Water is leaking out of an inverted conical tank at a rate of 13,000 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 13,000 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 (in cm³/min) at which water is being pumped into the tank. (Round your answer to the nearest integer.)

Answer 1

Lets call the volumetric flow rate into the tank, F (solving for this)

F - 13000 = dV/dt (in - out = accumulation)

V = (1/3)πr²h for an inverted cone with h being the height from peak to base

We also have a constraint due to the geometry that h/r = 6/2 or r = h/3

Substitute in for r

V = (1/27)πh³

Take the derivative wrt t:

dV/dt = (1/9)πh² dh/dt

You can now calculate dV/dt from h and dh/dt given and then use dV/dt in the first equation to solve for F.

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