#### Question

A farmer has 70 m of fencing, with which he encloses three sides of a rectangular sheep pen; the fourth side being a wall. If the area of the pen is 600 sq. m, find the length of its shorter side.

#### Solution

Let the length and breadth of the rectangular sheep pen be x and y respectively.

From the given information,

x + y + x = 70

2x + y = 70 … (1)

Also, area = xy = 600

Using (1), we have:

x (70 – 2x) = 600

70x – 2x^{2} = 600

2x^{2} – 70x + 600 = 0

x^{2} – 35x + 300 = 0

x^{2} – 15x – 20x + 300 = 0

x(x – 15) – 20(x – 15) = 0

(x – 15)(x – 20) = 0

x = 15, 20

If x = 15, then y = 70 – 2x = 70 – 30 = 40

If x = 20, then y = 70 – 2x = 70 – 40 = 30

Thus, the length of the shorter side is 15 m when the longer side is 40 m. The length of the shorter side is 20 m when the longer side is 30 m.