Question

Sand pouring from a chute forms a conical pile whose height is always equal to the diameter. If the height increases at a constant rate of 5 feet per minute, at what rate is the sand pouring from the chute when the pile is 10 feet high

Answer 1

125π ft³/min

Explanation:

Volume of a cone = 1/3πr²h

r is the radius

h is the height.

r = d/2

Volume = 1/3π(d/2)²h

Volume = 1/3π(d²/4)h

Volume = 1/3πd²h/4

Since the diameter is equal to height, hence d = h

V = 1/3πh³/4

V = 1/12πh³

dV/dt = dV/dh × dh/dt

Given

dh/dt = 5ft/min

dV/dh = 3/12πh²

dV/dh = πh²/4

Given h = 10ft

dV/dh = π(10)²/4

dV/dh = 25πft²

Next is to get dV/dt

dV/dt = 25π × 5

dV/dt = 125π ft³/min

Hence the rate at which the sand is pouring from the chute when the pile is 10 feet high is 125π ft³/min