Gravel is being dumped from a conveyor belt at a rate of 15 ft^3/hr and its coarseness is such that is forms a pile in the shape of an inverted right cone whose height is three times its base radius. How fast is the height of the pile increasing when the pile has a height of 10 feet?

Answer 1

Q: Gravel is being dumped from a conveyor belt at a rate of 25 ft3/min, and its coarseness is such that it forms a pile in the shape of a cone whose base diameter and height are always equal. How fast is the height of the pile increasing when the pile is 6 ft high? (Round your answer to two decimal places.)

A: since h = 2r, r = h/2
v = pi/3 r^2h = pi/12 h^3
So,
dv/dt = pi/4 h^2 dh/dt

When h=6, we have

25 = 9pi dh/dt

the units is a bit different but the same idea. Hope it helps

Answer 2

h=3r, so r = h/3

v = π/3 r^2 h = π/3 (h/3)^2 h = π/27 h^3

dv/dt = π/9 h^2 dh/dt

So, plug in your numbers to get dh/dt.