Rate of change of the depth of water in a conical tank

A conical tank with vertex down is 8 metres in diameter and 12 metres deep. Water flows into the tank at 10 m³ per minute. Find the rate of change of the depth of the water at the instant when the water is 6 metres deep.

Solution

Ratio between height and radius: h/r = 12/4 → h = h/3

Volume of conical = 1/3 x π x r² x h
                                 = 1/3 x π x (h/3)² x h

                                 = 1/27 x π x h³

dV/dh = 1/9 x π x h²

Rate of change of volume dV/dt = dV/dh x dh/dt

                                                    10  = 1/9 x  π x h² x dh/dt

When the deep of water is 6 cm

                                                    10  = 1/9 x  π x 6² x dh/dt

                                                    dh/dt = 90/(36π) = 5/(2 π)