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4. A rectangular enclosure is formed by using 800m of fencing. Find the greatest possible area that can be enclosed in this way and the corresponding dimensions of the rectangle.

  • Engineering -

    4 x = 800
    x = 200 feet on a side square

    prove that square is max area

    A = x y
    P = 2 x + 2 y perimeter
    so x + y = P/2
    A = x (P/2 - x) = -x^2 +P/2 x

    x^2 - (P/2) x = A find vertex,complete square
    x^2 - (P/2)x + P^2/16 = -A + P^2/16

    (x-P/4)^2 = -(A-P^2/16)

    vertex at x= P/4 (then y = P/4 too :)
    A = (P/4)^2
    so square with side = Perimeter/4 is max area

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