Question

A conical water tank with vertex down has a radius of 10 feet at the top and is 22 feet high. If water flows into the tank at a rate of 30 {\rm ft}^3{\rm /min}, how fast is the depth of the water increasing when the water is 14 feet deep?

I have tried several times but continue to mess up somewhere. Please help!

Answer 1

at a time of t min, height of water level = h, radius of water level = r

V = (1/3)πr^2 h
by ratios:
r/h = 10/22 = 5/11
11r = 5h
r = (5h/11)
V = (1/3)π(25h^2/121)h
= (25/363)π h^3
dV/dt = (25/121)π h^2 dh/dt

given: dV/dt = 30 ft^3/min
find: dh/dt when h = 14

30 = (25/121)π(196) dh/dt
dh/dt = appr .38197 ft/min