Calculus Practice Problems

A filter filled with liquid is in the shape of a vertex-down cone with a height of 12 inches and a diameter of 18 inches at its open (upper) end. If the liquid drips out the bottom of the filter at the constant rate of 3 cubic inches per second, how fast is the level of the liquid dropping when the liquid is 2 inches deep?

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  1. r=9
    h=12
    so, r = 3/4 h

    v = 1/3 pi r^2 h
    = 1/3 pi (3/4 h)^2 h
    = 3/16 pi h^3

    dv/dt = 9/16 pi h^2 dh/dt
    so, just solve

    -3 = 9/16 pi 2^2 dh/dt

    for dh/dt

    -3 = 4/3 pi (3/2)^2

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