calculus

At a sand and gravel plant, sand is falling off a conveyor and onto a conical pile at the rate of 10 cubic feet per minute. The diameter of the base of the cone is approx­ imately three times the altitude. At what rate is the height of the pile changing when it is 15 feet high?

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  1. let the height be h ft
    then diameter = 3h ft, and the radius is 3h/2

    V = (1/3)π r^2 h
    = (1/3)π (9h^2/4)(h) = (3/4)π h^3

    dV/dt = (9/4)π h^2 dh/dt

    when h = 15, dV/dt = 10
    10 = (9/4)π (15) dh/dt
    dh/dt = 40/(135π) ft/s
    = 8π/27 ft/s

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