Two perpendicular cross roads of equal width run through the middle of a rectangular field of length 80 m and breadth 60 m. If the area of the cross roads is 675 square meters, then the width of each road is:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → 5 m
Step 1: Let the width of each road be x meters
- There are two perpendicular roads of width x.
- Area of cross roads:
$\text{Area} = (\text{length} \times \text{width of road}) + (\text{breadth} \times \text{width of road}) - (\text{overlap})$
- Overlap is a square of side x, counted twice.
$\text{Area} = 80x + 60x - x^2 = 140x - x^2$
Step 2: Equate to given area
$140x - x^2 = 675$
$x^2 - 140x + 675 = 0$
Step 3: Solve the quadratic equation
$x = \frac{140 \pm \sqrt{140^2 - 4 \cdot 675}}{2}$
$x = \frac{140 \pm \sqrt{19600 - 2700}}{2} = \frac{140 \pm \sqrt{16900}}{2}$
$\sqrt{16900} = 130$
$x = \frac{140 \pm 130}{2}$
$x = \frac{140 + 130}{2} = \frac{270}{2} = 135 \quad \text{(not possible, larger than field!)}$
$x = \frac{140 - 130}{2} = \frac{10}{2} = 5$
Width of each road: 5 m