Question

A farmer with 1670 meters of fencing wants to enclose a rectangular plot that borders on a straight highway. If the farmer does not fence the side along the highway, what is the largest area that can be enclosed? Give your answer correct to the nearest square meter.

Answer 1

The largest area that can be enclosed to the nearest square meter will be 348613 m²

Explanation

Suppose, the length and width of the rectangular plot are  $l$ and $w$ respectively.

The farmer does not fence the side along the highway. Lets assume, the farmer does not fence across one length side. So, the total fence needed $= l+2w$

Given that, the length of the fence is 1670 meters. So, the equation will be.....

$l+2w=1670\\ \\ l= 1670-2w$

Now, the area of the plot.....

$A= l*w\\ \\ A= (1670-2w)w \\ \\ A=1670w-2w^2$

Taking derivative on both sides of the above equation in respect of $w$ , we will get.....

$\frac{dA}{dw}=1670-2(2w)\\ \\ \frac{dA}{dw}=1670-4w$

Now $A$ will be maximum when $\frac{dA}{dw}=0$ . So....

$1670-4w=0\\ \\ 4w=1670\\ \\ w=\frac{1670}{4}=417.5$

Thus, the area will be:  $A= 1670w-2w^2=1670(417.5)-2(417.5)^2= 348612.5 \approx 348613$

So, the largest area that can be enclosed to the nearest square meter will be 348613 m²