$($This question is meant to be solved without a calculator.$)$

A coffee pot has the shape of a cylinder with a radius of $5$ inches, as shown in the figure below. Let $h$ be the depth of the coffee in the pot, measured in inches, where $h$ is a function of time $t,$ measured in seconds. The volume $V$ of coffee ith the pot is changing at a rate of $-5\sqrt[\phantom{2}]{h}i\frac{n^2}{sec}.$

a$)$ Show that $\frac{dh}{dt}=-\frac{\sqrt[\phantom{2}]{h}}{5_{\text{I}}}.$

b$)$ Given that $h=17$ at time $t=0,$ solve the differential equation $\frac{dh}{dt}=-\frac{\sqrt[\phantom{2}]{h}}{5}$ for $b$ as a function of $t.$

c$)$ At what time $t$ is the coffee pot empty?