Water is leaking out of an inverted conical water tank at a rate of 12,000 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 water tank at a rate of 12,000 cm³/min at the same time that water is being pumped into the tank at a constant rate.

The tank has height H=18�=18m and the diameter at the top is d=10�=10m. If the water level is rising at a rate of 20 cm/min when the height hℎ of the water is 7 m, find the rate at which water is being pumped into the tank.

(Note that the proportions shown in the diagram may not match the randomly generated parameters of your question.)

 

Give your answer in cm³/min to the nearest thousand.

Answer 1

Draw a diagram.

Let h = height of water at time t and r = radius of water surface at time t.

Using similar triangles, 5 / 18 = r / h

5h = 18r So, r = (5/18)h

V = (1/3)πr²h = (1/3)π(5/18)²h³

dV/dt = π(5/18)²h²(dh/dt)

c - 12000 = π(5/18)²(49)(20)

Solve for c.