Question

1. A farmer has 70 m of fencing with which, he encloses 3 side of a rectangular sheep pen. the fourth side being a wall. if the area of the pen is 600 sq m find the lenght of its shorter side.

Answer 1

Suppose x and y be the sides of rectangular sheep pen.
Since 3 side of a rectangular sheep pen is enclosed by 70 m.
So; total length fenced = $x+x+y=70\mathrm{m}$
$$\begin{gathered} \Rightarrow 2x+y=70 \\ \Rightarrow y=70-2x...\left(\mathrm{i}\right) \end{gathered}$$
And area of pen = $xy=600{\mathrm{m}}^2$
$$\begin{gathered} \Rightarrow x\left(70-2x\right)=600 \\ \Rightarrow 70x-2x^2=600 \\ \Rightarrow 2x^2-70x+600=0 \\ \Rightarrow x^2-35x+300=0 \\ \Rightarrow x^2-15x-20x+300=0 \\ \Rightarrow x\left(x-15\right)-20\left(x-15\right)=0 \\ \Rightarrow \left(x-20\right)\left(x-15\right)=0 \\ \Rightarrow x=20\mathrm{or}x=15 \end{gathered}$$
When x = 15; then y = $70-2\times 15=40$
When x = 20; then y = $70-2\times 20=30$
Therefore the shorter side is either 15 m or 20 m.