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grain is being poured through a chute at rate of 8 ft^3/min and forms a conical pile whose radius of the bottom is always 1/2 the height of pile. How fast is circumference increasing when the pile is 10 feet high?

  • calculus -

    r = (1/2)h
    2r = h

    Let V be the volume

    V = (1/3)π r^2 h
    V = (1/3)π(r^2)(2r) = (2/3)π r^3
    dV/dt = 2πr^2 dr/dt
    when h = 10, r = 5
    8 = 2π(25)dr/dt
    dr/dt = 4/(25π)

    Let circumference be C

    C = 2πr
    dC/dt = 2π dr/dt
    so at that instance,
    dC/dt = 2π(4/(25π) = 8/25 ft/min

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