Question

A footpath of uniform width runs all around the inside of a rectangular garden of 40 m x 30 m. If the path occupies 136 m², find the width of the path.

Answer 1

Let the given rectangular field be ABCD with length AB = 40m and width BC = 30m.
If the width of uniform path = x m,
the length of rectangle excluding path is EF = (40 - 2x) m and the width of rectangle excluding path is FG = (30 - 2x) m.
Now,
Area of rectangle ABCD -Area of rectangle EFGH = Area of path
⇒ (40 x 30) - [(40 - 2x) x (30 - 2x)] = 136
⇒ 1200 - [200 - 80x - 60x + 4x²] = 136
⇒ 1200 - 1200 + 140x - 4x² = 136
⇒ 4x² - 140x + 136 = 0
⇒ x² - 35x + 34 = 0
⇒ x² - 34x - x + 34 = 0
⇒ x(x - 34) -1(x - 34) = 0
⇒ (x - 34)(x - 1) = 0
⇒ x = 34 or x = 1
Rejected x = 34 (because it does not satisfy the calculation of the area of the path i.e. 136m²), we get x = 1
Thus, the width of the path is 1m.