A conical paper cup 4inches across the top and 6 inches deep leaks water at a rate of 1 cubic inch per minute. at what rate does the water level drop (a) when the water is 3 inches deep? (b) when the cup is half full?
a) It's given that $\displaystyle \frac{dV}{dt}=-1$. We know that $\displaystyle V = \frac{2\pi rh}{3}$ and $\displaystyle \frac{dV}{dh} = \frac{2\pi r}{3}$. We also know that, by similar triangles, $\displaystyle h=3$ implies that $\displaystyle r=1$.
Putting it all together, we have:
$\displaystyle \frac{dV}{dt}=\frac{dV}{dh}\cdot\frac{dh}{dt} \implies -1 = \frac{2\pi\cdot 1}{3}\cdot\frac{dh}{dt} \implies \frac{dh}{dt}=-\frac{3}{2\pi}$