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6 March, 22:39

A bucket that weighs 6 lb and a rope of negligible weight are used to draw water from a well that is 80 ft deep. The bucket is filled with 40 lb of water and is pulled up at a rate of 2 ft/s, but water leaks out of a hole in the bucket at a rate of 0.2 lb/s. Find the work done in pulling the bucket to the top of the well.

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  1. 6 March, 22:46
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    The work done in pulling the bucket to the top of the well is 3,360 ft-lb

    Explanation:

    Given

    Weight = 6 lb

    Depth = 80ft

    Weight of Water = 40lb

    Rate = 2ft/s

    Leak Rate = 0.2ft/s

    Calculating Workdone to lift the bucket

    Work = Force * Distance

    Work = 6 * 80

    Work = 480ft-lb

    At time t, the bucket is xi = 2t above the original depth of 80ft.

    t = ½xi

    But it now holds 40lb - 0.2t lb of water

    = 40 - 0.2 (½xi)

    = 40 - 0.1xi.

    This is the size of the water when it is x ft above the original depth.

    To move this amount of water, we need (40 - 0.1xi) ∆x

    So, W = ∫ (40 - 0.1xi) ∆x {1, n}

    Where n = 80

    W = ∫ (40 - 0.1x) dx {0,80}

    W = 40x - ½ (0.1x²) {0,80}

    W = 40x - x²/20 {0,80}

    W = 40 (80) - 80²/20

    W = 3200 - 320

    W = 2880 ft-lb

    The work done in pulling the bucket to the top of the well = 2880 + 480

    = 3,360 ft-lb
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