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 $600 m$ , find the dimensions of the plot to have maximum area.

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Solution

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 = $L + 2 B = 600$
Hence, $L = 600 - 2 B$
The area is the product of length and breadth.
$A r e a = L \times B$
$= (600 - 2 B) B$
$= - 2 B^{2} + 600 B$
The maximum of a quadratic expression $a x^{2} + b x + c$ is at $x = \frac{- b}{2 a}$ if $a < 0$ .
Calculate the maximum point of the expression $- 2 B^{2} + 600 B$ .
$B = \frac{- 600}{2 (- 2)} \Rightarrow B = \frac{600}{4} ∴ B = 150 m$
Step 2: Find the value of the length.
Substitute value of $B$ in the equation $L = 600 - 2 B$
$\begin{array}{rcl} L & = & 600 - 2 \times 150 \\ ∴ L & = & 300 \end{array}$
Hence, the length of rectangular plot is $300 m$ and the breadth is $150 m$ .

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