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Question

Sand is being poured onto a conical pile at the constant rate of 50 cm3/ minute such that the height of the cone is always one half of the radius of its base. How fast is the height of the pile increasing when the sand is 5 cm deep.

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Solution

Let r be the radius, h be the height and V be the volume of the conical pile at any time t. Then,V=13πr2hV=13π2h2h h=r2V=43πh3dVdt=4πh2dhdt50=4πh2dhdtdhdt=504π52dhdt=12π cm/min

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