In the centre of a rectangular lawn of dimensions \(50\,\text{m} \times 40\,\text{m}\), a rectangular pond is to be constructed so that the area of grass surrounding the pond is \(1184\,\text{m}^2\) (see Fig. 4.1). Find the length and breadth of the pond.
Length = 34 m, Breadth = 24 m
Step 1: The whole lawn is a rectangle with size \(50 \times 40\).
Total area of lawn = \(50 \times 40 = 2000\,\text{m}^2\).
Step 2: A pond is made in the centre. The grass is the part outside the pond.
Grass area is given as \(1184\,\text{m}^2\).
Step 3: This means:
Grass area = Total lawn area – Pond area
\(1184 = 2000 - \text{Pond area}\)
So, Pond area = \(2000 - 1184 = 816\,\text{m}^2\).
Step 4: Now, let the width of the grass border (the part outside pond) be \(x\) metres.
Then, length of pond = \(50 - 2x\) (because \(x\) is cut from both left and right).
Breadth of pond = \(40 - 2x\) (because \(x\) is cut from top and bottom).
Step 5: Pond area = \((50 - 2x)(40 - 2x)\).
We know Pond area = 816, so:
\((50 - 2x)(40 - 2x) = 816\).
Step 6: Expand:
\(2000 - 100x - 80x + 4x^2 = 816\).
\(2000 - 180x + 4x^2 = 816\).
\(4x^2 - 180x + 1184 = 0\).
Divide by 4: \(x^2 - 45x + 296 = 0\).
Step 7: Solve the quadratic:
\(x^2 - 45x + 296 = 0\).
By factorisation: \((x - 8)(x - 37) = 0\).
So, \(x = 8\) or \(x = 37\).
Step 8: Can \(x = 37\)?
No, because if border = 37, then pond size becomes negative. So only \(x = 8\) is correct.
Step 9: Pond dimensions:
Length = \(50 - 2(8) = 34\,\text{m}\).
Breadth = \(40 - 2(8) = 24\,\text{m}\).
Final Answer: Length = 34 m, Breadth = 24 m.