Question 26

The footpath of uniform width runs all around a rectangular field 28 meters long and 22 meters wide. If the path occupies 600 m² area, find the width of the path.

Solution:

Let "x" be the width of the path

Length of the rectangular field = 28 m

Width of the rectangular field = 22 m

Area of the path = 600

Length of the larger rectangle = 28 + x + x = (28 + 2 x) m

Width of the larger rectangle = 22 + x + x = (22 + 2 x) m

Area of the path = Area of larger rectangle - Area of smaller rectangle

600 = (28 + 2 x)  (22 + 2 x) - 28 x 22

600 = 616 + 56 x + 44 x + 4 x² - 616

600 = 56 x + 44 x + 4 x²

600 = 4 x² + 100 x

4 x² + 100 x = 600

Now we are going to divide the whole equation by 4,so we get

x² + 25 x = 150

x² + 25 x - 150 = 0

x² + 30 x  - 5 x - 150 = 0

x (x + 30) - 5 (x + 30) = 0

(x - 5) (x + 30) = 0

x - 5 = 0         x + 30 = 0

x = 5               x = -30

Negative value is not possible. Because here x represents width of the path.

Therefore width of the path is 5 m

Verification:

Length of larger rectangle = 28 + 2 (5) = 38 m

Width of the larger rectangle = 22 + 2 (5) = 32 m

Area of path = Area of larger rectangle - Area of smaller rectangle

600 = 38 (32) - 28 (22)

600 = 1216 - 616

600 = 600