Question

The length and breadth of rectangular garden of rehana are 14 m and 12 m . If the cost of constructing an equally wide path around the garden is ₹ 1,380 at the rate of Rs. 20 per sq.meter , then let us write by calculating how much wide is the path.

Answer 1

The area of the path is :

1380 / 20 = 69m²

Let width of the path be x then the length of the outer rectangle is :

14 + 2x

Width = 12 + 2x

Area of the path = Area of outer rectangle - Area of inner triangle

Area of outer rectangle :

(2x + 12)(2x + 14) = 4x² + 52x + 168

Area of inner rectangle is :

12 × 14 = 168

4x² + 52x +168 - 168 = 69

4x² + 52x = 69

4x² + 52x - 69 = 0

We will use quadratic formula :

{-52 +/- √52² - 4 × 4 ×-69} / 8

(-52 +/- 61.71) /8

x = 1.21m or - 14.21m

We take the positive value :

x = 1. 21m

Answer 2

Step-by-step explanation:

The Dimensions of the inner rectangle will be (14 – 2x) × (12 – 2x)

And area of inner rectangular ground = (14 – 2x)(12 – 2x)

Now, area of path = area of the whole garden – an area of the garden without a path

$\frac{cost \: of \: construction}{rate \: of cost \: \: construction} = \: 168 – 168 + 28x + 24x – 4x2$

$⇒ \frac{1380}{20} = 168 – 168 + 28x + 24x – 4x2$

$⇒ 69 = 168 – 168 + 28x + 24x – 4x2$

$⇒ 4x2 – 52x + 69 = 0$

$⇒ 4x2 - 46x – 6x + 69 = 0$

$⇒ 2x(2x – 23) – 3(2x – 23) = 0$

$⇒ (2x – 3)(2x – 23) = 0$

$⇒ 2x – 3 = 0 \:or\: 2x – 23 = 0$

$⇒ x = 1.5 m \:or\: x = 11.5 m$

Now, logically 11.5 m is not possible, because in that case, the length of inner rectangular ground will be negative,

Hence, width of path = x = 1.5 m