the 4^4 possible paths (these are not excluded by the first term).
# Test cases
1 -> 1
2 -> 0.75
3 -> 0.5625
4 -> 0.390625
5 -> 0.27734375
6 -> 0.1904296875
7 -> 0.132568359375
8 -> 0.09027099609375
9 -> 0.0620574951171875
10 -> 0.042057037353515625
11 -> 0.02867984771728515625
12 -> 0.0193674564361572265625
• Related: OEIS A001411 Commented Dec 23, 2019 at 20:14
• @Downvoter How can I improve this challenge, or future challenges? Any advice you’d like to share? Commented Dec 24, 2019 at 11:52
• I guess the downvoter tripped over their own path when reading this challenge. Commented Dec 24, 2019 at 14:34
• why not allow the output to be the number of paths (like in OEIS)?
– qwr
Commented Dec 26, 2019 at 22:37
• @qwr That was the other option I considered. I settled down on proportion rather than number because proportion has a clearer meaning. ("There are 5916 non-intersecting paths of length 8. Is that a lot? Well, it's a fraction 0.09 of all exiting paths"). Anyway it's too late to change now... Commented Dec 26, 2019 at 22:52
# Haskell, 153 149 144 140 126 118 92 bytes
Here we represent the 2d coordinates as a single integer: We start at 0, and the directions N,E,S,W correspond to adding +n,+1,-n,-1 where n is the input (we could also use any larger number). Using this we generate all possible paths, and then just check for duplicate numbers in those paths.
Thanks @H.PWiz for -26 bytes!
g n|a<-scanl(+)0<$>mapM(\_->[1,-1,n,-n])[1..n]=sum[1|x<-a,[b|b<-x,c<-x,b==c]==x]/sum[1|x<-a] Try it online! ### Explanation --generate all possible paths a<-scanl(+)0<$>mapM(\_->[1,-1,n,-n])[1..n]
--count the non-self intersecting paths, compute the ratio to the total
g n|a<- ... =sum[1|x<-a,[b|b<-x,c<-x,b==c]==x]/sum[1|x<-a]
• Very clever way to reduce to 1D! Commented Dec 23, 2019 at 22:24
• 92 bytes. I think it can be shorter Commented Dec 24, 2019 at 15:46
• thanks, I feel stupid that I didn't notice the mapM opportunity, and the uniqueness check is neat, maybe something to add for the tips! Commented Dec 24, 2019 at 15:55
• @H.PWiz Sorry, I mistakenly attributed your suggestion to another wizard! Commented Dec 24, 2019 at 19:26
# Python 2, 89 bytes
f=lambda n,S=[0]:n>=len(S)and sum(f(n,S+[S[-1]+d])for d in[-n,-1,1,n])/4.or len(set(S))>n
Try it online!
A recursive approach inspired by flawr's lovely Haskell answer. Outputs a float.
# Jelly, 15 12 bytes
‘ı×Ƭ¤ṗÄQƑ€Æm
Try it online!
A monadic link taking N as its argument and returning a float representing the proportion of non-self-intersecting walks of length N. Generates all walks of that length and then checks for intersections, using complex numbers to represent the 2D coordinates.
Thanks to @LuisMendo for saving a byte, and @Mr.XCoder for saving 2 more!
## Explanation
‘ | Increment by 1
¤ | Following as a nilad
ı | - 1i
×Ƭ | - Multiply (by 1i) until no new values, returning all intermediate values
ṗ | Cartesian power (with N+1)
Ä | Cumulative sums of innermost lists
QƑ€ | Check whether each list is invariant when uniquified
Æm | Arithmetic mean
• Good idea :-) I also used complex numbers in my code to generate the test cases Commented Dec 23, 2019 at 22:25
• @LuisMendo thanks. Nice challenge! Commented Dec 23, 2019 at 22:27
• 12 bytes with ‘ı×Ƭ¤ṗÄQƑ€Æm. Commented Dec 24, 2019 at 9:03
• @Mr.Xcoder thanks! You also brought my R answer under 100 bytes. Commented Dec 24, 2019 at 9:11
# MATL, 16 bytes
Done with a lot of help from Luis Mendo in MATL CHATL.
QJ4:!^Z^!YsSdAYm
Try it at MATL Online!
Alternative:
Q8BPZFZ^!YsSdAYm
Try it at MATL Online!
## Explanation
QJ4:!^Z^!YsSdAYm – Full program. Receives an integer N as input.
4: – Range 1...4.
J !^ – Raise J (imaginary unit) to those powers*. Yields j, -1, -j, 1.
Q Z^ – Cartesian power N+1.
!Ys – Transpose and take the cumulative sums.
Sd – Sort and get the deltas (consecutive differences).
A – All. For each column, if it contains 0, then 0, else 1.
Ym – Arithmetic mean.
(*): ! is needed there because Octave is weird.
# R + gtools, 111 88 bytes
mean(!apply(diffinv(t(expand.grid(rep(list(c(1,-1,n<-scan(),-n)),n)))),2,anyDuplicated))
Try it online!
Test suite
Reads N from STDIN and implicitly prints the answer as a float.
Thanks to @LuisMendo for a fab challenge and saving 6 bytes! Thanks to @Mr.XCoder for saving 3 bytes (indirectly via my Jelly answer). Thanks to @Giuseppe for saving 10 bytes with an excellent suggestion partially inspired by @flawr’s Haskell answer!
• any(duplicated(...)) -> anyDuplicated(...)? Commented Dec 24, 2019 at 14:11
• Or doing the 1-d reduction inspired by flawr, you can get to 91 bytes Commented Dec 24, 2019 at 14:19
• @Giuseppe thanks. The 1-d reduction fails for N=1; I can’t see an obvious way around that that costs fewer than 5 bytes. Commented Dec 24, 2019 at 18:30
• perhaps a nice base R expand.grid will work: 88 bytes Commented Dec 24, 2019 at 18:49
• @Giuseppe thanks, I’d not used diffinv or anyDuplicated before; both very handy! Commented Dec 24, 2019 at 19:41
# 05AB1E, 16 bytes
Uses flawr's method of reducing to 1D.
1‚D(«Iãε.¥DÙQ}ÅA
Try it online!
# Haskell, 69 bytes
(%[])
-1%_=1
n%l=sum[(n-1)%map(+d)(0:l)|d<-[1,-1,pi,-pi],all(/=0)l]/4
Try it online!
Uses a similar idea to flawr's solution of storing 2D coordinates as 1D values: $$\(x,y)\$$ is represented as $$\x+\pi y\$$. Maybe this is unfair because finite precision means that eventually two different coordinates will be represented as the same value. This will take an extremely large input though because toRational pi equals 884279719003555 % 281474976710656.
Instead of updating the position after each move, we map the position change onto the list of previously visited coordinates. That way, we can detect a collision by seeing a 0 among the previous coordinates. Because we only check a coordinate the loop after it's added, one extra loop is done, for n+1 total.
# Python 3, 76 bytes
f=lambda n,p=0,*S:n<1or sum(f(n-1,q,p,*S)for q in{p-1,p+1,p-1j,p+1j}-{*S})/4
Try it online!
# Charcoal, 56 52 bytes
Ｎθ≔⁰η⊞υ⟦⁰⟧ＦυＦΦ⁺⟦¹θ±¹±θ⟧§ι±¹¬№ικ¿⁼Ｌιθ≦⊕η⊞υ⁺ι⟦κ⟧Ｉ∕ηＸ⁴θ
Try it online! Link is to verbose version of code. Explanation:
Ｎθ
Input N.
≔⁰η
Initialise the number of paths to 0.
⊞υ⟦⁰⟧
Start off with a path with no steps.
Ｆυ
Perform a breadth-first search of the paths.
ＦΦ⁺⟦¹θ±¹±θ⟧§ι±¹¬№ικ
Take @flawr's list 1, N, -1, -N and add the last position on the current path to each value. Filter out results that already appear in that path, and loop over the remaining self-avoiding values.
¿⁼Ｌιθ
If this path already has N steps...
≦⊕η
... then increment the count of found paths.
⊞υ⁺ι⟦κ⟧
... otherwise construct a new path and add it to the search list.
Ｉ∕ηＸ⁴θ
Print the proportion of self-avoiding paths.
| 2,518
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https://www.hackmath.net/en/examples/grade?page_num=102
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# Grade - examples - page 102
1. Pins
Sarah bought 9 pins, Eva bought 4 pins and saved 2 euros. How many pins buy Dana, when she have 3 euro?
2. Alcohol mixing
How much 55% alcohol we must pour into 1500 g 80% alcohol to form a 60% alcohol? How much 60% alcohol created?
3. Salary in enterprise
The average salary in the company is 27 000 CZK, 30% of workers have the lowest average income of 19 thousand CZK. There were an increase in the salary in this group by 2%. How much % increased the average salary across the company?
4. Rhombus and diagonals
The a rhombus area is 150 cm2 and the ratio of the diagonals is 3:4. Calculate the length of its height.
5. Master and apprentice
Master painted the roof in 3 hours and apprentice for 4 hours. How many of roof they painted in hour and how many in three quarters of an hour?
6. The rod
The rod is painted in four colors. 55% of the bar is painted in blue, green 0.2 of rod, 1/8 is brown and the remaining 45 cm of white. How long is rod?
7. Area of square
Calculate the content area of the square whose perimeter is 24 dm.
8. QuizQ2
The square of the first number is equal to three-fifths of the second number. Determine both numbers if you know that the second number is 5 times greater than the first, and neither of numbers is not equal to zero.
9. Unknown number 2
I think the number. When he reduces it four times, I'll get 11. What number am I thinking?
10. Draw a trapezoid
Draw a trapezoid if given a = 7 cm, b = 4 cm, c = 3.5 cm, diagonal AC = 5cm. Solve as a construction task.
11. The fence
I'm building a fence. Late is rounded up in semicircle. The tops of late in the field between the columns are to copy an imaginary circle. The tip of the first and last lath in the field is a circle whose radius is unknown. The length of the circle chord i
12. Chord 3
What is the radius of the circle where the chord is 2/3 of the radius from the center and has a length of 10 cm?
13. Compound interest
Compound interest: Clara deposited CZK 100,000 in the bank with an annual interest rate of 1.5%. Both money and interest remain deposited in the bank. How many CZK will be in the bank after 3 years?
14. Divide money 2
Ben and Dan had the same amount of money at the start. When Ben gave 300 to Dan, the ratio of Ben 's money to Dan's money became 2:3. How much money did each have at first?
15. Fan
The fan has a speed of 210 RPM. Calculate for time of one fan period.
16. Kitchen
Kitchen roller has a diameter 70 mm and width of 359 mm. How many square millimeters roll on one turn?
17. Park
In the park is marked diamond shaped line connecting locations A, D, S, C, B, A. Calculate its length if |AB| = 108 m, |AC| = 172.8 m.
18. Class
The class has 18 students. Everyone knows inline skating or skateboarding. Inline skating can ride 11 students on a skateboard 10. How many ride on inline skates and on skateboard?
19. Bookshelve
Bookshelve with an original price of € 200 twice become cheaper. After the second discounted by 15% the price was € 149.60. Determine how much % become cheaper for the first time.
20. Juice box
In the box is 0.3 liters of juice. How many liters of juice contains 3 these boxes?
Do you have an interesting mathematical example that you can't solve it? Enter it, and we can try to solve it.
To this e-mail address, we will reply solution; solved examples are also published here. Please enter e-mail correctly and check whether you don't have a full mailbox.
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# Polynomial Functions
• Oct 20th 2011, 09:23 AM
richtea9
Polynomial Functions
Hi people, hoping you can help me understand polynomial functions.
I have been given an example which is the following:
p(x) = 2x^3 + 3x + 1
Coefficient of x^3: 2 - I understand why this is a 2 because it is in front of x^3.
Coefficient of x^2: 0 - I understand this as there is no x to the power of 2.
Coefficient of x^1: 3 - I see why as 3 is in front of the single x.
Coefficient of x^0: 1 - However, I do not see why this is?
• Oct 20th 2011, 09:32 AM
Youkla
Re: Polynomial Functions
Because \$\displaystyle x^0 = 1\$. So what you really have is
\$\displaystyle p(x) = 2x^3 + 3x +1x^0\$ which is just \$\displaystyle p(x) = 2x^3 + 3x +1\$
There is no need to actually state \$\displaystyle x^0 = 1\$ in the polynomial since it just equals 1.
• Oct 20th 2011, 10:13 AM
richtea9
Re: Polynomial Functions
Quote:
Originally Posted by Youkla
Because \$\displaystyle x^0 = 1\$. So what you really have is
\$\displaystyle p(x) = 2x^3 + 3x +1x^0\$ which is just \$\displaystyle p(x) = 2x^3 + 3x +1\$
There is no need to actually state \$\displaystyle x^0 = 1\$ in the polynomial since it just equals 1.
So the 1 and the end of the expression is representing \$\displaystyle x^0\$
• Oct 20th 2011, 10:21 AM
Youkla
Re: Polynomial Functions
Quote:
Originally Posted by richtea9
So the 1 and the end of the expression is representing a single X?
The 1 at the end of the polynomial is just a constant. It's technically the coefficient of \$\displaystyle x^0\$, but like I said, we don't write \$\displaystyle x^0\$ in the actual polynomial itself since it just equals 1.
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Napier’s Logarithm
This post is a response to Mr. Cornally’s post on logarithms. However, since it is intended for his students, and I am not one of his students, I thought I would post my ideas in a separate location to allow their dialog to be untainted by my thoughts. After all, I am a strong believer that the discussion is much more important than the conclusion. [edit: Doh! I forgot about that whole ping back thing. Now I just have to pray that his students aren't the type to click on links.]
The Description of the Wonderful Canon of Logarithms,
and the use of which not only in Trigonometry, but also in all
Mathematical Calculations, most fully and easily explained in the most
expeditious manner.
By the author and discoverer
John Napier.
Baron of Merchiston, etc. Scotland.
ON THE AMAZING CANON OF LOGARITHMS.
Preface.
Since nothing is more tedious, fellow mathematicians, in the practice of the mathematical arts, than the great delays suffered in the tedium of lengthy multiplications and divisions, the finding of ratios, and in the extraction of square and cube roots– and in which not only is there the time delay to be considered, but also the annoyance of the many slippery errors that can arise: I had therefore been turning over in my mind, by what sure and expeditious art, I might be able to improve upon these said difficulties.
In the end after much thought, finally I have found an amazing way of shortening the proceedings, and perhaps the manner in which the method arose will be set out elsewhere: truly, concerning all these matters, there could be nothing more useful than the method that I have found. For all the numbers associated with the multiplications, and divisions of numbers, and with the long arduous tasks of extracting square and cube roots are themselves rejected from the work, and in their place other numbers are substituted, which perform the tasks of these rejected by means of addition, subtraction, and division by two or three only. Since indeed the secret is best made common to all, as all good things are, then it is a pleasant task to set out the method for the public use of mathematicians. Thus, students of mathematics, accept and freely enjoy this work that has been produced by my benevolence Farewell.
So Napier came up with the concept of the logarithm in the fifteen hundreds.* Back in the day, things like taking the square root of a number required much more than pressing a few buttons on your TI-83. Not only did they not have graphing calculators, but they didn’t even have any calculators at all. What a hard life it must have been. Sure we can do multiplication and division with relatively small numbers, but what about topics like astronomy where we’re dealing with distances of millions of miles or more. I would hate to have to perform calculations with those numbers by hand. I think Napier hated that too.
let us consider a basic algebra problem for a minute. what is $x^{13} \times x^{16}$ ? easy, you say… All we have to do is add the $13$ and $16$ to get $x^{29}$. We use this fact so often, but rarely give it very much thought. Let us put a number in for $x$, say $2$. Now we have the equation: $2^{13} \times 2^{16}=2^{29}$. I know, this doesn’t really seem like a big deal. However, this tells me something that I would have had a very difficult time figuring out any other way. Namely, that $8,192 \times 65536=536870912$. How do I know? Well if I know that $2^{13}=8192$ and that $2^{16}=65536$, and that $2^{29}=536870912$ then I can just substitute those numbers into my equation and I have the product of two really large numbers (without having to rely on a calculator). This is an novel idea. I have just reduced the task of multiplying two numbers in the thousands to the job of adding two 2-digit numbers.
Let’s try another example. What about computing $\sqrt[3]{2^{33}}$ No problem, I just have to divide $33$ by $3$ to get $11$. So we have $\sqrt[3]{2^{33}} = 2^{11}$. Nothing special, right? Wrong. I now know that $\sqrt[3]{8589934592} = 2048$. Imagine trying to calculate that cubed root by hand.
This is the power of a logarithm. It reduces multiplication and division of giant numbers to addition and subtraction with small numbers. Exponents and roots become as simple as multiplication and division.
Let’s try an example:
Compute the area of a rectangle with a base of 132874 and a height of 324938 without using a calculator.
Well that problem is easy enough to set up. we know formula for the area of a rectangle is base times height, or in this case:
$A=132874 \times 324938$.
Here is where things get difficult. Try computing that by hand and you’ll start to understand the difficulties of being a mathematician in the fourteen hundreds. However, if we know for a fact that $132874 \approx 10^{5.12344}$ and that $324938 \approx 10^{5.51180}$ then we can rewrite our equation into something easier to calculate.
$A=10^{5.12344} \times 10^{5.51180}$
Well if we just add the exponents, we get $10^{10.63524} = 43175760898$ (the answer is actually $43175811812$ but I was pretty close. If I used more decimals I would have been much closer).
But wait says the skeptic. How do you know that $10^{10.63524} = 43175760898$? Doesn’t that seem much harder than computing the basic multiplication? Yes, that is a very difficult calculation to make. This is where common logarithms comes in. Let’s make a pact to always use a base ten system. Now if some kind mathematician would spend their lives writing a book of logarithms – a dictionary if you will – translating simple numbers into what they are as a power of ten, then we can be rid of multiplication and powers of large numbers once and for all. All we will have to do to multiply two numbers is look up what they are as a power of ten, add the exponents, and translate the result back into a regular number with the same book. For hundreds of years that is exactly what people would do.
*It should be mentioned that Napier’s logarithms were much more cumbersome than the logs we use today. Of course, the subject of algebra was much more cumbersome than it is today. His logarithms weren’t based as much off of exponents, but he thought of the topic in a dynamic, geometric-esque way. Imagine two parallel tracks, each with a train at one end. both trains start off at the same place, but on one track a train is moving at a constant speed and on the other track, a train is constantly accelerating. Napier’s version of a logarithm was essentially a way to jump back and forth between those two trains.
On a side note, I have not been writing too much educationally related stuff in my blog recently. It isn’t that I have nothing to say, but more so that last Friday I turned in a 28 page paper and today I submitted a 40 page paper. I am pretty burnt out on writing about education right now and decided it would be much more fun to dive into the math world that I love so much. Maybe this weekend I will get back to writing about education.
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# CBSE CLASS 11 PHYSICS SAMPLE PAPER - 1
CBSE Class 11
Physics Sample Paper – 1
Time: 3 Hrs. Maximum Marks: 70
General Instructions:
• All questions are compulsory.
• There are 30 questions in total. Questions 1 to 8 carry one mark each, questions 9 to 18 carry two marks each, questions 19 to 27 carry three marks each and questions 28 to 30 carry five marks each.
• There is no overall choice. However, an internal choice has been provided in one question of two marks, one question of three marks and all three questions of five marks each. You have to attempt only one of the given choices in such questions.
• Use of calculator is not permitted.
• You may use the following physical constants wherever necessary
$\begin{gathered}e=1.6\times10^{-19}\mathrm{C}\\\mathrm{c}=3\times10^{8}\mathrm{~ms}^{-1}\\\mathrm{~h}=6.6\times10^{-34}\mathrm{JS}\\\mu_{\mathrm{o}}=4\pi\times10^{-7}\mathrm{NA}^{-2}\\\mathrm{k}_{\mathrm{B}}=1.38\times10^{23}\mathrm{JK}^{-1}\\\mathrm{~N}_{\mathrm{A}}=6.023\times10^{23}/\mathrm{mole}\\\mathrm{m}_{\mathrm{n}}=1.6\times10^{-27}\mathrm{~kg}\end{gathered}$
1. Give the number of significant figures in $5.300 \times 10^{3}$ (1)
2. The dimension $\mathrm{ML}^{-1} \mathrm{~T}^{-2}$ corresponds to ….? (1)
3. Why do we use a wrench of long arm to unscrew a nut tightly fitted to a bolt? (1)
4. Can kinetic energy be negative? What about potential energy? (1)
5. Does the spring constant of a spring depend on its length? (1)
6. Is the Young’s modulus of rubber greater than that of steel? (1)
7. State the SI unit of angular velocity. (1)
8. Explain, why a cricketer moves his hands back while holding a catch. (1)
9. Find the angle of projection for which horizontal range and maximum height are equal.
OR
Is acceleration vector in uniform circular motion a constant vector? (2)
1. Differentiate between wave velocity and particle velocity for a mechanical wave in the medium. (2)
2. A light body and a heavy body have same momentum. Which one has greater kinetic energy? Support your answer with an explanation (2)
3. State the law of equipartition of energy. (2)
4. What is an adiabatic process? How is it different from an isothermal process? (2)
5. A ball of mass 5 kg strikes against a wall at an angle of $45^{\circ}$ and is reflected at the same angle. Find the change in momentum. (2)
6. Check the dimensional consistency of the following equation (2)
$\frac{1}{2} \mathrm{mv}^{2}=\mathrm{mgh}$
where m is the mass of the body, v is its velocity, g is acceleration due to gravity and h is the height.
1. Why is it is easier to pull a lawn mower than to push it?
$\mathrm{S}(\mathrm{t})=5 \mathrm{t} \hat{\mathrm{\imath}}+6 \mathrm{t}^{2} \hat{\mathrm{j}}-10 \hat{\mathrm{k}}$
where t is in seconds. Find the velocity v t( ) and acceleration a t( ) of the particle at t = 1s. (2)
1. Why is it is easier to pull a lawn mower than to push it? (2)
2. Three particles of mass m are placed at the corners of an equilateral triangle. Find the position of centre of mass in terms of coordinates. (2)
3. The kinetic energy of a satellite is E. Find the total energy of the satellite. (3)
4. State Bernoulli’s theorem. Explain the lift on an airfoil using the theorem (3)
5. Explain why (3)
(i) a body with large reflectivity is a poor emitter
(ii) heating systems based on circulation of steam are more efficient in warming a building than those based on circulation of hot water.
1. What is a Carnot’s engine? What is its efficiency? (3)
2. A cylinder of fixed capacity 44.8 litres contains helium gas at standard temperature and pressure. What is the amount of heat needed to raise the temperature of the gas in the cylinder by $15.0^{\circ} \mathrm{C}$? Given R = 8.32J/mol/K.
(3)
1. A particle executes SHM according to the equation
$x=\mathrm{A} \cos . \omega \mathrm{t}$
Draw graphs to represent the displacement, velocity and acceleration of the particle. (3)
1. A sound wave traveling along a string is described by (3)
$y=5 \times 10^{-3} \sin (80 x-3 t)$
Calculate
(i) the amplitude
(ii) the wavelength
(iii) frequency of the wave.
1. What is a conservative force? Prove that gravitational force is conservative and frictional force is non-conservative
(3)
1. Springs A and B are identical except that A is, stiffer than B, i.e. force constant $\mathrm{K}_{\mathrm{A}}>\mathrm{K}_{\mathrm{B}}$. In which spring is more work expended if they are stretched by the same amount. (3)
2. Draw the first three harmonics in an open organ pipe. Two piano strings A and B are playing slightly out of tune and produce beats of frequency 5Hz. The tension in string B is slightly increased and the beat frequency is found to decrease to 3Hz. What is the original frequency of B if the frequency of A is 500Hz?
OR
Two identical springs each of force constant K are connected in (a) series (b) parallel, so that they support a mass m. Find the ratio of the time periods of the mass in the two systems. (5)
1. Two bodies A and B of masses 5 kg and 10 kg in contact with each other rest on a table against a rigid wall as shown in the given figure. The coefficient of friction between the bodies and the table is 0.15. A force of 200 N is applied horizontally to A. What are (a) the reaction of the partition (b) the action-reaction forces between A and B? (c) What happens when the wall is removed? (5)
1. What is a projectile? Derive the expressions for the time of flight, and maximum height for the projectile thrown upwards at an angle θ with the horizontal direction.
OR
The ceiling of a long hall is 25 m high. What is the maximum horizontal distance that a ball thrown with a speed of $40 \mathrm{~ms}^{-1}$ can go without hitting the ceiling of the hall? (5)
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https://www.eduzip.com/ask/question/displaystyleint-0pi2-sin-x-cos-x-dx160-579595
| 1,627,168,561,000,000,000
|
text/html
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crawl-data/CC-MAIN-2021-31/segments/1627046151531.67/warc/CC-MAIN-20210724223025-20210725013025-00611.warc.gz
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Mathematics
# $\displaystyle\int_{0}^{\pi/2} \sin x \cos x\ dx$
##### SOLUTION
$\displaystyle\int_{0}^{\pi/2} \sin x \cos x\ dx$
Let $t=\sin x \implies dt=\cos x dx$
$t_1 =sin {\dfrac {\pi}{2}}=1=upper \ limit$
$t_2=sin {0}=0=lower \ limit$
Replacing we get,
$\displaystyle \int _0^1 t \ dt \\ \left.\dfrac {t^2}2 \right] _0^1 =\dfrac 12$
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Subjective Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 105
#### Realted Questions
Q1 Subjective Medium
Solve: $\int\limits_0^\infty {\left( {{a^{ - x}} - {b^{ - x}}} \right)dx}$
1 Verified Answer | Published on 17th 09, 2020
Q2 Single Correct Medium
Solve $\displaystyle \int\frac{1-\sqrt{x}}{1+\sqrt{x}} dx$
• A. $3\displaystyle \sqrt{x}+\frac{x}{2}-3\log(1+\sqrt{x})+c$
• B. $3\displaystyle \sqrt{x}+3\log(1+\sqrt{x})-\frac{1}{2}x+c$
• C. $3\displaystyle \sqrt{x}-\frac{1}{2}x-3\log(1+\sqrt{x})++c$
• D. $4\sqrt{x}-x-4\log(1+\sqrt{x})+c$
1 Verified Answer | Published on 17th 09, 2020
Q3 Subjective Medium
Evaluate: $\displaystyle \int _{\pi/6}^{\pi/3}\dfrac{\sin{x}+\cos{x}}{\sqrt{\sin{2x}}}dx$
1 Verified Answer | Published on 17th 09, 2020
Q4 Subjective Medium
Obtain as the limit of sum $\displaystyle\overset{log_e^7}{\underset{log_e^3}{\displaystyle\int}}e^xdx$.
1 Verified Answer | Published on 17th 09, 2020
Q5 Single Correct Medium
If I =$\overset { 2 }{ \underset { -3 }{ \int } } (|x + 1| + |x + 2| +|x -1|) dx$, then i equals
• A. $\dfrac{35}{2}$
• B. $\dfrac{47}{2}$
• C. $\dfrac{39}{2}$
• D. $\dfrac{31}{2}$
| 641
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# Welcome to the LEAP Q&A Forum
## GRE - Quantitative Reasoning - Algebra
### (1-x)/(x-1) = 1/x Quantity A : x Quantity B : -1/2
(1-x)/(x-1) = 1/x
Quantity A : x
Quantity B : -1/2
A)The quantity in Column A is greater.
B)The quantity in Column B is greater.
C)The two quantities are equal.
D)The relationship cannot be determined from the information given.
(1-x) / (x - 1) = 1/x
- (x - 1) / (x - 1) = 1/x
for x ≠ 1
-1 = 1/x
x = -1 < -1/2
Quantity B is greater
Option B is correct
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| 492
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http://www.enotes.com/homework-help/verify-identity-1-secx-tanx-secx-tanx-307420
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# Verify the identity 1/secx-tanx=secx+tanx .
sciencesolve | Teacher | (Level 3) Educator Emeritus
Posted on
You need to transform both sides expanding the function sec x into fraction such that:
`1/(1/(cos x) - tan x) = 1/(cos x)+ tan x`
Cross multiplying yields:
`1 = (1/(cos x) - tan x)(1/(cos x)+ tan x)`
You may transform the special product to the right in difference of squares such that:
`1 = 1/(cos^2 x) - tan^2 x`
This last line expresses one of the three forms of basic trigonometric identity, hence the identity `1/(sec x-tan x)=sec x+tan x` is established.
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https://www.topperlearning.com/forums/home-work-help-19/abcd-is-a-parallelogram-the-circle-through-a-b-and-c-inter-mathematics-circles-63411/reply
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Question
Sat February 23, 2013 By:
# ABCD is a parallelogram. The circle through A, B and C intersects CD produced at E, prove that AE=AD
Sat February 23, 2013
Answer : Given :ABCD is a parallelogram. The circle through A, B and C intersects CD produced at E
=> angle AED + angle ABC = 180 ... (1) (Sum of opposite angles of cyclic quadrilaterals is 180)
=> angle ADE + angle ADC = 180 ... (2) (linear pair)
=> angle ABC = angle ADC ... ........(3) (opposite angles of parallelogram are equal)
From (1) and (2)
=> angle AED + angle ABC = angle ADE + angle ADC
=> angle AED = angle ADE (using (3))
In triangle AED,
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https://plainmath.net/53049/solving-frac-58-cot-36-circ-equal-cos-3-x-without-substituting-the-trig
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Solving \frac 58 \cot 36^{\circ}=\cos^3 x without substituting the trig
Solving ${\frac{58}{\mathrm{cot}36}}^{\circ }={\mathrm{cos}}^{3}x$ without substituting the trig values for ${36}^{\circ }$
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otoplilp1
Since
${\mathrm{cos}}^{3}x=\frac{3\mathrm{cos}x+\mathrm{cos}3x}{4}$
we need to prove that:
$\frac{5{\mathrm{cos}36}^{\circ }}{8{\mathrm{sin}36}^{\circ }}=\frac{3{\mathrm{cos}18}^{\circ }+{\mathrm{cos}54}^{\circ }}{4}$
or
$5{\mathrm{cos}36}^{\circ }=3{\mathrm{sin}54}^{\circ }+3{\mathrm{sin}18}^{\circ }+{\mathrm{sin}90}^{\circ }-{\mathrm{sin}18}^{\circ }$
or
$2{\mathrm{cos}36}^{\circ }+2{\mathrm{cos}108}^{\circ }=1$
which is true because
$2{\mathrm{cos}36}^{\circ }+2{\mathrm{cos}108}^{\circ }=\frac{2{\mathrm{sin}36}^{\circ }{\mathrm{cos}36}^{\circ }+2{\mathrm{sin}36}^{\circ }{\mathrm{cos}108}^{\circ }}{{\mathrm{sin}36}^{\circ }}=$
$=\frac{{\mathrm{sin}72}^{\circ }+{\mathrm{sin}144}^{\circ }-{\mathrm{sin}72}^{\circ }}{{\mathrm{sin}36}^{\circ }}=1$
Not exactly what you’re looking for?
Jordan Mitchell
$\frac{5}{8}\cdot \mathrm{cot}{36}^{\circ }=\frac{5\mathrm{cos}{36}^{\circ }}{8\mathrm{sin}{36}^{\circ }}=\frac{5{\mathrm{cos}}^{2}{36}^{\circ }}{4\mathrm{cos}{18}^{\circ }}$ Using Proving trigonometric equation $\mathrm{cos}\left({36}^{\circ }\right)-\mathrm{cos}\left({72}^{\circ }\right)=\frac{1}{2}$ $\mathrm{cos}{36}^{\circ }-\left(2{\mathrm{cos}}^{2}{36}^{\circ }-1\right)=\frac{1}{2}⇔5{\mathrm{cos}}^{2}{36}^{\circ }=\left(1+\mathrm{cos}{36}^{\circ }{\right)}^{2}=\left(2{\mathrm{cos}}^{2}{18}^{\circ }{\right)}^{2}$
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http://math.stackexchange.com/questions/124143/question-about-continuity/124147
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This is a question from my math book:
Let $a< b < c$. Suppose that $f$ is continuous on $[a,b]$, $g$ is continuous on $[b,c]$, and $f(b) = g(b)$. Define $h$ on $[a,c]$ by $h(x) = f(x)$ for $x\in [a,b]$ and $h(x) = g(x)$ for $x \in [b,c]$. Prove that $h$ is continuous on $[a,c]$.
What I want to do is prove that $h$ is continuous on $[a,c]$ but not at $b$.
I'm thinking that I have to pick an $α$ such that $a<α< b$, and then show that $h(x)$ is continuous at $α$. So basically, I want to show that $∀\epsilon>0, ∃ δ>0$ such that if $x\in [a,c]$ and $|x-α|<δ$ then $|h(x)-h(α)|<\epsilon$. And from the given information, I know that $f:[a,b]\to \mathbb{R}$, and $∀ \epsilon>0, ∃ δ>0$ such that if $x\in [a,b]$ and $|x-α|<δ_f$ then $|f(x)-f(α)|<\epsilon$.
How do I connect these two definitions to find $δ$? And is there anything else I need to prove? Like are there any cases I should be making?
-
What you want to show is false; $h$ will be continuous at $b$. For points in $[a,c]$ that are actually in $[a,b)$, you can use the continuity of $f$ directly, taking care not to "go" past $b$; for points that are in $(b,c]$, you can use the continuity of $g$ (again, taking care not to go "past" $b$); for the point $b$, you'll need to combine the fact that $f$ is continuous at $b$ (remember that this means that $f$ is continuous from the left at $b$) and that $g$ is continuous at $b$ from the right. – Arturo Magidin Mar 25 '12 at 4:07
What you have to prove is precisely that $h$ is continuous at $b$. Because at any point $d$ other than $b$, if $d<b$ then $h(x)=f(x)$ for all $x$ in a small interval around $d$, and so the continuity of $f$ implies the continuity of $h$ for every $d\in[a,b)$. Similarly one deduces that $h$ is continuous on $(b,c]$ by using the continuity of $g$.
For the continuity at $b$. Fix $\varepsilon>0$. By the continuity of $f$ at $b$, there exists $\delta_1>0$ such that if $x<b$ and $b-x<\delta_1$, then $|f(x)-f(b)|<\varepsilon$. Similarly, there exists $\delta_2>0$ such that if $x>b$ and $x-b<\delta_2$, then $|g(x)-g(b)|<\varepsilon$.
Let $\delta=\min\{\delta_1,\delta_2\}$. Now, if $|x-b|<\delta$, we consider two cases: first, if $x<b$, then $b-x=|x-b|<\delta\leq\delta_1$, and so $$|h(x)-h(b)|=|f(x)-f(b)|<\varepsilon;$$ if $x>b$, then $x-b=|x-b|<\delta\leq\delta_2$, and so $$|h(x)-h(b)|=|g(x)-g(b)|<\varepsilon.$$ So $h$ is continuous at $b$.
-
We can prove that $h$ is continuous on $[a,c]$ in three steps:
First, we prove that $h$ is continuous on $[a,b)$. Since $f$ is continuous on $[a,b)$, given any $x\in [a,b)$ and $\epsilon>0$ we have some $\delta>0$ such that for $y\in [a,b)$ we have $|x-y|<\delta\implies |f(x)-f(y)|<\epsilon$. Let $\delta'=\min\{\delta,b-x\}$. Since $h=f$ on $[a,b)$, we have that for $y\in [a,c]$, $|x-y|<\delta'$ implies that $y\in [a,b)$ and $|x-y|<\delta$ so $|h(x)-h(y)|<\epsilon$, thus $h$ is continuous at $x$. Hence $h$ is continuous on $[a,b)$.
Second, we prove that $h$ is continuous on $(b,c]$. This proof is nearly identical to the previous one.
Finally, we prove that $h$ is continuous at $b$. Since $f$ and $g$ are continuous at $b$, given any $\epsilon>0$ we have some $\delta,\delta'>0$ such that for $y\in [a,b]$ if $|b-y|<\delta$ then $|f(b)-f(y)|<\epsilon$ and for $y\in [b,c]$ if $|b-y|<\delta'$ then $|g(b)-g(y)|<\epsilon$. Thus we can let $\delta''=\min\{\delta,\delta'\}$ and we get that if $y\in [a,c]$ and $|b-y|<\delta''$ then either $y\in [a,b]$ and $|b-y|<\delta$ in which case $|h(b)-h(y)|=|f(b)-f(y)|<\epsilon$, or $y\in [b,c]$ and $|b-y|<\delta'$ in which case $|h(b)-h(y)|=|g(b)-g(y)|<\epsilon$. Thus $h$ is continuous at $b$.
-
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#### What is 130 percent of 539?
How much is 130 percent of 539? Use the calculator below to calculate a percentage, either as a percentage of a number, such as 130% of 539 or the percentage of 2 numbers. Change the numbers to calculate different amounts. Simply type into the input boxes and the answer will update.
## 130% of 539 = 700.7
Calculate another percentage below. Type into inputs
Find number based on percentage
percent of
Find percentage based on 2 numbers
divided by
Calculating one hundred and thirty of five hundred and thirty-nine How to calculate 130% of 539? Simply divide the percent by 100 and multiply by the number. For example, 130 /100 x 539 = 700.7 or 1.3 x 539 = 700.7
#### How much is 130 percent of the following numbers?
130 percent of 539.01 = 70071.3 130 percent of 539.02 = 70072.6 130 percent of 539.03 = 70073.9 130 percent of 539.04 = 70075.2 130 percent of 539.05 = 70076.5 130 percent of 539.06 = 70077.8 130 percent of 539.07 = 70079.1 130 percent of 539.08 = 70080.4 130 percent of 539.09 = 70081.7 130 percent of 539.1 = 70083 130 percent of 539.11 = 70084.3 130 percent of 539.12 = 70085.6 130 percent of 539.13 = 70086.9 130 percent of 539.14 = 70088.2 130 percent of 539.15 = 70089.5 130 percent of 539.16 = 70090.8 130 percent of 539.17 = 70092.1 130 percent of 539.18 = 70093.4 130 percent of 539.19 = 70094.7 130 percent of 539.2 = 70096 130 percent of 539.21 = 70097.3 130 percent of 539.22 = 70098.6 130 percent of 539.23 = 70099.9 130 percent of 539.24 = 70101.2 130 percent of 539.25 = 70102.5
130 percent of 539.26 = 70103.8 130 percent of 539.27 = 70105.1 130 percent of 539.28 = 70106.4 130 percent of 539.29 = 70107.7 130 percent of 539.3 = 70109 130 percent of 539.31 = 70110.3 130 percent of 539.32 = 70111.6 130 percent of 539.33 = 70112.9 130 percent of 539.34 = 70114.2 130 percent of 539.35 = 70115.5 130 percent of 539.36 = 70116.8 130 percent of 539.37 = 70118.1 130 percent of 539.38 = 70119.4 130 percent of 539.39 = 70120.7 130 percent of 539.4 = 70122 130 percent of 539.41 = 70123.3 130 percent of 539.42 = 70124.6 130 percent of 539.43 = 70125.9 130 percent of 539.44 = 70127.2 130 percent of 539.45 = 70128.5 130 percent of 539.46 = 70129.8 130 percent of 539.47 = 70131.1 130 percent of 539.48 = 70132.4 130 percent of 539.49 = 70133.7 130 percent of 539.5 = 70135
130 percent of 539.51 = 70136.3 130 percent of 539.52 = 70137.6 130 percent of 539.53 = 70138.9 130 percent of 539.54 = 70140.2 130 percent of 539.55 = 70141.5 130 percent of 539.56 = 70142.8 130 percent of 539.57 = 70144.1 130 percent of 539.58 = 70145.4 130 percent of 539.59 = 70146.7 130 percent of 539.6 = 70148 130 percent of 539.61 = 70149.3 130 percent of 539.62 = 70150.6 130 percent of 539.63 = 70151.9 130 percent of 539.64 = 70153.2 130 percent of 539.65 = 70154.5 130 percent of 539.66 = 70155.8 130 percent of 539.67 = 70157.1 130 percent of 539.68 = 70158.4 130 percent of 539.69 = 70159.7 130 percent of 539.7 = 70161 130 percent of 539.71 = 70162.3 130 percent of 539.72 = 70163.6 130 percent of 539.73 = 70164.9 130 percent of 539.74 = 70166.2 130 percent of 539.75 = 70167.5
130 percent of 539.76 = 70168.8 130 percent of 539.77 = 70170.1 130 percent of 539.78 = 70171.4 130 percent of 539.79 = 70172.7 130 percent of 539.8 = 70174 130 percent of 539.81 = 70175.3 130 percent of 539.82 = 70176.6 130 percent of 539.83 = 70177.9 130 percent of 539.84 = 70179.2 130 percent of 539.85 = 70180.5 130 percent of 539.86 = 70181.8 130 percent of 539.87 = 70183.1 130 percent of 539.88 = 70184.4 130 percent of 539.89 = 70185.7 130 percent of 539.9 = 70187 130 percent of 539.91 = 70188.3 130 percent of 539.92 = 70189.6 130 percent of 539.93 = 70190.9 130 percent of 539.94 = 70192.2 130 percent of 539.95 = 70193.5 130 percent of 539.96 = 70194.8 130 percent of 539.97 = 70196.1 130 percent of 539.98 = 70197.4 130 percent of 539.99 = 70198.7 130 percent of 540 = 70200
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] |
https://www.jobilize.com/online/course/1-4-vectors-motion-by-openstax?qcr=www.quizover.com&page=3
| 1,623,900,000,000,000,000
|
text/html
|
crawl-data/CC-MAIN-2021-25/segments/1623487626465.55/warc/CC-MAIN-20210617011001-20210617041001-00296.warc.gz
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# 1.4 Vectors (Page 4/5)
Page 4 / 5
## Co-planar vectors
A pair of vectors determines an unique plane. The pair of vectors defining the plane and other vectors in that plane are called coplanar vectors.
## Axial vector
Motion has two basic types : translational and rotational motions. The vector and scalar quantities, describing them are inherently different. Accordingly, there are two types of vectors to deal with quantities having direction. The system of vectors that we have referred so far is suitable for describing translational motion and such vectors are called “rectangular” or "polar" vectors.
A different type of vector called axial vector is used to describe rotational motion. Its graphical representation is same as that of rectangular vector, but its interpretation is different. What it means that the axial vector is represented by a straight line with an arrow head as in the case of polar vector; but the physical interpretation of axial vector differs. An axial vector, say $\mathbf{\omega }$ , is interpreted to act along the positive direction of the axis of rotation, while rotating anti –clockwise. A negative axial vector like, $-\mathbf{\omega }$ , is interpreted to act along the negative direction of axis of rotation, while rotating clockwise.
The figure above captures the concept of axial vector. It should be noted that the direction of the axial vector is essentially tied with the sense of rotation (clockwise or anti-clockwise). This linking of directions is stated with "Right hand (screw) rule". According to this rule ( see figure below ), if the stretched thumb of right hand points in the direction of axial vector, then the curl of the fist gives the direction of rotation. Its inverse is also true i.e if the curl of the right hand fist is placed in a manner to follow the direction of rotation, then the stretched thumb points in the direction of axial vector.
Axial vector is generally shown to be perpendicular to a plane. In such cases, we use a shortened symbol to represent axial or even other vectors, which are normal to the plane, by a "dot" or "cross" inscribed within a small circle. A "dot" inscribed within the circle indicates that the vector is pointing towards the viewer of the plane and a "cross" inscribed within the circle indicates that the vector is pointing away from the viewer of the plane.
Axial vector are also known as "pseudovectors". It is because axial vectors do not follow transformation of rectangular coordinate system. Vectors which follow coordinate transformation are called "true" or "polar" vectors. One important test to distinguish these two types of vector is that axial vector has a mirror image with negative sign unlike true vectors. Also, we shall learn about vector or cross product subsequently. This operation represent many important physical phenomena such as rotation and magnetic interaction etc. We should know that the vector resulting from cross product of true vectors is always axial i.e. pseudovectors vector like magnetic field, magnetic force, angular velocity, torque etc.
List the application of projectile
pls explain what is dimension of 1in length and -1 in time ,what's is there difference between them
what are scalars
show that 1w= 10^7ergs^-1
what's lamin's theorems and it's mathematics representative
if the wavelength is double,what is the frequency of the wave
What are the system of units
A stone propelled from a catapult with a speed of 50ms-1 attains a height of 100m. Calculate the time of flight, calculate the angle of projection, calculate the range attained
58asagravitasnal firce
Amar
water boil at 100 and why
what is upper limit of speed
what temperature is 0 k
Riya
0k is the lower limit of the themordynamic scale which is equalt to -273 In celcius scale
Mustapha
How MKS system is the subset of SI system?
which colour has the shortest wavelength in the white light spectrum
if x=a-b, a=5.8cm b=3.22 cm find percentage error in x
x=5.8-3.22 x=2.58
| 877
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# Square and Cube Practice Set – Mathematics
1 . 136√? + 2841 = 8961 A. 2116 B. 1936 C. 2025 D. 2209 Answer: Option C Explanation: 136√? + 2841 = 8961 or 136√? = 8961-2841 or √? = 6120/136 = 45 ? = (45)2 = 2025 View Answer
2 . √? + 4 1/5 of 980 = 4180 A. 4096 B. 4225 C. 3969 D. 3844 Answer: Option A Explanation:√? + 21/5 of 980 = 4180 or √? + 4116 = 4180 or √? = 4180 – 4116 = 64 ? = (64)2 = 4096 View Answer
3 . 19√? + 291 = 340 * 2/5 + 364 A. 100 B. 121 C. 144 D. 81 Answer: Option B Explanation:19√? + 291 = 340 * 2/5 + 364 or 19√? + 291 = 136 + 364 or 19√? = 500 – 291 = 209 √? = 290/19 = 11 ? = (11)2 = 121 View Answer
4 . (√77284)2 A. 278 B. 268 C. 272 D. 262 Answer: Option A Explanation:(√77284)2 = ?2 or ? = √77284 = 278 View Answer
5 . √5776/√289 = ?/493 A. 2404 B. 2104 C. 2304 D. 2204 Answer: Option D Explanation:√5779/√289 = ?/493 or 76/17 = ?/493 ? = 493*76 / 17 = 2204 View Answer
6 . 48+√49+25-√25 / √144-√81 A. 26 B. 25 C. 24 D. 23 Answer: Option B Explanation:? = 48+√49+25-√25 / √144-√81 = 48+7+25-5 / 12-9 = 75/3 = 25 View Answer
Grouping of Identical Figures Practice Set
# Grouping of Identical Figures Practice Set
Grouping of Identical Figures Practice Set
# Grouping of Identical Figures Practice Set
Grouping of Identical Figures Practice Set
# Grouping of Identical Figures Practice Set
Grouping of Identical Figures Practice Set
# Grouping of Identical Figures Practice Set
Grouping of Identical Figures Practice Set
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# Triangle vertex
• Jul 10th 2008, 09:29 PM
euclid2
Triangle vertex
The points A(5,1) and B(-3,6) represent one of the equal sides of an isosceles triangle. Determine one of the possible points that would represent the third vertex of the triangle. Provide calculations to support your answer.
Thanks
• Jul 10th 2008, 09:44 PM
earboth
Quote:
Originally Posted by euclid2
The points A(5,1) and B(-3,6) represent one of the equal sides of an isosceles triangle. Determine one of the possible points that would represent the third vertex of the triangle. Provide calculations to support your answer.
Thanks
The third vertex of the triangle is located on a circle around A with radius $\displaystyle r = \sqrt{89}$ , that means the circle passes through B.
OR
the third vertex of the triangle is located on a circle around B with radius $\displaystyle r = \sqrt{89}$ , that means the circle passes through A.
EDIT: I've repaired a silly mistake. Thank you, kalagota!
• Jul 10th 2008, 09:46 PM
kalagota
first, compute the distance between A and B.
second, the set of all points that satisfies you condition is the set of (x,y) such that the distance between A (or B) and (x,y) is equal to the distance between A and B.
so for example, $\displaystyle d^2 = (x_B-x_A)^2 + (y_B-y_A)^2 = 89$
we consider: from point A, we will find a point (x,y) such that the distance of A and (x,y) is $\displaystyle d$.
so $\displaystyle d^2 = 89 = (x-5)^2 + (y-1)^2$ (this equation represents a circle) the point is, you can find any (x,y) that satisfies that equation..
NOTE: if you find an (x,y), be sure that A,B and (x,y) are not collinear.
you can also do this with respect to point B. so, there will be a new equation.. with same d.
• Jul 10th 2008, 09:48 PM
kalagota
Quote:
Originally Posted by earboth
The third vertex of the triangle is located on a circle around A with radius $\displaystyle r = \sqrt{91}$ , that means the circle passes through B.
OR
the third vertex of the triangle is located on a circle around B with radius $\displaystyle r = \sqrt{91}$ , that means the circle passes through A.
91? hmm, $\displaystyle (5 - -3)^2 + (1 - 6)^2 = 8^2 + (-5)^2 = 64 + 25$
• Jul 10th 2008, 09:52 PM
earboth
In addition to my previous post I've attached a sketch of the situation.
I've choosen a point on the circle $\displaystyle c_A$ (approximately (-8, -2) check if this point lies exactly on this circle!)
and
a point on the circle $\displaystyle c_B$ (approximately (0,-7) check if this point lies exactly on this circle!)
• Jul 10th 2008, 10:03 PM
kalagota
Quote:
Originally Posted by earboth
a point on the circle $\displaystyle c_B$ (approximately (-7, 0) check if this point lies exactly on this circle!)
that should be (0,-7)
• Jul 10th 2008, 10:15 PM
euclid2
are you suggesting that either one of those points are appropriate for the third vertex, if so how did you come up with those? and how do i verify they are correct?
• Jul 10th 2008, 10:21 PM
earboth
Quote:
Originally Posted by euclid2
are you suggesting that either one of those points are appropriate for the third vertex, if so how did you come up with those? and how do i verify they are correct?
You only have to use the equations kalagota has posted in this thread:
Take (-8, -2) and plug in the coordinates into the appropriate equation:
$\displaystyle (x+3)^2+(y-6)^2=89$ will yield:
$\displaystyle (-8+3)^2+(-2-6)^2=89$ Check! So (-8,-2) is indeed a vertex of the isosceles triangle.
• Jul 10th 2008, 10:24 PM
euclid2
Ok thanks.
I'm just not sure where you get the (-8,-2) from that's all?
• Jul 11th 2008, 01:18 AM
earboth
Quote:
Originally Posted by euclid2
Ok thanks.
I'm just not sure where you get the (-8,-2) from that's all?
If you have a look at the drawing I posted in one of my previous post you'll probably find some points with integer coordinates which lie on one of the circles. These points are my first guess.
Since a drawing isn't accurate enough I advised you to check, if the point lies exactly on the circle by using the equations of the circle. That's all, nothing mysterious about it.
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https://developer.apple.com/documentation/accelerate/working_with_matrices?changes=_8
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Article
# Working with Matrices
Solve simultaneous equations and transform points in space.
## Overview
A matrix is a 2D array of values arranged in rows and columns. The simd library provides support for matrices of up to four rows and four columns, containing 16 elements. It uses a column major naming convention; for example, a `simd_double4x2` is a matrix containing four columns and two rows.
The simd library provides initializers that include options for creating matrices from either rows or columns from the appropriately sized vectors. For example, the following code uses two vectors of four elements to create a 2 x 4 matrix and a 4 x 2 matrix:
The following examples show a few common uses of matrices.
### Solve Simultaneous Equations
You can use matrices to solve simultaneous equations of the form AX = B; for example, to find x and y in the following equations:
You first create a 2 x 2 matrix containing the left-side coefficients:
Then create a vector containing the right-side values:
To find the values of x and y, multiply the inverse of the matrix `a` with the vector `b`:
The result, `x`, is a two-element vector containing `(x = -2.6, y = 1.8)`.
### Transform Vectors with Matrix Multiplication
Matrices provide a convenient way to transform (translate, rotate, and scale) points in 2D and 3D space.
The following image shows point A translated to B, rotated to C, and scaled to D:
By representing 2D coordinates as a three-element vector, you can transform points using matrix multiplication. Typically, the third component of the vector, `z`, is set to 1, which indicates that the vector represents a position in space.
For example, the vector shown as A in the preceding illustration is defined as a `simd_float3` with the following code:
Transform matrices for 2D coordinates are represented by 3 x 3 matrices.
#### Translate
A translate matrix takes the following form:
The simd library provides constants for identity matrices (matrices with ones along the diagonal, and zeros elsewhere). The 3 x 3 `Float` identity matrix is `matrix_identity_float3x3`.
The following function returns a `simd_float3x3` matrix using the specified `tx` and `ty` translate values by setting the elements in an identity matrix:
To apply a translate to the position vector, you multiply the pair together:
The resulting `translatedVector` has the values `(x: 4.0, y: 5.0, z: 1.0)`, shown as B in the illustration above.
#### Rotate
A rotation matrix around the z-axis (that is, on the xy plane) takes the following form:
The following function returns a `simd_float3x3` matrix using the specified rotation angle in radians:
To apply a rotation to the previously translated vector, you multiply the pair together:
The resulting `rotatedVector` has the values `(x: 0.964102, y: 6.33013, z: 1.0)`, shown as C in the illustration above.
#### Scale
A scale matrix takes the following form:
The following function returns a `simd_float3x3` matrix using the specified x and y scale values:
To apply a scale to the previously rotated vector, you multiply the pair together:
The resulting `scaledVector` has the values `(x: 7.71282, y: 7.91266, z: 1.0)`, shown as D in the illustration above.
The three transform matrices can be multiplied together and the product multiplied with the position vector to get the same result:
### Vectors, Matrices, and Quaternions
Working with Vectors
Use vectors to calculate geometric values, calculate dot products and cross products, and interpolate between values.
Working with Quaternions
Rotate points around the surface of a sphere, and interpolate between them.
Rotating a Cube by Transforming Its Vertices
Rotate a cube through a series of keyframes using quaternion interpolation to transition between them.
simd
Perform computations on small vectors and matrices.
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