In the centre of a rectangular lawn of dimensions 50 m × 40 m, a rectangular pond has to be constructed so that the area of the grass surrounding the pond would be 1184 m2.Find the length and breadth of the pond.

Problem : In the centre of a rectangular lawn of dimensions 50 m × 40 m, a rectangular pond has to be constructed so that the area of the grass surrounding the pond would be 1184 m². Find the length and breadth of the pond.

Doubt by Ananya

Solution :

Length of Rectangular Lawn = 50 m (Given)
Breadth of Rectangular Lawn = 40 m (Given)
Area of the Rectangular Lawn = 50 × 40 = 2000 m²

Area of the Grass surrounding the Pond = 1184 m² (Given)

Let the distance between the pond and the boundary of Lawn = x

Now
Length of the Pond = (50-2x) m
Breadth of the Pond = (40-2x) m
Area of the Pond = (50-2x)(40-2x) — (1)

Also,
Area of Pond = Area of Lawn - Area of Grass surrounding the PondArea of Pond 
= 2000-1184
Area of Pond = 816 m²  — (2)

From (1) and (2)
(50-2x) (40-2x) = 816
50(40 - 2x) - 2x (40 - 2x) = 816
2000 - 100x - 80x + 4x² = 816
4x² - 180x + 2000 -816 = 0
4x² - 180x + 1184 = 0
4(x² - 45x + 296) = 0
x² - 45x + 296 = 0
x² - (37+8)x + 296 = 0
x² - 37x - 8x + 296 = 0
x(x-37) - 8(x-37) = 0
(x - 8) (x - 37) = 0
x - 8 = 0 OR x - 37 = 0
x = 8 OR x = 37

When x = 37 thenLength of the Pond = 50-2x
= 50 - 2(37)= 50 - 74
= -24
But length can't be -ve.
∴ Rejected.

When x = 8

Length of the Pond = 50 - 2x
= 50 - 2(8)
= 50-16
= 34 m

Breadth of the Pond = 40-2x= 40 - 2(8)
= 40 -16
= 24 m

∴ Length of the Pond = 34 m
   Breadth of the Pond = 24 m