*Related rates problem* Gravel is being poured onto a pile at the rate of 180 m3

Question

*Related rates problem*
Gravel is being poured onto a pile at the rate of 180 m3/min. The pile is in the shape of a cone whose base diameter is always three times its height. Find the rate at which the diameter of the base is increasing when the pile is 6m high. *Related rates problem*
Gravel is being poured onto a pile at the rate of 180 m3/min. The pile is in the shape of a cone whose base diameter is always three times its height. Find the rate at which the diameter of the base is increasing when the pile is 6m high. *Related rates problem*

Explanation / Answer

For any cone:
V = ?r²h/3

If we require that d = 2r = 3h, then h = (2/3)r
V = ?r²(2/3)r/3
V = ?r³(2/9)

Rate is a parameter that is "with respect to time". So we differentiate with respect to 't'.
V' = 3?r²(2/9) r'
V' = ?r²(2/3) r'

We already know the volumetric flow rate V' = 180 m³/min.
180 = ?r²(2/3) r'
270 = ?r² r'

270 = ?(d/2)² r'
270 = ?d² r' / 4
1080 = ?d² r'
1080 = ?(3h)² r'
1080 = 9?h² r'
120 = ?h² r'

When height is 6m
120 = ?(6)² r'
120 = 36? r'
10/(3?) = r'

But that is the rate the radius is increase. The diameter is twice that.
d' = 2r' = 20/(3?)

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