A spherical balloon with radius r inches has a volume V(r)=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...

A spherical balloon with radius r inches has a volume V(r)=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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Textbook Question

Chapter 1, 1.1 - Problem 26 - Calculus: Early Transcendentals (7th Edition, James Stewart).
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embizze | High School Teacher | (Level 1) Educator Emeritus

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The amount of air required to inflate a spherical balloon from radius r to radius (r+1) is required:

The difference in the volumes is:

`V_2-V_1=4/3 pi (r+1)^3-4/3 pi r^3`

`=4/3 pi[r^3+3r^2+3r+1-r^3]`

`=4/3 pi[3r^2+3r+1]`

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Given an initial radius r, the amount of air A needed to increase the radius of a spherical balloon is:

`A=4/3 pi(3r^2+3r+1)`

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