I need help with this cal 1 related rates question?

Question

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?

6527 views around the world

You can reuse this answer
Creative Commons License

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)#