Gravel is being dumped from a conveyor belt at a rate of 10 cubic feet per minut

Question

Gravel is being dumped from a conveyor belt at a rate of 10 cubic feet per minute. It forms a pile in the shape of a right circular cone whose base diameter and height are always the same. How fast is the height of the pile increasing when the pile is 19 feet high?
Recall that the volume of a right circular cone with height h and radius of the base r is given by

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Explanation / Answer

volume of cone v=(1/3)r2h where r is radius , h is height

base diameter and height are always the same

d =h

diameter=2*radius

2r =h

r =h/2

volume of cone v=(1/3)(h/2)2h

v=(1/12)h3

differentiate with respect to time t

dv/dt=(1/12)3h2 dh/dt

dv/dt=(1/4)h2 dh/dt

Gravel is being dumped from a conveyor belt at a rate of 10 cubic feet per minute =>dv/dt =10

pile is 19 feet high =>h =19

10=(1/4)192 dh/dt

dh/dt =40/(192)

dh/dt=0.03527

height is increasing at 0.03527 ft /min

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