Answer to Question #256176 in Calculus for Alunsina

Question

A spherical balloon with radius "r" has a volume as shown. 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.

V(r)=4/3pi(r³)

Answer 1

For $r$ in inches and $V(r)$ in inches³

$V(r) =\dfrac{4}{3}\pi r^3\\\ \\V(r+1) =\dfrac{4}{3}\pi (r+1)^3$

The amount of air to add is the difference between V(r) and V(r+1). Subtract V(r):

$V(r+1)-V(r)=\dfrac{4}{3}\pi (r+1)^3-\dfrac{4}{3}\pi (r)^3$

$=\dfrac{4}{3}\pi[(r+1)^3-(r)^3]\\\ \\= \dfrac{4}{3}\pi[r^3+3r^2+3r+1-r^3]\\\ \\=\dfrac{4}{3}\pi [3r^2+3r+1]$$\ \ inches^3$

This is the final function.