If is proposed to build a single circular park equal in area to the sum of areas of two circular parks of diameters 16 m and 12 m in a locality. The radius of the new park would be

Options

A. 10 m
B. 15 m
C. 20 m
D. 24 m

Explanation

To find the radius of the new circular park that is equal in area to the sum of the areas of two circular parks with diameters of 16 m and 12 m, we can follow these steps:

### Step-by-Step Solution:

1. **Calculate the radius of the first park:**
- The diameter of the first park is given as 16 m.
- The radius \( r_1 \) is calculated as:
\[ r_1 = \frac{\text{diameter}}{2} = \frac{16 \, \text{m}}{2} = 8 \, \text{m} \]

2. **Calculate the radius of the second park:**
- The diameter of the second park is given as 12 m.
- The radius \( r_2 \) is calculated as:
\[ r_2 = \frac{\text{diameter}}{2} = \frac{12 \, \text{m}}{2} = 6 \, \text{m} \]

3. **Calculate the area of the first park:**
- The area \( A_1 \) of the first park is given by the formula:
\[ A_1 = \pi r_1^2 = \pi (8 \, \text{m})^2 = \pi \times 64 \, \text{m}^2 \]

4. **Calculate the area of the second park:**
- The area \( A_2 \) of the second park is given by the formula:
\[ A_2 = \pi r_2^2 = \pi (6 \, \text{m})^2 = \pi \times 36 \, \text{m}^2 \]

5. **Calculate the total area of both parks:**
- The total area \( A \) of both parks is:
\[ A = A_1 + A_2 = \pi \times 64 \, \text{m}^2 + \pi \times 36 \, \text{m}^2 = \pi (64 + 36) \, \text{m}^2 = \pi \times 100 \, \text{m}^2 \]

6. **Set the area of the new park equal to the total area:**
- Let the radius of the new park be \( r \).
- The area of the new park is:
\[ A_{\text{new}} = \pi r^2 \]
- Setting the areas equal:
\[ \pi r^2 = \pi \times 100 \]

7. **Cancel \( \pi \) from both sides:**
- This simplifies to:
\[ r^2 = 100 \]

8. **Calculate the radius of the new park:**
- Taking the square root of both sides:
\[ r = \sqrt{100} = 10 \, \text{m} \]

### Final Answer:
The radius of the new circular park is **10 m**.