Solve the problem.
-You have 64 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: 48 feet, width: 16 feet
B) length: 16 feet, width: 16 feet
C) length: 32 feet, width: 32 feet
D) length: 32 feet, width: 16 feet

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 other three sides. Let the width be $w$ and the length be $l$. The total fencing used is $2w + l = 64$ feet (since the length is only fenced on one side). To maximize the area $A = l \times w$, we express $l$ in terms of $w$ from the fencing constraint: $l = 64 - 2w$. Substituting into the area formula gives $A = w(64 - 2w)$. Differentiating $A$ with respect to $w$ and setting the derivative to zero to find the maximum area, we get $64 - 4w = 0$, solving this gives $w = 16$ feet. Substituting $w = 16$ back into the equation for $l$, we get $l = 64 - 2(16) = 32$ feet. Therefore, the dimensions that maximize the area are a length of 32 feet and a width of 16 feet.

Solve the Problem. -You Have 64 Feet of Fencing to Enclose a Rectangular