A cable that weighs 6 lb/ft is used to lift 500 lb of coal up a mine shaft 400 ft deep. Find the work done. Show how to approximate the requ

Question

A cable that weighs 6 lb/ft is used to lift 500 lb of coal up a mine shaft 400 ft deep. Find the work done. Show how to approximate the required work by a Riemann sum. (Let x be the distance in feet below the top of the shaft. Enter xi* as xi.)

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Anna 1 week 2021-11-23T17:37:09+00:00 1 Answer 0 views 0

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    2021-11-23T17:38:36+00:00

    Answer:

    8.2 *10^5 ft lb

    Step-by-step explanation:

    Wcoal = 800*500  = 400000 = 4*10^5 ft lb

    Wrope =\int\limits^{400}_0 {6(400-y)} \, dy

    Using a right Riemann sum

    The width of the entire region to be estimated = 400 – 0 = 400

    Considering 8 equal subdivisions, then the width of each rectangular division is 400/8 = 50

    F(x) = 6(400-y)

    \left[\begin{array}{cccccccccc}x&0&50&100&150&200&250&300&350 &400\\f(x)&2400&2100&1800&1500&1200&900&600&300&0\end{array}\right]

    Wrope = 50(2100) + 50(1800) + 50(1500) + 50(1200) + 50(900) + 50(600)

    + 50(300) + 50(0) = 420000 = 4.2 * 10^5 ft lb

    Note: Riemann sum is an approximation, so may not give a accurate value

    work done =  Wcoal + Wrope= 4*10^5+ 4.2 * 10^5 = 8.2 *10^5 ft lb

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