Answer
a. $\frac{dS}{dt}=(4\pi r+2\pi h)\frac{dr}{dt}$
b. $\frac{dS}{dt}=2\pi r\frac{dh}{dt}$
c. $\frac{dS}{dt}=(4\pi r+2\pi h)\frac{dr}{dt} +2\pi r\frac{dh}{dt} $
d. $\frac{dr}{dt}=-\frac{r}{2r+h} \frac{dh}{dt} $
Work Step by Step
a. Given $S=2\pi r^2+2\pi rh$, we have $\frac{dS}{dt}=(4\pi r+2\pi h)\frac{dr}{dt}$ where $h$ is constant.
b. $\frac{dS}{dt}=2\pi r\frac{dh}{dt}$ where $r$ is constant.
c. $\frac{dS}{dt}=(4\pi r+2\pi h)\frac{dr}{dt} +2\pi r\frac{dh}{dt} $
d. $4\pi r\frac{dr}{dt}+2\pi h\frac{dr}{dt}+2\pi r \frac{dh}{dt} =0$ which gives
$(2 r+ h)\frac{dr}{dt}+ r \frac{dh}{dt} =0$ and $\frac{dr}{dt}=-\frac{r}{2r+h} \frac{dh}{dt} $
