A conical water tank with vertex down has a radius of 12 feet at the top and is 23 feet high. If water flows into the tank at a rate of 20 {\rm ft}^3{\rm /min}, how fast is the depth of the water increasing when the water is 12 feet deep?

Answer 1

Let the radius of the water level be r
let the height of the water level be h
by ratios,
r/h = 12/23
12h = 23r
r = 12h/23

V = (1/3)πr^2 h
= (1/3)π(144h^2/529)h
= (48π/529) h^3

dV/dt = (144π/529) h^2 dh/dt
for the given data:
20 = (144π/529)(144) dh/dt

I will leave it up to you to do the buttonpushing to find dy/dt