Question

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.

Answer 1

It is given that the dimensions of rectangular park is 50 m × 40 m.
∴ Area of the rectangular park = 50 × 40 = 2000 m²
Area of the grass surrounding the pond = 1184 m²
Now,
Area of the rectangular pond
= Area of the rectangular park − Area of the grass surrounding the rectangular pond
= 2000 − 1184
= 816 m²
Let the uniform width of the surrounding grass be x.
∴ Length of the rectangular pond = (50 − 2x) m
Breadth of the rectangular pond = (40 − 2x) m
Now,
Area of rectangular pond = 816 m²
∴ (50 − 2x) × (40 − 2x) = 816
⇒ 2000 − 80x − 100x + 4x² = 816
⇒ 4x² − 180x + 2000 − 816 = 0
⇒ 4x² − 180x + 1184 = 0
⇒ x² − 45x + 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
For x = 37,
Length of rectangular pond = 50 − 2 × 37 = −24 m, which is not possible
So, x ≠ 37
Therefore, x = 8.
When x = 8,
Length of the rectangular pond = 50 − 2 × 8 = 50 − 16 = 34 m
Breadth of the rectangular pond = 40 − 2 × 8 = 40 − 16 = 24 m.