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13 January, 00:41

Model a water tank by a cone 4040 ft high with a circular base of radius 2020 feet at the top. Water is flowing into the tank at a constant rate of 8080 cubic ft/min. How fast is the water level rising when the water is 1212 ft deep

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  1. 13 January, 00:51
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    dh / dt =.13 ft / s

    Explanation:

    Let at height h, radius be r within the cone.

    20 / 40 = r / (40-h)

    2r = 40 - h

    r = (40 - h) / 2

    Volume of upper cone

    V = 1/3 πr² (40 - h)

    = 1/3 π (40 - h) ² x (40 - h) / 4

    = 1/12 x π (40 - h) ³

    Differentiating on both sides

    dV / dt = 1/12 x3 x π (40 - h) ² dh / dt

    = π (40 - h) ² / 4 dh / dt

    Given

    dV / dt = 80, h = 12

    80 = π / 4 x (40 - 12) ² dh / dt

    dh / dt =.13 ft / s
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