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Calculate work to wind a cable

Given a 50 foot cable, weight 4 pounds per foot. Calculate the work required to wind the cable 25 ft, assuming the only gravity is the only force in the problem.


The force function is f(x) = 200 - 4x, i.e., the total weight of the cable minus 4 pounds per foot as we wind the cable up. So then,

    \[ W = \int_0^{25} (200-4x) \, dx = 200 (25) - 2 (625) = 5000 - 1250 = 3750 \text{ ft-lbs}. \]

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