Water is leaking out of an inverted conical tank at a rate of 14000.0 cubic centimeters per min at the same time that water is being pumped into the tank at a constant rate. The tank has height 8 meters and the diameter at the top is 6.5 meters. If the water level is rising at a rate of 17 centimeters per minute when the height of the water is 2 meters, find the rate at which water is being pumped into the tank in cubic centimeters per minute.

Question

Water is leaking out of an inverted conical tank at a rate of 14000.0 cubic centimeters per min at the same time that water is being pumped into the tank at a constant rate. The tank has height 8 meters and the diameter at the top is 6.5 meters. If the water level is rising at a rate of 17 centimeters per minute when the height of the water is 2 meters, find the rate at which water is being pumped into the tank in cubic centimeters per minute. 
Honestly have no clue on this one... please help!

Steve · Answer

Draw a side-view diagram. If the water is at depth y, then the surface of the water has radius 325/800 y. Thus the volume is

v = 1/3 pi (325/800 y)^2 y = 0.1729 y^3

dv/dt = 0.5187 y^2 dy/dt

Now just plug in your numbers. If water is being pumped in at c cm^3/min, then

c-14000 = 0.5187 * 200^2 * 17
c = 366716 cm^3/min