Calculus...please help solve. Sand falls from an overhead hopper to form a right circular cone.?

Question

If the cone formed has an angle a find the rate of change of volume with respect to height? If the height is changing at 2cms per minute, what volume of sand is falling from the hopper when the height of the cone is 3 meters? (The volume of a cone is (1/3)pr2h where h is the height and r is the radius.)

Answer 1

If the angle of the cone is a, then we have tan a = h/r. So r = h/tan a = h cot a.
V = (1/3) πr^2h = (1/3) π h^3 cot^2 a.
So dV/dh = π h^2 cot^2 a.

dh/dt = 2 cm/min = 0.02 m/min, h = 3m:
dV/dt = dV/dh dh/dt = π h^2 cot^2 a dh/dt
= (0.18π cot^2 a) m^3/min.