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March 25, 2017

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The Maximum Garden Problem. A farmer has 230 ft
of fence to enclose a rectangular garden. What is the
largest garden area that can be enclosed with the 230 ft
of fence?

  • Math - ,

    The quick answer is that for a given perimeter, the largest area possible is a square. Thus each side equals 1/4 of the perimeter = 230/4 = 57.5'
    Area = 57.5² sq.ft.
    = 3306.25 sq.ft.

    The mathematical method is to calculate the area in terms of the permimeter and one of the sides (x). Differentiate with respect to x and equate the derivative to zero to find the maximum area.

    Length of one side = x
    Length of the other side (of rectangle)
    = 230/2-x
    Area,
    A(x)
    = x(115-x)
    = 115x -x²
    A'(x) = 115-2x
    x=57.5
    A"(x) = -2 <0 so x=57.5 is a maximum.
    Proceed to calculate the area.

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