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14 May, 22:49

A rectangular plot of farmland will be bounded on one side by a river and on the other three sides by a single-strand electric fence. With 2400 m of wire at your disposal, what is the largest area you can enclose, and what are its dimensions?

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  1. 14 May, 23:51
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    x = 600 m

    y = 1200 m

    Amax = 720000 m²

    Step-by-step explanation:

    Let call x the smaller side of the rectangular plot and y the largest (we assume we have one y side bounded by a river: Then

    A (p) Area of the plot x*y

    A (p) = x*y

    And perimeter of the plot (to be fenced) is:

    P (p) = 2*x + y = 2400 ⇒ y = 2400 - 2*x

    Area of rectangular plot as function of x:

    A (x) = x * (2400 - 2x)

    Taking derivatives on both sides of the equation

    A' (x) = (2400 - 2x) + (-2) * x ⇒ A' (x) = (2400 - 2x) - 2x

    A' (x) = 0 ⇒ 2400 - 4x = 0 ⇒ 4x = 2400

    x = 600 m

    And y = 2400 - 2*x

    y = 2400 - 1200

    y = 1200 m

    And the largest enclosed area is Amax = 1200*600

    Amax = 720000 m²
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