Math CentralQuandaries & Queries


Question from Gisela, a student:

As sand leaks out of a hole in a container, it forms a conical pile whose
altitude is always the same as its radius. If the height of the pile is increasing
at a rate of 6 in/min, find the rate at which the sand is leaking out when the
altitude is 10in.


The volume of a right circular cone is 1/3 π r2 h where r is the radius and h is the height. In your cone the height and the radius are the same so r =h and thus the volume is given by V = 1/3 π h3. V and h are both functions of time so if you differentiate both sides with respect to t you will have an equation involving dV/dt, h and dh/dt. You are told that dh/dt = 6 in/min and you are asked to find dV/dt when h = 10 in.


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