Find the rational number.

Options

A. Area of a circle with radius `(1)/(pi)`
B. Radius of circle with area ` (1)/(pi)`
C. Circumference of circle with radius ` (1)/(pi)`
D. Radius of circle with circumference `(1)/(pi)`

Answer

To solve the problem of finding a rational number among the given statements, we will analyze each statement step by step.

### Step-by-Step Solution:

1. **Understanding the Definition of Rational Numbers**:
A rational number is any number that can be expressed as the quotient or fraction \( \frac{p}{q} \), where \( p \) and \( q \) are integers and \( q \neq 0 \).

2. **Analyzing Statement 1**:
- **Statement**: Area of a circle with radius \( r = 1/\pi \).
- **Formula**: The area \( A \) of a circle is given by \( A = \pi r^2 \).
- **Calculation**:
\[ A = \pi \left(\frac{1}{\pi}\right)^2 = \pi \cdot \frac{1}{\pi^2} = \frac{1}{\pi} \]
- **Conclusion**: \( \frac{1}{\pi} \) is an irrational number (since \( \pi \) is irrational). Therefore, this statement is **false**.

3. **Analyzing Statement 2**:
- **Statement**: Radius of a circle with area \( A = 1/\pi \).
- **Formula**: From the area formula, we have \( A = \pi r^2 \).
- **Calculation**:
\[ \pi r^2 = \frac{1}{\pi} \implies r^2 = \frac{1}{\pi^2} \implies r = \frac{1}{\pi} \]
- **Conclusion**: Again, \( \frac{1}{\pi} \) is irrational. Thus, this statement is **false**.

4. **Analyzing Statement 3**:
- **Statement**: Circumference of a circle with radius \( r = 1/\pi \).
- **Formula**: The circumference \( C \) is given by \( C = 2\pi r \).
- **Calculation**:
\[ C = 2\pi \left(\frac{1}{\pi}\right) = 2 \]
- **Conclusion**: \( 2 \) is a rational number. Therefore, this statement is **true**.

5. **Analyzing Statement 4**:
- **Statement**: Radius of a circle with circumference \( C = 1/\pi \).
- **Formula**: From the circumference formula, we have \( C = 2\pi r \).
- **Calculation**:
\[ 2\pi r = \frac{1}{\pi} \implies r = \frac{1}{2\pi^2} \]
- **Conclusion**: \( \frac{1}{2\pi^2} \) is irrational (since \( \pi \) is irrational). Thus, this statement is **false**.

### Final Conclusion:
The only rational number found in the statements is from **Statement 3**, which gives us the circumference of a circle with radius \( r = \frac{1}{\pi} \) as \( 2 \).

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