The sun subtends an angle of `(1//2)^(@)` on earth. The image of sun is obtained on the screen with the help of a convex lens of focal length 100 cm the diameter of the image obtained on the screen will be

Options

A. `1`
B. `3`
C. `5`
D. `9`

Explanation

To solve the problem, we need to find the diameter of the image of the sun formed by a convex lens. The sun subtends an angle of \( \frac{1}{2}^\circ \) at the earth, and the focal length of the lens is given as 100 cm.

### Step-by-Step Solution:

1. **Convert the angle from degrees to radians**:
\[ \theta = \frac{1}{2}^\circ = \frac{1}{2} \times \frac{\pi}{180} \text{ radians} \]
\[ \theta = \frac{\pi}{360} \text{ radians} \]

2. **Use the formula for the diameter of the image**:
The diameter of the image \( x \) formed by the lens can be calculated using the formula:
\[ x = f \cdot \theta \]
where \( f \) is the focal length of the lens.

3. **Substitute the values into the formula**:
Given that the focal length \( f = 100 \text{ cm} \), we can substitute:
\[ x = 100 \cdot \frac{\pi}{360} \]

4. **Calculate the value of \( x \)**:
\[ x = \frac{100\pi}{360} = \frac{100\pi}{360} = \frac{5\pi}{18} \text{ cm} \]
To get a numerical approximation, we can use \( \pi \approx 3.14 \):
\[ x \approx \frac{5 \times 3.14}{18} \approx \frac{15.7}{18} \approx 0.872 \text{ cm} \]

5. **Convert to centimeters**:
Since the diameter is usually expressed in centimeters, we can round our answer:
\[ x \approx 0.872 \text{ cm} \approx 0.89 \text{ cm} \]

6. **Final answer**:
The diameter of the image of the sun obtained on the screen is approximately \( 0.89 \text{ cm} \).