To follow up:

Implicitly differentiating with respect to \(t\), we obtain:

\(\displaystyle \d{A}{t}=2\pi\left(2r\d{r}{t}+\d{r}{t}h+r\d{h}{t}\right)\)

Hence:

\(\displaystyle \d{r}{t}=\frac{\d{A}{t}-2\pi r\d{h}{t}}{2\pi(2r+h)}\)

Plug in the given data:

\(\displaystyle \d{r}{t}=\frac{20\pi\frac{\text{cm}^2}{\text{s}}-2\pi (4\text{ cm})\left(-2\frac{\text{cm}}{\text{s}}\right)}{2\pi(2(4)+1)\text{ cm}}=\frac{18}{9}\,\frac{\text{cm}}{\text{s}}=2\,\frac{\text{cm}}{\text{s}}\)