Math CentralQuandaries & Queries


Question from Malik, a student:

Sand is leaking out of a hole at the bottom of a container at a rate of 90cm3/min. As it leaks out, it forms a pile in the shape of a right circular cone whose base is 30cm below the bottom of the container. The base radius is increasing at a rate of 6mm/min. If, at the instant that 600cm3 have leaked out, the radius is 12cm, find the amount of leakage when the pile touches the bottom of the container.


I find this problem strange. I don't think the rates given are necessary to solve it. The physical fact that is the key to the solution is that slope of the right circular cone of sand is a constant, that is if r is the radius of the pile in centimeters at time t minutes and h is the height in centimeters at the same time then r/h is a constant, regardless of the time.

The volume of a right circular cone is 1/3 π r2 h. You are told that the volume is 600 cm3 when r = 12 cm at some specific time and hence at this time

600 = 1/3 π 122 h

Solve for h and then find r/h at this time.

When the pile touches the bottom of the container h = 30 cm. At this time r/h has the same value as you found in the previous step. Solve for r. Now you know r and h when the pile touches the container so you can calculate the volume of the pile.

I hope this helps,

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