A square lamina maintains its shape
as it expands. Find the rate of change in its area
with respect to side length when its side measures four centimeters.
We have a square lamina, which is a
two-dimensional flat object whose thickness is negligible. We’re asked to find the rate of
change of its area with respect to side length. And that’s given a particular side
length of four centimeters. Let’s begin by calling the side
length 𝑥. Then we can find an equation for
the area with respect to side length. Since our lamina is a square and we
know that the area of a square is the square of the side length, as a function of
side length 𝑥, the area 𝐴 of 𝑥 is equal to 𝑥 squared.
Now, normally, when we think of
rates of change, we think of the rate of change with respect to time. But we’re not given any information
about the rate of change with respect to time. All we have is the area as a
function of side length.
To find the rate of change with
respect to side length then, we differentiate the area function with respect to side
length 𝑥. That is, we find d𝐴 by d𝑥. And to do this, we use the power
rule for derivatives. This tells us that for any real
power 𝑛 and constant coefficient 𝑎, d by d𝑥 of the function 𝑎 times 𝑥 to the
power 𝑛 is 𝑛 multiplied by 𝑎 times 𝑥 raised to the power 𝑛 minus one. That is, we bring the power 𝑛 down
to the front and multiply by it and subtract one from the power or exponent. In our case, this means bringing
down the exponent two and multiplying by it. And then our new exponent is two
minus one. This is one, and since anything to
the power one is simply itself, we have d𝐴 by d𝑥 equals two 𝑥.
Now, to find the rate of change in
the lamina’s area when its side length is four, we substitute four for 𝑥 in d𝐴 by
d𝑥. So we have two times four, which is
eight. Since our area is measured in
square centimeters and our side length in centimeters, the units are square
centimeters per centimeter. That is the number of square
centimeters resulting from a one-centimeter change in side length. Hence, the rate of change of the
lamina’s area with respect to side length when its side measures four centimeters is
eight square centimeters per centimeter.