Pre Calculus

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A contractor is to fence off a rectangular field along a straight river, the side along the river requiring no fence.
What is the least amount of fencing needed to fence of 30,000 meters squared?

  • Pre Calculus -

    perimeter = L + 2 W = p

    L W = 30,000
    so
    L = 30,000/W
    then
    p = 30,000/W + 2 W

    dp/dW = -30,000/W^2 + 2
    for in or max, dp/dW = 0
    30,000 = 2 W^2
    W^2 = 15,000
    W = 122.5
    then L = 30,000/122.5 = 244.9

  • Pre Calculus -

    above gives p = 244.9+2*122.5 = 490

    If you do not know derivatives then

    p = 30,000/W + 2 W
    2 W^2 - Wp +30,000 = 0
    W^2 - (p/2)W = -15,000
    W^2 - (p/2)W + p^2/16 = -15,000 + p^2/16

    (W-p/4)(W-p/4) = 1/16( p^2 - 240,000)
    that is vertex at W = p/4 and p = sqrt(240,000) or 490 sure enough

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