### Video Transcript

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.

Weโll begin by recalling the average rate of change formula. The average rate of change over continuous function ๐ over the closed interval ๐ to ๐ is given by ๐ of ๐ minus ๐ of ๐ over ๐ minus ๐. Now the problem we have here is we donโt 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 letโs define a function for the area with respect to its side length.

We know that if ๐ฅ is the side length in centimeters of a square, the area of that square is defined by ๐ด of ๐ฅ, which is simply equal to ๐ฅ squared. So, that is our function. And in fact, we know this is indeed a continuous function as required. ๐ฅ squared is a monomial. In other words, itโs 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 weโre interested in the rate of change of the area when its side length changes from 81 centimeters to 81.2 centimeters. So, weโll let ๐ be equal to 81 and ๐ be equal to 81.2. Substituting what we know into the average rate of change formula โ and, of course, our function is defined as ๐ด โ and we see the average rate of change is ๐ด of 81.2 minus ๐ด of 81 over 81.2 minus 81. Now, of course, seeing as our function ๐ด of ๐ฅ is equal to ๐ฅ squared, ๐ด of 81.2 is 81.2 squared and ๐ด 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. Itโs 162.2 square centimeters per centimeter.