Answer
a. $\frac{dS}{dt}=\frac{2\pi r^2+\pi h^2}{\sqrt {r^2+h^2}}\frac{dr}{dt}$
b. $\frac{dS}{dt}=\frac{\pi rh}{\sqrt {r^2+h^2}}\frac{dh}{dt}$
c.$\frac{dS}{dt}=\frac{2\pi r^2+\pi h^2}{\sqrt {r^2+h^2}}\frac{dr}{dt}+\frac{\pi rh}{\sqrt {r^2+h^2}}\frac{dh}{dt}$
Work Step by Step
a. Given $S=\pi r\sqrt {r^2+h^2}$, we have $\frac{dS}{dt}=(\pi \sqrt {r^2+h^2}+\frac{\pi r(2r)}{2\sqrt {r^2+h^2}})\frac{dr}{dt}=(\frac{\pi(r^2+h^2)+\pi r^2}{\sqrt {r^2+h^2}})\frac{dr}{dt}=\frac{2\pi r^2+\pi h^2}{\sqrt {r^2+h^2}}\frac{dr}{dt}$ where $h$ is constant.
b. $\frac{dS}{dt}=(\frac{\pi r(2h)}{2\sqrt {r^2+h^2}})\frac{dh}{dt}=\frac{\pi rh}{\sqrt {r^2+h^2}}\frac{dh}{dt}$ where $r$ is constant.
c. Combine the results above, $\frac{dS}{dt}=\frac{2\pi r^2+\pi h^2}{\sqrt {r^2+h^2}}\frac{dr}{dt}+\frac{\pi rh}{\sqrt {r^2+h^2}}\frac{dh}{dt}$
