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26 April, 05:23

A fence must be built to enclose a rectangular area of 20,000 square feet. fencing material cost $2.50 per foot for the two sides facing north and south and $3.50 per foot for the other two sides. find the cost of the least expensive fence.

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  1. 26 April, 05:31
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    The area is:

    A = x * y = 20000

    The cost function is:

    C = 2.50 (2x) + 3.50 (2y)

    Rewriting we have:

    C = 5x + 7y

    Writing as a function of x we have:

    C = 5x + 7 (20000 / x)

    Rewriting:

    C (x) = 5x + 140000 / x

    We derive:

    C ' (x) = 5-140000 / x ^ 2

    We equal zero and clear x:

    0 = 5-140000 / x ^ 2

    140000 / x ^ 2 = 5

    x ^ 2 = 140000/5

    x = root (140000/5)

    x = 167.33 feet

    Therefore the cost is:

    C (167.33) = 5 * (167.33) + 7 * (20000 / 167.33)

    C (167.33) = 1673.32 $

    Answer:

    The cost of the least expensive fence is:

    $ 1673.32
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