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Gravel is being dumped from a conveyor belt at a rate of 30 cubic feet per minute. It forms a pile in the shape of a right circular cone whose base diameter and height are always equal to each other. How fast is the height of the pile increasing when the pile is 14 feet high?

  • calculus - ,

    radius of cone --- r
    height of cone ---h

    vol = (1/3)πr^2 h
    but h = 2r or r = h/2
    Vol = (1/3)π (h^2/4)(h)
    = (1/12)π h^3

    d(Vol)/dt = (1/4)π h^2 dh/dt
    so when h = 14

    30 = (1/12)π(196) dh/dt

    solve for dh/dt

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