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26 September, 05:36

Gravel is being dumped from a conveyor belt at a rate of 35 ft3/min, and its coarseness is such that it forms a pile in the shape of a cone whose base diameter and height are always equal. How fast is the height of the pile increasing when the pile is 9 ft high? (Round your answer to two decimal places.)

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  1. 26 September, 05:42
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    dh/dt ≈ 0.55 ft/min

    Step-by-step explanation:

    The volume is given by the formula ...

    V = (1/3) πr²h

    We have r = h/2, so the volume as a function of height is ...

    V = (1/3) π (h/2) ²h = (π/12) h³

    Then the rates of change are related by ...

    dV/dt = (π/4) h²·dh/dt

    dh/dt = (4·dV/dt) / (πh²) = 4 (35 ft³/min) / (π (9 ft) ²)

    dh/dt ≈ 0.55 ft/min
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