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Rate of change of volume of a cone

Sand falls from an overhead hopper to form a right circular cone. If the cone formed has an angle theta 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)pir2h where h is the height and r is the radius.)

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Solution Preview

Please see the attached file.

If the cone has angle α, then we have the relation
r = h tanα
This essentially solves you problem, as follows:
The volume V = ...