Question

A rectangular plot of land is to be enclosed by fencing. One side is along a river and so needs no fence. If the total fencing available is $600m$, find the dimensions of the plot to have maximum area.

Answer 1

Step 1: Find the value of the breadth:

Since, one side of a rectangular plot is along river , so it will not require any fence.

Consider the length is $L$ and breadth is $B$.

Therefore, Perimeter = $L+2B=600$

Hence, $L=600-2B$

The area is the product of length and breadth.

$Area=L\times B$

$=\left(600-2B\right)B$

$=-2B^2+600B$

The maximum of a quadratic expression $a{x}^2+bx+c$ is at $x=\frac{-b}{2a}$ if $a<0$.

Calculate the maximum point of the expression $-2B^2+600B$.

$$\begin{gathered} B=\frac{-600}{2\left(-2\right)} \\ \Rightarrow B=\frac{600}{4} \\ \therefore B=150m \end{gathered}$$

Step 2: Find the value of the length.

Substitute value of $B$ in the equation $L=600-2B$

$\begin{array}{rcl}L& =& 600-2\times 150\\ \therefore L& =& 300\end{array}$

Hence, the length of rectangular plot is $\mathbf{300}\mathbf{}\mathit{m}$ and the breadth is $\mathbf{150}\mathbf{}\mathit{m}$.