The sun's angular diameter is measured to be 1920''. The distance of the sun from the earth is `1.496xx10^(11)m.` What is the diameter of the sun?

Options

A. `A. 1.39xx10^(9)m`
B. `B. 1.39xx10^(10)m`
C. `C. 1.39xx10^(11)m`
D. `D.1.39xx10^(12)`m

Explanation

To find the diameter of the sun based on its angular diameter and the distance from the Earth, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Given Values**:
- The angular diameter of the sun, \( \theta = 1920'' \) (arcseconds).
- The distance from the Earth to the sun, \( R = 1.496 \times 10^{11} \, \text{m} \).

2. **Convert Angular Diameter from Arcseconds to Radians**:
- First, convert arcseconds to degrees:
\[ \theta_{\text{degrees}} = \frac{1920 \, \text{arcseconds}}{3600 \, \text{arcseconds per degree}} = \frac{1920}{3600} \, \text{degrees} \]
- Next, convert degrees to radians:
\[ \theta_{\text{radians}} = \theta_{\text{degrees}} \times \frac{\pi}{180} \]
- Therefore:
\[ \theta_{\text{radians}} = \left(\frac{1920}{3600}\right) \times \frac{\pi}{180} \]

3. **Calculate the Diameter of the Sun**:
- The formula for the diameter \( d \) based on the angular diameter \( \theta \) and distance \( R \) is:
\[ d = R \cdot \theta \]
- Substitute the values:
\[ d = 1.496 \times 10^{11} \, \text{m} \times \left(\left(\frac{1920}{3600}\right) \times \frac{\pi}{180}\right) \]

4. **Perform the Calculations**:
- Calculate \( \theta_{\text{radians}} \):
\[ \theta_{\text{radians}} = \left(\frac{1920}{3600}\right) \times \frac{\pi}{180} \approx 0.000533 \, \text{radians} \]
- Now calculate \( d \):
\[ d = 1.496 \times 10^{11} \, \text{m} \times 0.000533 \approx 1.39 \times 10^{9} \, \text{m} \]

5. **Final Result**:
- The diameter of the sun is approximately \( 1.39 \times 10^{9} \, \text{m} \).

### Conclusion:
The correct answer is option A: \( 1.39 \times 10^{9} \, \text{m} \).