A spherical balloon with radius r inches has a volume Vr43pi

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.

Solution

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]

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)