Related Rates!?

Question

Sand is falling off a conveyor onto a conical pile at the rate of 15 cubic feet per minute. The diameter of the base of the cone is approximately twice the altitude. At what rate is the height of the pile changing when it is 10 feet high?


Please answer soon i have a test tomorrow!

Answer 1

Let
h = diameter cone
r = radius cone
V = volume cone

dV/dt = 15 ft³/min
h = r

Find dh/dt when h = 10 ft.

V = (1/3)πr²h = (1/3)πh²*h = πh³/3

Take the derivative.

dV/dh = πh²

dh/dt = (dV/dt) / (dV/dh) = 15 / (πh²) = 15/(πh²)

dh/dt = 15/(π*10²) = 15/(100π) = 3/(20π) ft/min