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A spherical balloon with radius $ r $ inches has volume $ V(r) = \frac{4}{3} \pi r^3 $. Find a function that represents the amount of air required to inflate the balloon from a radius of $ r $ inches to a radius of $ (r + 1) $ inches.

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4$\pi\left(r^{2}+r+\frac{1}{3}\right)$

01:51

Jeffrey Payo

Calculus 1 / AB

Calculus 2 / BC

Calculus 3

Chapter 1

Functions and Models

Section 1

Four Ways to Represent a Function

Functions

Integration Techniques

Partial Derivatives

Functions of Several Variables

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A multivariate function is a function whose value depends on several variables. In contrast, a univariate function is a function whose value depends on only one variable. A multivariate function is also called a multivariate expression, a multivariate polynomial, a multivariate series, or a multivariate function of several variables.

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In calculus, partial derivatives are derivatives of a function with respect to one or more of its arguments, where the other arguments are treated as constants. Partial derivatives contrast with total derivatives, which are derivatives of the total function with respect to all of its arguments.

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So we're given the formula for the volume of a sphere and that represents the volume of air in the balloon that has a radius of our. So if your balloon had a radius of R plus one, the volume would be V of R plus one equals 4/3 pi times are plus one quantity. Cute. Now the question asks us for the function that would represent the amount of volume increase. So this would be the difference between these two volumes. So the increase in volume would be the volume with a radius of R plus one minus the volume with a radius of our. So let's calculate that. So the volume with the radius of R plus one is 4/3 pi times are plus one quantity cubed and the volume with the radius of our is 4/3 pi times are cubed. Notice that we have a factor of 4/3 pi in both of these terms. So let's factor that out 4/3 pi times and now our job is to simplify this quantity are plus one quantity cubed minus R cubed Now, Perhaps sometime in your past, you learned to multiply Ah binomial using the binomial theorem. But just in case you didn't let's go ahead and work it out the regular old long way, the algebraic way. All right, so what we need to do is multiply our plus one times are plus one times are plus one and then don't forget we're going to subtract r cubed from that. So let's start by multiplying the first to our plus ones together using the foil method. And we have our squared plus two are plus one that will then be multiplied by R plus one, the 3rd 1 that we still haven't used yet. So now we still are carrying that 4/3 pi along for the ride. Now what I would do, there's different ways to do this. But what I would do is I would multiply r squared by both of these factors and that gives me r cubed plus r squared. And then I would multiply to our by both of these factors and that gives me to R squared plus two are And then I would multiply one by both of these factors and that would give me are plus one and then we don't want to forget. We still have minus R cubed at the end. So now is a good time. We could cancel the Ark loot cubed and the minus are cute. And now we'll combine some like terms. So we have 4/3 pi times What? Well, we have r squared and to our squared. So those add together to be three R squared we have to our and are. So those add together to be three are and we have one, so that's just one. Now I see nothing wrong with leaving the answer just like that. But you know, there are times when you get your answer. You look up your answer in the book and there's looks a little bit different, and you don't know if you're right or not. Sometimes you're right there, right, and your answers just look different. So here's another option for simplifying the answer. What if we factored three out of all the terms inside the brackets? It's a little bit strange to factor three out of one, but it could be done if we factored three out of all of those, we would have three times The quantity R squared plus are plus 1/3 and The reason for doing that is just to reduce it with the three on the bottom of the fraction. So this form of the answer looks like this four pi times the quantity R squared plus are plus 1/3 again, a nun s unnecessary step, in my opinion. But that's how you would get your answer to look like that.

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