Question

A spherical balloon with radius r inches has volume V(r) = 4 3 πr3. Find an expression that represents the amount of air required to inflate the balloon from a radius of r inches to a radius of r + 1 inches. (Express your answer in terms of π and r.)

Answer 1

\[
V(r)= \frac 43 \pi r^3\\
V(r+1)= \frac 43 \pi(r+1)^3\\

V(r+1) - V(r)= \frac 43 \pi(r+1)^3-\frac 43 \pi r^3=??
\]

Answer 2

After expanding and simplifying, one gets
\[

V(r+1)-V(r)=\frac{4}{3} \pi \left(3 r^2+3
r+1\right)
\]

Answer 3

In the above, we used
\[
(r+1)^3=r^3+3 r^2+3 r+1
\]