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Question Video: Finding the Average Rate of Change of the Area of a Rectangle When Its Length Changes between Two Given Values Mathematics • Higher Education

A rectangular lamina has length 4 times its width. As it is heated, it expands, preserving this shape. What is the average rate of change of its area when its length changes from 21 to 23.5 cm?

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Video Transcript

A rectangular lamina has length four times its width. As it is heated, it expands, preserving this shape. What is the average rate of change of its area when its length changes from 21 to 23.5 centimeters?

We begin by recalling the average rate of change formula. For continuous function 𝑓, the average rate of change over a closed interval π‘Ž to 𝑏 is 𝑓 of 𝑏 minus 𝑓 of π‘Ž over 𝑏 minus π‘Ž. And so, it should be quite clear to us that we are going to need to define a function representing the area of the rectangular lamina. Let’s draw a sketch of our lamina. We’re going to define its width to be equal to π‘₯ centimeters. Now, since its length is four times its width, its length must be four π‘₯ centimeters. Its area is the product of these two dimensions. In square centimetres, it’s four π‘₯ times π‘₯, which is four π‘₯ squared. And so, π‘Ž as a function of π‘₯ is four π‘₯ squared.

Now, we need to be really careful. We defined π‘₯ to be equal to the width of the rectangular lamina. And we’re actually given information about its length, which we define to be equal to four π‘₯. Now, the value of π‘Ž will be the value of π‘₯ when four π‘₯ is equal to 21. If we divide both sides of this equation by four, we get π‘₯ equals 21 over four. Similarly, the value we’re going to use for 𝑏 is going to be the value of π‘₯ when four π‘₯, the length, is equal to 23.5 centimeters. So, dividing through by four and we get π‘₯ is equal to 23.5 over four or forty-seven eighths.

And we now have everything we need in order to be able to use the average rate of change formula. Before we do, we did say that the average rate of change formula is used for continuous functions. Now, our function four π‘₯ squared is a monomial. It’s a single-termed polynomial. And of course, we know that polynomial functions are continuous over their entire domain. And so, π‘Ž of π‘₯ itself must be a continuous function. And so, we can use the average rate of change formula. Now, the numerator of that formula will be 𝐴 of 𝑏 minus 𝐴 of π‘Ž. So that’s 𝐴 of 47 over eight minus 𝐴 of 21 over four. Since our function 𝐴 of π‘₯ is four π‘₯ squared, this is four times 47 over eight squared minus four times 21 over four squared, which is 445 all divided by 16.

Now, we need to be really careful with the denominator of our average rate of change function. Remember, we’re interested in the average rate of change of the area when the length changes from 21 to 23.5. And so, we’re not going to subtract the values of π‘₯. We are, however, going to subtract the values of four π‘₯. In other words, we’re going to subtract the lengths. We’re going to work out 23.5 minus 21. And so, the average rate of change of the area with respect to its length is 445 over 16 all divided by 23.5 minus 21. 23.5 minus 21 is 2.5. So, we calculate 445 over 16 divided by 2.5, and we get 89 over eight.

The average rate of change then of the area when its length changes from 21 to 23.5 centimeters is 89 over eight. And since area is square centimeters and length is centimeters, the units are square centimeter per centimeter.

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