The difference between the circumference and the diameter of the circle is 15 m . What is the area of the circle ?

Options

A. `225 m^(2) `
B. `165 m^(2) `
C. `156 m^(2) `
D. none of these

Answer

To solve the problem, we need to find the area of the circle given that the difference between the circumference and the diameter is 15 m.

### Step-by-Step Solution:

1. **Understand the given information**:
We know that the difference between the circumference (C) and the diameter (D) of the circle is 15 m.
\[ C - D = 15 \]

2. **Write the formulas for circumference and diameter**:
The formulas are:
- Circumference \( C = 2\pi r \)
- Diameter \( D = 2r \)

3. **Substitute the formulas into the equation**:
\[ 2\pi r - 2r = 15 \]

4. **Factor out the common term**:
\[ 2r(\pi - 1) = 15 \]

5. **Solve for the radius \( r \)**:
\[ r = \frac{15}{2(\pi - 1)} \]
Using \( \pi \approx \frac{22}{7} \):
\[ r = \frac{15}{2\left(\frac{22}{7} - 1\right)} \]
Simplifying \( \frac{22}{7} - 1 = \frac{22}{7} - \frac{7}{7} = \frac{15}{7} \):
\[ r = \frac{15}{2 \times \frac{15}{7}} = \frac{15 \times 7}{2 \times 15} = \frac{7}{2} \text{ m} \]

6. **Calculate the area of the circle**:
The area \( A \) of the circle is given by the formula:
\[ A = \pi r^2 \]
Substituting the value of \( r \):
\[ A = \pi \left(\frac{7}{2}\right)^2 = \pi \times \frac{49}{4} = \frac{49\pi}{4} \]
Using \( \pi \approx \frac{22}{7} \):
\[ A = \frac{49 \times \frac{22}{7}}{4} = \frac{49 \times 22}{28} = \frac{1078}{28} = 38.5 \text{ m}^2 \]

### Final Answer:
The area of the circle is \( 38.5 \text{ m}^2 \).