# calcalus

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Water is leaking out of an inverted conical tank at a rate of 12200 cubic centimeters per minute 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 65 meters. If the water level is rising at a rate of 21 centimeters per minute when the height of the water is 10 meters, find the rate at which water is being pumped into the tank in cubic centimeters per minute

• calcalus -

when the water is at depth y, the radius r of the surface can be found using similar triangles:

r/y = (65/2)/15
r = 13/6 y

at depth y, the volume of water is

v = 1/3 pi r^2 y
= 1/3 pi (13/6)^2 y^3
= 169pi/108 y^3

dv/dt = -12200+C = 169pi/36 y^2 dy/dt

at y=10m=1000cm, dy/dt = 21cm/min

-12200 + C = 169pi/36 * 1000000 * 21 = 309708675
C = 309696475

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