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calculus

A farmer wants to make three identical rectangular enclosures along a straight river, as in the diagram shown below. If he has 1200 yards of fence, and if the sides along the river need no fence, what should be the dimensions of each enclosure if the total area is to be maximized?

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  1. Let the length be 3x, and the width be y.
    Maximum area is achieved when 3x = 4y = 600
    So the dimensions are 600 by 150
    To see this, note that
    3x + 4y = 1200
    The area is
    a = 3xy = 3x(1200-3x)/4
    find that da/dx = 0 when x = 200

    In all these area problems, maximum area is when the available fencing is divided equally among the lengths and the widths.

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  2. Let,
    x = length
    y = width
    We know that the total length fences is 1200 m
    l 3x + y = 1200
    l y = 1200-3x (1)
    l A = xy (2)
    Substitute (1) in (2)
    l A = x(1200-3x)
    l A = 1200x-3x2
    Domain: 0 ≤ x ≤
    Using derivatives,
    l = 1200-6x
    Optimum: l 1200-6x=0
    l 6x = 1200
    Answer:
    x = 200m
    y = 1200 - 3(200)
    y = 600m

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