Question

A spherical balloon with radius r inches has volume defined by the function below. 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 + 2 inches. (Give the answer in terms of π and r.)
V(r) = 4/3 pi r^3

Answer 1

The amount of the air needed is exactly \(V(r+2)-V(r)\)
\((4/3)\pi(r+2)^3-(4/3)\pi r^3=(4/3)\pi((r+2)^3-r^3)\)
\(=(4/3)\pi(6r^2+12r+8)\)