Write the word or phrase that best completes each statement or answers the question.
Provide the missing information.
-Ramon wants to fence in a rectangular portion of his back yard against the back of his garage for a vegetable garden. He plans to use 40 feet of fence, and needs fence on only three sides. Find the
Maximum area he can enclose. (Hint: The lengths of the 3 fenced sides of the rectangle must add
Up to 40.)
A) 200 sq. ft.
B) 400 sq. ft.
C) 100 sq. ft.
D) 225 sq. ft.

Question

Quizplus · Accepted Answer

The maximum area that can be enclosed by a fixed perimeter is achieved with a square shape. However, since only three sides need fencing, the shape becomes a rectangle with two sides equal. Let the length of the two equal sides be $x$ and the third side (along the garage) be $y$. The perimeter is $2x + y = 40$. To maximize the area $A = xy$, we set $y = 40 - 2x$, and the area becomes $A = x(40 - 2x)$. Differentiating with respect to $x$ and setting the derivative to zero gives the maximum area. Solving this, $x = 10$ feet, so the maximum area is $10 \times 20 = 200$ sq. ft.

Write the Word or Phrase That Best Completes Each Statement