# Math

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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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