Answer
$\frac{dr}{dt}=1ft/min$
$\frac{dA}{dt}=40\pi ft^2/min$
Work Step by Step
Step 1. We are given that the balloon is inflated at a rate of $\frac{dV}{dt}=100\pi ft^3/min$
Step 2. Using the formula $V=\frac{4}{3}\pi r^3$, we have $\frac{dV}{dt}=4\pi r^2\frac{dr}{dt}=100\pi$
Step 3. At $r=5$, we have $4\pi 5^2\frac{dr}{dt}=100\pi$ and $\frac{dr}{dt}=\frac{100\pi}{100\pi}=1ft/min$
Step 4. For the surface area $A=4\pi r^2$, we have $\frac{dA}{dt}=8\pi r\frac{dr}{dt}=8\pi (5)(1)=40\pi ft^2/min$
