What is the radius of circular field whose area is equal to the sum of the areas of three smaller circular fields of radii 8 m ,9m and 12 m respectively ?

Options

A. 17 m
B. 20 m
C. 21 m
D. 29 m

Answer

To find the radius of a circular field whose area is equal to the sum of the areas of three smaller circular fields with given radii, we can follow these steps:

### Step 1: Calculate the areas of the smaller circles
The area \( A \) of a circle is given by the formula:
\[ A = \pi r^2 \]
where \( r \) is the radius of the circle.

For the three smaller circles:
- Radius \( r_1 = 8 \, m \)
- Radius \( r_2 = 9 \, m \)
- Radius \( r_3 = 12 \, m \)

Calculating the areas:
1. Area of Circle 1:
\[ A_1 = \pi (8^2) = \pi (64) = 64\pi \]
2. Area of Circle 2:
\[ A_2 = \pi (9^2) = \pi (81) = 81\pi \]
3. Area of Circle 3:
\[ A_3 = \pi (12^2) = \pi (144) = 144\pi \]

### Step 2: Sum the areas of the smaller circles
Now, we add the areas of the three circles:
\[ A_{\text{total}} = A_1 + A_2 + A_3 = 64\pi + 81\pi + 144\pi = (64 + 81 + 144)\pi = 289\pi \]

### Step 3: Set the area of the larger circle equal to the total area
Let \( R \) be the radius of the larger circle. The area of the larger circle is:
\[ A_{\text{large}} = \pi R^2 \]
Setting the area of the larger circle equal to the total area of the smaller circles:
\[ \pi R^2 = 289\pi \]

### Step 4: Solve for \( R^2 \)
Dividing both sides by \( \pi \):
\[ R^2 = 289 \]

### Step 5: Take the square root to find \( R \)
Taking the square root of both sides:
\[ R = \sqrt{289} = 17 \]

### Conclusion
The radius of the circular field is:
\[ \boxed{17 \, m} \]

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