The ratio between the length and the breadth of a rectangular park is 3 : 2. If a man cycling along the boundary of the park at the speed of 12 km/hr completes one round in 8 minutes, then the area of the park is :

Question

Accepted Answer

To determine the area of the rectangular park, we need to follow the given information and calculations:

1. **Understanding the Problem:**
   - The ratio of the length to the breadth of the park is 3:2.
   - A man cycles around the boundary of the park at a speed of 12 km/hr and completes one round in 8 minutes.

2. **Calculating the Perimeter:**
   - First, convert the speed from km/hr to m/min:
     $$
     12 \ \text{km/hr} = 12000 \ \text{m/hr} = \frac{12000}{60} \ \text{m/min} = 200 \ \text{m/min}
     $$
   - In 8 minutes, the man covers:
     $$
     200 \ \text{m/min} \times 8 \ \text{min} = 1600 \ \text{m}
     $$
   - Therefore, the perimeter of the park is 1600 meters.

3. **Using the Ratio to Find Dimensions:**
   - Let the length be $3x$ and the breadth be $2x$.
   - The perimeter of a rectangle is given by:
     $$
     2(\text{Length} + \text{Breadth}) = 1600
     $$
   - Substituting the expressions for length and breadth:
     $$
     2(3x + 2x) = 1600
     $$
     $$
     2 \times 5x = 1600
     $$
     $$
     10x = 1600
     $$
     $$
     x = 160
     $$

4. **Calculating the Length and Breadth:**
   - Length $= 3x = 3 \times 160 = 480 \ \text{m}$
   - Breadth $= 2x = 2 \times 160 = 320 \ \text{m}$

5. **Calculating the Area:**
   - The area of the rectangle is given by:
     $$
     \text{Area} = \text{Length} \times \text{Breadth} = 480 \ \text{m} \times 320 \ \text{m} = 153600 \ \text{m}^2
     $$

6. **Conclusion:**
   - The calculated area of the park is $153600 \ \text{m}^2$, which matches Option 2.

7. **Explanation of Incorrect Options:**
   - **Option 1: $152600 \ \text{m}^2$**: This value is incorrect as it does not match the calculated area based on the given dimensions and perimeter.
   - **Option 3: $9600 \ \text{m}^2$**: This value is significantly lower than the calculated area, indicating a miscalculation or misunderstanding of the problem.
   - **Option 4: $25600 \ \text{m}^2$**: This value is also incorrect as it is much smaller than the calculated area, suggesting an error in the calculation process.

Thus, the correct answer is Option 2: $153600 \ \text{m}^2$.