In a rectangular park of dimensions 50 m × 40 m, a rectangular pond is constructed so that the area of grass strip of uniform width surrounding the pond would be 1184 m2. Find the length and breadth of the pond.
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.
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