Water is leaking out of an inverted conical tank at a rate of 12,500 cm3/min at the same time that water is being pumped into the tank at a constant rate.

Question

The tank has height 6 m and the diameter at the top is 4 m. If the water level is rising at a rate of 20 cm/min when the height of the water is 2 m, find the rate at which water is being pumped into the tank. (Round your answer to the nearest integer.)

Answer 1

Volume of cone, V = πr²h/3

r/h = 2/6 given

r = h/3

V=π(h/3)²h/3 = πh³/27

dV/dt = 3(π/27)h²dh/dt + 12500 cm³/min

At h = 2m = 200 cm and dh/dt = 20 cm/min

dV/dt = π (40000/9)20+12500 cm³/min You have to add the 12,500 to compensate for the leakage.

= π(800000/9)+12500

≈ 291753 cm³/min