Question

You have$200\mathrm{feet}$ 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. What is the largest area that can be enclosed?

Answer 1

Find the largest area that can be enclosed.

Given the length of the total fence $=200\mathrm{feet}$.

Let the length of the rectangular plot is $l$.

The width of the rectangular plot is $w$.

Total perimeter that fence has covered $=l+2w$ ( since only one side of the length is covered by fence.)

$\Rightarrow l+2w=200...\left(1\right)$

Area of the rectangular plot $=lw$

$\Rightarrow A=(200-2w)\times w$

$\Rightarrow A=200w-2w^2$

For maximum area, the derivative of $A$with respect to $w$ will be zero.

$$\begin{gathered} \frac{dA}{dw}=200-4w \\ \Rightarrow 0=200-4w \\ \Rightarrow 4w=200 \\ \Rightarrow w=50 \end{gathered}$$

Now,

$$\begin{gathered} \Rightarrow l=200-2\left(50\right)[\mathrm{Form}(1\left)\right] \\ \Rightarrow l=100 \end{gathered}$$

The maximum area that can be covered $=lw$

$$\begin{gathered} \Rightarrow A_{max}=100\times 50 \\ \Rightarrow A_{max}=5000{\mathrm{feet}}^2 \end{gathered}$$

Hence, the largest area that can be enclosed is $5000{\mathrm{feet}}^2$.