Related Rates: Water is leaking out of an inverted conical tank

Question

Water is leaking out of an inverted conical tank at a rate of 8400 cubic centimeters per min at the same time that water is being pumped into the tank at a constant rate. The tank has height 7 meters and the diameter at the top is 3.5 meters. If the water level is rising at a rate of 24 centimeters per minute when the height of the water is 2.5 meters, find the rate at which water is being pumped into the tank in cubic centimeters per minute.

Answer 1

Tough one!! Ok, normally volume of cone V = pi/3(r^2)h. However, in this case we can relate r and h because their ratios are constant. If you draw picture of the cone from the side, the radius would be 1.75m on the top and the total height is 7. Now draw a water level somewhere in the middle and label distance from bottom as H and radius at that level r. Through similar triangles, R/H = 1.75/7 or h = R*(7/1.75).

Let X = rate of flow into tank.

V = pi/3(r^2)h - now sub in for R

V = (pi/3)(1.75/7)^2* h^3 then differentiate wrt time t

dv/dt = (pi/3)(1.75/7)^2*3h^2*dh/dt then put in all the values making sure all units are in cubic meters (example 8400 cm^3 = 0.0084m^3)

(x - 0.0084) = (pi/3)*(175/2)^2*3*(2.5)^2(0.24)

Solve for x = 0.302974 m^3 / min. convert back to cm^3/min