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A conical water tank with vertex down has a radius of 12 feet at the top and is 23 feet high. If water flows into the tank at a rate of 20 {\rm ft}^3{\rm /min}, how fast is the depth of the water increasing when the water is 17 feet deep?

  • calculus - ,

    at a depth of h ft, the radius of the water surface is r=(12/23)*h ft

    v = 1/3 pi r^2 h
    = pi/3 * (12/23)^2 * h^3

    dv/dt = pi(12/23)^2 h^2 dh/dt
    20 = pi(12/23)^2 (17)^2 dh/dt
    dh/dt = 2645 / 10404pi = 0.081 ft/min = 0.97 in/min

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