Question

Differential Equations

The right-circular conical tank shown in the figure below loses water out of a circular hole of area A at its bottom. When water leaks through a hole, friction and contraction of the stream near the hole reduce the volume of water leaving the tank per second to
cA ×sqrt(2gh), where c is an empirical constant.

Determine a differential equation for the height of the water h at time t > 0. The radius of the hole is 3 in., g = 32 ft/s2, and the friction/contraction factor is
c = 0.6

I've found dV/dt to be (8pi/15)sqrt(h), but I can't figure out how to get dh/dt

Answer 1

Since you have the rate of change of the volume, you need to express the volume in the conical container in terms of the radius and height, and since there is a direct relation between height and radius, you can get V as a function of h and convert dV/dt to dh/dt.

Answer 2

Ah, turns out I was relating dh/dt right but had dV/dt wrong. Thanks anyways!

Answer 3

Thanks, too.