Question

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 will maximize the area.

• A. length: $33$ feet, width: $32$ feet
• B. length: $16$ feet, width: $16$ feet
• C. length: $48$ feet, width: $16$ feet
• D. length: $32$ feet, width: $16$ feet

Answer 1

The correct option is D

length: $32$ feet, width: $16$ feet

Explanation of correct option:

Finding the length and the width
Perimeter of given rectangle $=L+2W$ (No fencing along river side )
$$\begin{gathered} \therefore W=\frac{64}{2}-\frac{L}{2} \\ =32-\frac{L}{2} \end{gathered}$$
Here $L$is Length and $W$ is width of rectangle.
Area of rectangle $=L\times W$
Area should be maximum for that we take derivative of area to zero with respect to $L$
$$\begin{gathered} \Rightarrow \frac{dA}{dL}=0 \\ \Rightarrow \frac{d\left({\displaystyle 32L-\frac{L^2}{2}}\right)}{dL}=0 \\ \Rightarrow 32-\frac{1}{2}.2L=0 \\ \Rightarrow L=32\text{feet} \end{gathered}$$
$$\begin{gathered} \therefore W=32-16 \\ =16\text{feet} \end{gathered}$$

Hence, option D is correct.