The sun subtends an angle half a degree at the pole of a concave mirror which has a radius of curvature of 15 m. Then the size (diameter) of the image of sun formed by the concave mirror is

Options

A. 8.55 cm
B. 7.55 cm
C. 6.55 cm
D. 5.55 cm

Answer

To solve the problem, we need to find the size (diameter) of the image of the Sun formed by a concave mirror with a radius of curvature of 15 m, given that the Sun subtends an angle of half a degree at the pole of the mirror.

### Step-by-Step Solution:

1. **Understand the Geometry**:
The Sun is considered to be at infinity with respect to the concave mirror. The angle subtended by the Sun at the pole of the mirror is given as half a degree.

2. **Convert the Angle to Radians**:
The angle in degrees needs to be converted to radians for calculations.
\[ \text{Angle in radians} = \frac{\text{Angle in degrees} \times \pi}{180} \]
For half a degree:
\[ \theta = \frac{0.5 \times \pi}{180} = \frac{\pi}{360} \text{ radians} \]

3. **Determine the Focal Length**:
The radius of curvature (R) of the concave mirror is given as 15 m. The focal length (f) is related to the radius of curvature by:
\[ f = \frac{R}{2} = \frac{15}{2} = 7.5 \text{ m} \]

4. **Calculate the Size of the Image**:
The size of the image can be calculated using the formula for the arc length of a circle, which is:
\[ \text{Arc length} = \text{Radius} \times \text{Angle} \]
Here, the radius is the focal length (f), and the angle is \(2\theta\) (since the angle subtended is at the pole, we consider both sides):
\[ \text{Arc length} = f \times 2\theta = 7.5 \times 2\left(\frac{\pi}{360}\right) \]
Simplifying this:
\[ \text{Arc length} = 7.5 \times \frac{2\pi}{360} = 7.5 \times \frac{\pi}{180} \approx 7.5 \times 0.01745 \approx 0.1309 \text{ m} \]

5. **Convert to Centimeters**:
To find the diameter of the image, we need to multiply the arc length by 100 (to convert meters to centimeters):
\[ \text{Diameter of the image} = 0.1309 \times 100 \approx 13.09 \text{ cm} \]

6. **Final Result**:
The size (diameter) of the image of the Sun formed by the concave mirror is approximately:
\[ \text{Diameter} \approx 13.09 \text{ cm} \]