The area of a ring is `16.94 cm^(2)` and the area of the outer circle is `55.44 cm^(2)` .Find the perimeter of the inner circle .

Options

A. 22 cm
B. `26.4` cm
C. `38.5` cm
D. `29.04` cm

Answer

To solve the problem, we need to find the perimeter of the inner circle given the area of the ring and the area of the outer circle.

### Step-by-Step Solution:

1. **Identify the Areas**:
- Area of the ring = \( 16.94 \, \text{cm}^2 \)
- Area of the outer circle = \( 55.44 \, \text{cm}^2 \)

2. **Calculate the Area of the Inner Circle**:
- The area of the ring is the difference between the area of the outer circle and the area of the inner circle.
\[ \text{Area of the inner circle} = \text{Area of the outer circle} - \text{Area of the ring} \]
\[ \text{Area of the inner circle} = 55.44 \, \text{cm}^2 - 16.94 \, \text{cm}^2 \]
\[ \text{Area of the inner circle} = 38.50 \, \text{cm}^2 \]

3. **Use the Area Formula to Find the Radius of the Inner Circle**:
- The area of a circle is given by the formula \( A = \pi r^2 \).
- We can rearrange this to find \( r^2 \):
\[ r^2 = \frac{\text{Area}}{\pi} \]
\[ r^2 = \frac{38.50}{\pi} \]
- Using \( \pi \approx 3.14 \):
\[ r^2 = \frac{38.50}{3.14} \approx 12.25 \]

4. **Calculate the Radius**:
- To find \( r \), take the square root of \( r^2 \):
\[ r = \sqrt{12.25} \approx 3.5 \, \text{cm} \]

5. **Calculate the Perimeter of the Inner Circle**:
- The perimeter (circumference) of a circle is given by the formula \( C = 2\pi r \):
\[ C = 2 \times \pi \times r \]
\[ C = 2 \times 3.14 \times 3.5 \]
\[ C \approx 21.98 \, \text{cm} \]
- Rounding this, we can say:
\[ C \approx 22 \, \text{cm} \]

### Final Answer:
The perimeter of the inner circle is approximately \( 22 \, \text{cm} \).