Chapter 4.4, Problem 9E

To determine

To find: The largest area of rectangular plot of farmland enclosed by a single-strand electric fence and its dimension.

Answer to Problem 9E

The dimension of the fence is 400 m by 200 m.

The largest area of the plot is $80,000{\text{ m}}^2$ .

Explanation of Solution

Given information: The length of the wire is given by 800 m.

Formula used:

Area of the rectangle, $A=l\cdot w$ , where l is the length and w is the width.

Perimeter of the rectangle, $P=2\left(l+w\right)$

Calculation:

Suppose a and b be the sides of the rectangular fence.

Since, one side of the farmland is bounded by a river.

So, by considering only three sides, the perimeter will be, $P=a+2b$ .

The perimeter should be 800 m.

Thus,

  $\begin{array}{c}a+2b=800\\ a=800-2b\end{array}$

Also, the area of the rectangle,

  $\begin{array}{c}A=a\cdot b\\ =\left(800-2b\right)\cdot b\\ =800b-2b^2\end{array}$

Taking the first derivative,

  $\frac{dA}{db}=800-4b$

Assume $\frac{dA}{db}=0$

  $\begin{array}{c}0=800-4b\\ 4b=800\\ b=200\end{array}$

Thus,

  $\begin{array}{c}a=800-2\left(200\right)\\ =400\end{array}$

Thus, the dimension will be $a=400\text{ m},b=200\text{ m}$ .

The largest area of the rectangular plot is given by,

  $\begin{array}{c}A=200\cdot 400\\ =80000{\text{m}}^2\end{array}$

Hence, the largest area of rectangular plot of farmland is 80,000 square meters.