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

Find the average rate of change in the volume of a cube when its side length changes from 8 cm to 9.5 cm.

02:39

Video Transcript

Find the average rate of change in the volume of a cube when its side length changes from eight centimeters to 9.5 centimeters.

We’ll begin by recalling the average rate of change formula. The average rate of change of a continuous function 𝑓 over the closed interval π‘Ž to 𝑏 is given by 𝑓 of 𝑏 minus 𝑓 of π‘Ž over 𝑏 minus π‘Ž. And so, it’s clear to us that we’re going to need to find some function that describes the volume of a cube with respect to its side length. Well, given a cube, we know that its volume is its side length cubed. And so let’s define the side length of our cube to be equal to π‘₯ centimeters, and then volume is a function of π‘₯. We say that 𝑉 of π‘₯ is equal to π‘₯ cubed.

Now, when we defined the average rate of change formula, we did say that 𝑓 was a continuous function. So, we need to ask ourselves, is 𝑉 of π‘₯ a continuous function? 𝑉 of π‘₯ is, in fact, a monomial. In other words, it’s a polynomial with just one term. All polynomials, of course, are continuous over their entire domain. So we can say that yes, 𝑉 of π‘₯ equals π‘₯ cubed is indeed a continuous function. Next, let’s define the closed interval over which we’re finding the average rate of change. The side length and hence the value of π‘₯ is changing from eight to 9.5. So our closed interval will be the closed interval from eight to 9.5, meaning we could define π‘Ž to be equal to eight and 𝑏 to be equal to 9.5.

Substituting everything we know into the average rate of change formula, and we get 𝑉 of 9.5 minus 𝑉 of eight over 9.5 minus eight. Now, of course, we’re actually going to need to work out the value of 𝑉 of 9.5 and 𝑉 of eight. And since 𝑉 of π‘₯ is π‘₯ cubed, 𝑉 of 9.5 is 9.5 cubed, which is 857.375. Then, 𝑉 of eight is eight cubed, which is equal to 512. And so the average rate of change in the volume of the cube will be 857.375 minus 512 all over 9.5 minus eight which is of course equal to 1.5. That gives us a value of 230.25. And so that’s the average rate of change in the volume of the cube with respect to its side length. The answer then is 230.25.

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