Gravel is being dumped from a conveyor belt at a rate of 35 ft3/min,

Question

Gravel is being dumped from a conveyor belt at a rate of 35 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 13 ft high?

Answer 1

1. dv/dt = 35 ft³/min
2. h = diameter whereby h = 2r so r = 1/2h
3. Solving for dh/dt at h = 11 ft
4. V = 1/3πr²h
5. V = 1/3•π•(1/2h)²•h
6. V = 1/12•π•h³
7. dV/dt = 3/12•π•h²•dh/dt
8. dh/dt = (dV/dt)/(3/12•π•h²)
9. Plug in values
10. dh/dt = (35)/(3/12•π•13²)
11. dh/dt = 140/169π