Respuesta :
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.
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.