A square lamina maintains its shape as it expands. Find the average rate of change in its area when its side changes from 81 centimeters to 81.2 centimeters.
We’ll begin by recalling the average rate of change formula. The average rate of change over continuous function 𝑓 over the closed interval 𝑎 to 𝑏 is given by 𝑓 of 𝑏 minus 𝑓 of 𝑎 over 𝑏 minus 𝑎. Now the problem we have here is we don’t actually have a function that we can use. However, we are considering the rate of change of the area of the shape when its side length changes. So let’s define a function for the area with respect to its side length.
We know that if 𝑥 is the side length in centimeters of a square, the area of that square is defined by 𝐴 of 𝑥, which is simply equal to 𝑥 squared. So, that is our function. And in fact, we know this is indeed a continuous function as required. 𝑥 squared is a monomial. In other words, it’s a polynomial with a single term. And we know that all polynomials are continuous over their entire domain. And so we can use the average rate of change formula.
Now we’re interested in the rate of change of the area when its side length changes from 81 centimeters to 81.2 centimeters. So, we’ll let 𝑎 be equal to 81 and 𝑏 be equal to 81.2. Substituting what we know into the average rate of change formula — and, of course, our function is defined as 𝐴 — and we see the average rate of change is 𝐴 of 81.2 minus 𝐴 of 81 over 81.2 minus 81. Now, of course, seeing as our function 𝐴 of 𝑥 is equal to 𝑥 squared, 𝐴 of 81.2 is 81.2 squared and 𝐴 of 81 is 81 squared. So this becomes 81.2 squared minus 81 squared divided by 0.2, which gives us 162.2. And so the rate of change of area with respect to its side length is 162.2.
The area is measured in square centimeters and the side length in centimeters. So if we wish to give units, this would be centimeters squared per centimeter. It’s 162.2 square centimeters per centimeter.