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A gardener has 120 ft of fencing to fence in a rectangular garden. one side of the garden is bordered by a river and so it does not need any fencing.

1. what dimensions would guarantee a garden with an area of 1350 ft^2?
2. What dimensions would guarantee the greatest area? how much is the greatest area?

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  1. two width, each of x
    one length of y
    y + 2x = 120
    y = 120 - 2x

    area = xy
    = x(120-2x)
    = 120x - 2x^2

    1. 120x - 2x^2 = 1350
    x^2 - 60x + 675 = 0
    (x - 45)(x + 15) = 0
    x = 45 or x = -15, the last we will reject

    if x = 45 , y = 30

    2. area = 120 - 2x^2 is a parabola which opens downwards so the vertex will be the maximum .
    the x of the vertex is -120/(-4) = 30
    when x = 30
    y = 120-60 = 60

    A width of 30 and a length of 60 ft will produce the largest area of 1800 ft^2

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