A park is in the form of a one square one of the whose sides is 100 m . The area of the park excluding the circular lawn in the centre of the park is `8614 m^2` .The radius of the circular lawn is -----

Options

A. 21 m
B. 31 m
C. 41 m
D. None of these

Answer

To find the radius of the circular lawn in the park, we can follow these steps:

### Step 1: Calculate the area of the square park.
The area of a square is given by the formula:
\[ \text{Area} = \text{side}^2 \]
Given that the side of the square park is 100 m, we can calculate:
\[ \text{Area of the park} = 100^2 = 10000 \, \text{m}^2 \]

### Step 2: Set up the equation for the area excluding the circular lawn.
We know that the area of the park excluding the circular lawn is given as \(8614 \, \text{m}^2\). Therefore, we can set up the equation:
\[ \text{Area of the park} - \text{Area of the circular lawn} = 8614 \]
Substituting the area of the park:
\[ 10000 - \text{Area of the circular lawn} = 8614 \]

### Step 3: Solve for the area of the circular lawn.
Rearranging the equation gives:
\[ \text{Area of the circular lawn} = 10000 - 8614 = 1386 \, \text{m}^2 \]

### Step 4: Use the area of the circular lawn to find the radius.
The area of a circle is given by the formula:
\[ \text{Area} = \pi r^2 \]
Substituting the area of the circular lawn:
\[ \pi r^2 = 1386 \]
Using \( \pi \approx \frac{22}{7} \), we can rewrite the equation:
\[ \frac{22}{7} r^2 = 1386 \]

### Step 5: Solve for \( r^2 \).
Multiplying both sides by \( \frac{7}{22} \) to isolate \( r^2 \):
\[ r^2 = 1386 \times \frac{7}{22} \]

### Step 6: Calculate \( r^2 \).
Calculating the right side:
\[ r^2 = 1386 \times \frac{7}{22} = \frac{1386 \times 7}{22} = \frac{9702}{22} = 441 \]

### Step 7: Find \( r \).
Taking the square root of both sides:
\[ r = \sqrt{441} = 21 \, \text{m} \]

### Final Answer:
The radius of the circular lawn is \( 21 \, \text{m} \).

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