Models & Qualitative Behavior · Superchallenge problem

A water clock is a vessel shaped as a surface of revolution about a vertical axis, draining through a small hole at its bottom. By Torricelli's law, $A(h)\,h' = -c\sqrt{h}$, where $A(h)$ is the area of the water surface at depth $h$ and $c > 0$ is constant. The vessel is shaped so the water level falls at a constant rate. Its radius is $8$ cm at height $16$ cm above the hole. What is the volume (in cm$^3$) of the vessel up to that height? Give an exact answer.