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Gravel is being dumped from a conveyor belt at a rate of 20 ft^3/min. 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 ft high?

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Answer 1 · 2016-11-11T09:55:59

#80/(19^2pi)# ft/min

We know #d=h, =>r=h/2#

So #V=1/3pi(h/2)^2h\ \ \ =>\ \ \ V(t)=pi/12h^3(t)#

Differentiating over t

#V'(t)=pi/4h^2(t)h'(t)#

We know #V'(t)=20 \ \ AA t\ \ ,\ \ \ h(t_0)=19#
So

#20=pi/4*19^2h'(t_0)#

#h'(t_0)=80/(19^2pi)#