Rate of Change

The volume of the snowball is increasing at $\displaystyle \begin{align*} 72\pi \textrm{ cm}^3 / \textrm{min} \end{align*}$ , so $\displaystyle \begin{align*} \frac{dV}{dt} = 72\pi \end{align*}$ , as long as t is measured in minutes.

You also know that $\displaystyle \begin{align*} V = \frac{4}{3}\pi \, r^3 \end{align*}$ , which means $\displaystyle \begin{align*} \frac{dV}{dr} = 4\pi \, r^2 \end{align*}$ .

By the chain rule, $\displaystyle \begin{align*} \frac{dV}{dt} = \frac{dV}{dr} \cdot \frac{dr}{dt} \end{align*}$ , so

$\displaystyle \begin{align*} 72\pi &= 4\pi \, r^2 \, \frac{dr}{dt} \\ 72 &= 4r^2 \, \frac{dr}{dt} \\ 18 &= r^2 \,\frac{dr}{dt} \\ \frac{18}{r^2} &= \frac{dr}{dt} \end{align*}$

So now it's just a case of evaluating what the radius is when the volume is $\displaystyle \begin{align*} 36\pi \textrm{ cm}^3 \end{align*}$ and substituting in to evaluate $\displaystyle \begin{align*} \frac{dr}{dt} \end{align*}$ .