Water is leaking out of an inverted conical tank at a rate of 1100 cubic centimeters per min at the same time that water is being pumped into the tank at a constant rate. The tank has height 15 meters and the diameter at the top is 4.5 meters. If the water level is rising at a rate of 23 centimeters per minute when the height of the water is 3 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 1100 cubic centimeters per min at the same time that water is being pumped into the tank at a constant rate. The tank has height 15 meters and the diameter at the top is  4.5 meters. If the water level is rising at a rate of 23 centimeters per minute when the height of the water is 3 meters, find the rate at which water is being pumped into the tank in cubic centimeters per minute.

Damon · Answer

find the surface area of the water in the tank at 3 meters from the bottom (pointy end), pi r^2 (in CENTIMETERS squared)
then
dh (pi r^2 = dV
pi r^2 (dh/dt ) = dV/dt
so
dh/dt = (1/pir^2) dV/dt
23 = (1/pir^2)(flow rate in - 1100)