Math CentralQuandaries & Queries


Question from Tracy, a student:

Sand falls from a conveyor belt at the rate of 10 cubic feet per minute onto a conical pile. The radius of the base is always equal to half the pile's height. How fast is the height growing when the pile is 5ft high?

Hi Tracy.

1) Find the relationship between the quantities.

The involved quantities are volume and height in your question. So we turn to the equation for the volume of a cone: V = 1/3 (pi) r2 h. Since the radius of the base always equals half the height, we have r = (h/2), so the equation for the volume becomes V = h3 / 12.

2) Differentiate the equation with respect to time. This will relate the rates. This means you differentiate both sides of the equation:

dV/dt = d(h3/12)/dt

dV/dt = (1/12) d(h3)/dt

dV/dt = 3h2/12 (dh/dt) <-- remember that the chain rule applies!

dV/dt = h2 / 4 (dh/dt)

3) Substitute in the values you know are given in the question. dV/dt is the rate of change of the volume. That's 10 cubic feet/minute. h is the height: 5 ft. dh/dt is the rate of change of the height, which is what you are asked for. So we simply plug them in and solve for dh/dt:

10 = 52 / 4 (dh/dt)

dh/dt = 40 / 25

dh/dt = 1.6 ft / minute

Hope this helps. In the future, let us know where you are stuck on a problem, rather than just sending the question.
Stephen La Rocque.

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