Question

A gardener has 520 feet 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.

garden bordered by a river

What dimensions would guarantee that the garden has the greatest possible area?

shorter side: _____ft (feet)

longer side: ____ft (feet)

greatest possible area: ___ft2 (square-feet)

Answers

  1. The shortest side is 130 feet, the longest side is 260 feet and the greatest possible area is 33800 square feet

    What dimensions would guarantee that the garden has the greatest possible area?

    The given parameter is
    Perimeter, P = 520 feet
    Represent the shorter side with x and the longer side with y
    One side of the garden is bordered by a river:
    So the perimeter is:
    P = 2x + y
    Substitute P = 520
    2x + y = 520
    Make y the subject
    y = 520 – 2x
    The area is
    A = xy
    Substitute y = 520 – 2x in A = xy
    A = x(520 – 2x)
    Expand
    A = 520x – 2x^2
    Differentiate
    A’ = 520 – 4x
    Set to 0
    520 – 4x = 0
    Rewrite as:
    4x= 520
    Divide by 4
    x= 130
    Substitute x= 130 in y = 520 – 2x
    y = 520 – 2 *130
    Evaluate
    y = 260
    The area is then calculated as:
    A = xy
    This gives
    A = 130 * 260
    Evaluate
    A = 33800
    Hence, the shortest side is 130 feet, the longest side is 260 feet and the greatest possible area is 33800 square feet
    Read more about area at:
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