Jay'onia B.

asked • 05/11/15# Sand falls from a conveyer belt at a rate of 15m^3/min. How fast is the height changing when the pile is 2m high?

This is a related rates problem using cones. I am in Calculus.

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## 1 Expert Answer

Richard P. answered • 05/11/15

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This problem can be solved if an assumption is made about the angle of repose for a sand pile. According to Wikipedia, the angle of repose for dry sand is ~ 34°. A little bit of trig shows that with this angle, the relationship between the radius of the circular base,r, and the height, h , is r = h tan(56°)

The volume, V, is (1/3) π r

^{2}h. = (1/3) π [h tan(56°) ]^{2}h The time derivative of this is equal to the rate at which sand is being added. So

15 = π h

^{2}[tan(56°)]^{2}dh/dt Plugging in h =2 and solving for dh/dt yields:

dh/dt = 2.17 m/min.

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Jon P.

05/11/15