IN the centre of a rectangular lawn of dimensions `50 m xx 40 m,` a rectangular pond has to be cnstructed, so that the area of the grass surrounding the pond would be `1184m^(2).` Find the length and breadth of the pond.

Answer

To solve the problem of finding the dimensions of the pond in the rectangular lawn, we can follow these steps:

### Step 1: Understand the dimensions of the lawn
The dimensions of the rectangular lawn are given as:
- Length (L) = 50 m
- Breadth (B) = 40 m

### Step 2: Calculate the area of the lawn
The area of the rectangular lawn can be calculated using the formula:
\[ \text{Area of lawn} = \text{Length} \times \text{Breadth} = 50 \times 40 = 2000 \, \text{m}^2 \]

### Step 3: Set up the equation for the area of the pond
Let the width of the grass surrounding the pond be \(x\). Then, the dimensions of the pond can be expressed as:
- Length of the pond = \(50 - 2x\)
- Breadth of the pond = \(40 - 2x\)

### Step 4: Calculate the area of the grass surrounding the pond
The area of the grass surrounding the pond is given as \(1184 \, \text{m}^2\). Therefore, the area of the pond can be calculated as:
\[ \text{Area of pond} = \text{Area of lawn} - \text{Area of grass} = 2000 - 1184 = 816 \, \text{m}^2 \]

### Step 5: Set up the equation for the area of the pond
Using the dimensions of the pond, we can express the area of the pond as:
\[ \text{Area of pond} = (50 - 2x)(40 - 2x) \]
Setting this equal to the area of the pond we calculated:
\[ (50 - 2x)(40 - 2x) = 816 \]

### Step 6: Expand the equation
Expanding the left-hand side:
\[ 2000 - 100x - 80x + 4x^2 = 816 \]
This simplifies to:
\[ 4x^2 - 180x + 2000 - 816 = 0 \]
\[ 4x^2 - 180x + 1184 = 0 \]

### Step 7: Simplify the quadratic equation
We can divide the entire equation by 4 to simplify:
\[ x^2 - 45x + 296 = 0 \]

### Step 8: Solve the quadratic equation using the quadratic formula
The quadratic formula is given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \(a = 1\), \(b = -45\), and \(c = 296\):
\[ b^2 - 4ac = (-45)^2 - 4 \times 1 \times 296 = 2025 - 1184 = 841 \]
Now substituting back into the formula:
\[ x = \frac{45 \pm \sqrt{841}}{2 \times 1} = \frac{45 \pm 29}{2} \]
Calculating the two possible values for \(x\):
1. \(x = \frac{74}{2} = 37\)
2. \(x = \frac{16}{2} = 8\)

### Step 9: Calculate the dimensions of the pond
Using \(x = 8\) (since \(x = 37\) would yield negative dimensions):
- Length of the pond = \(50 - 2(8) = 34 \, \text{m}\)
- Breadth of the pond = \(40 - 2(8) = 24 \, \text{m}\)

### Final Answer
The dimensions of the pond are:
- Length = 34 m
- Breadth = 24 m