Solve the problem.
-You have 76 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: 38 ft, width: 38 ft
B)length: 57 ft, width: 19 ft
C)length: 19 ft, width: 19 ft
D)length: 38 ft, width: 19 ft

Question

Quizplus · Accepted Answer

To maximize the area of a rectangle when one side (the length along the river) is not fenced, you use the fencing for the two widths and one length. Let the width be $w$ and the length be $l$. You have $2w + l = 76$. To maximize the area ($A = w \times l$), set $l = 76 - 2w$, and substitute into the area formula to get $A = w(76 - 2w)$. The maximum area occurs when $w = 19$ ft (by setting the derivative of the area function to zero or using symmetry), making the length $l = 76 - 2(19) = 38$ ft.

Solve the Problem. -You Have 76 Feet of Fencing to Enclose a Rectangular