Solve the problem.
-You have 84 feet of fencing to enclose a rectangular plot that borders on a river. If you do not fence the side along the river, find the length and width of the plot that will maximize the area.
A) length: 42 feet, width: 21 feet
B) length: 63 feet, width: 21 feet
C) length: 42 feet, width: 42 feet
D) length: 21 feet, width: 21 feet

Question

Quizplus · Accepted Answer

To maximize the area of a rectangle with three sides (since one side is the river and doesn't need fencing), you use the fencing for two widths and one length. Let the width be $w$ and the length be $l$. The total fencing is $2w + l = 84$ feet. To maximize the area ($A = l \times w$), set $l = 84 - 2w$, and then substitute into the area formula to get $A = w(84 - 2w)$. The maximum area occurs when $w = 21$ feet (by setting the derivative of the area function to zero or using symmetry since it's a quadratic), making the length $l = 84 - 2(21) = 42$ feet.

Solve the Problem. -You Have 84 Feet of Fencing to Enclose a Rectangular