In a rectangle, opposite sides are congruent. Let one side have length x. The opposite side also has length x. The lengths of these two sides add to 2x. The lengths of all 4 sides add to 64, so the lengths of the other 2 sides add up to 64 - 2x. Each side measures 32 - x.
The rectangle has sides of length x and 32 - x.
The area of the rectangle is
A = LW
A = x(32 - x)
A = 32x - x^2
y = 32x - x^2 is a parabola that opens downward. The maximum value of the parabola is the vertex on top.
32x - x^2 = 0
(32 - x)x = 0
32 - x = 0 or x = 0
x = 32 or x = 0
Since the parabola is symmetric with respect to the vertical axis, the vertex has x-coordinate 16.
At x = 16, you get maximum area.
Two opposite sides measure 16 ft each. 32 - x = 32 - 16 = 16 The other two opposite sides also measure 16 ft.
Since all sides turned out to have length 16 ft, the rectangle is a square.
Answer: The maximum area is enclosed by a square with side 16 ft.
