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calculus

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Water is flowing freely from the bottom of a conical tank which is 12 feet deep and 6 feet in radius at the
top. If the water is flowing at a rate of 2 cubic feet per hour, at what rate is the depth of the water in the
tank going down when the depth is 3 feet.

  • calculus -

    When the water is x feet deep, and the surface has radius r, then by similar triangles,

    x/r = 12/6
    x = 2r
    r = x/2

    So, since the volume of water

    v = 1/3 pi r^2 x
    = 1/12 pi x^3

    dv/dt = pi/4 x^2 dx/dt

    -2 = pi/4 (9) dx/dt

    dx/dt = -8/(9pi) = -.28 ft^3/hr

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