Posted by **Anna** on Sunday, December 20, 2009 at 12:29pm.

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 -
**MathMate**, Sunday, December 20, 2009 at 1:10pm
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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