Question

A water tank has the shape of an inverted right circular cone with base radius 3 meters and height 6 meters. Water is being pumped into the tank at the rate of 12 meters3/sec. Find the rate, in meters/sec, at which the water level is rising when the water is 2 meters deep. Give 2 decimal places for your answer.

Answer 1

Given dV/dt , find dh/dt.

Use volume formula for cone.
Define radius in terms of height.
Differentiate with respect to 't'.
Sub in given values, dV/dt = 12, h=2.
Solve for dh/dt.


`V = (\pi)/(3) r^2 h \\ \\(r)/(h) = (3)/(6) \rightarrow r = (h)/(2) \\ \\ V = (\pi)/(3)((h)/(2))^2 h = (\pi)/(12) h^3 \\ \\ (dV)/(dt) = (\pi)/(12)(3h^2) (dh)/(dt) \\ \\ 12 = (\pi)/(12)(3*2^2) (dh)/(dt) \\ \\ (dh)/(dt) = (12)/(\pi)`