Water is leaking out of an inverted conical tank at a rate of 7,000 cm3/min at the same time that water is being pumped into the tank at a constant rate.

Question

Water is leaking out of an inverted conical tank at a rate of 7,000 cm³/min at the same time that water is being pumped into the tank at a constant rate. 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.

Answer 1

Let R is a rate of pamping water in tank. So rate dV/dt of water growth is equal: dV/dt = R - 7000.

Now volume of cone V = 1/3πr²h. No1/27οπππw we have that h/2r = 6/4, so r = h/3. We get V as function of h: V = 1/3πh³/9 = 1/27πh³. So R = 1/9πh²dh/dt + 7000 = 1/9π(200)²·20 + 7000 = 286252.68 cm³/min