Sand falls from an overhead bin, accumulating in a conical pile with a radius that is always three times its height. If the sand falls from the bin at a rate of 120 (ft^3) / min, how fast is the height of the sandpile changing when the pile is 10 feet high? [V = 1/3 (r^2) h]

Question

Mathematics

Zaiden Gardner

9 September, 16:14

Sand falls from an overhead bin, accumulating in a conical pile with a radius that is always three times its height. If the sand falls from the bin at a rate of 120 (ft^3) / min, how fast is the height of the sandpile changing when the pile is 10 feet high? [V = 1/3 (r^2) h]

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Answers (1)

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Camille Griffin · Answer 1

V = (1/3) Pi * (r^2) h

r = 3h = > V = (1/3) Pi*[ (3h) ^2]h = 3Pi * (h^3)

dV / dt = 9Pi (h^2) * [dh/dt]

dh/dt = [dV/dt] / [9Pi (h^2) ]

dV/dt = 120 (ft^3) / min

h = 10 ft

dh/dt = [120 (ft^3) / min] / [9Pi (10ft) ^2] = 0.042 ft/min

Answer: 0.042 ft/min