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 convace mirror is

Options

A. 7.5 cm
B. 6.55 cm
C. 3.5 cm
D. 1.31 cm

Explanation

To solve the problem of finding the diameter of the image of the sun formed by a concave mirror, we can follow these steps:

### Step 1: Understand the given data
- The angle subtended by the sun at the pole of the concave mirror is \( \theta = 0.5^\circ \).
- The radius of curvature of the concave mirror is \( R = 15 \, \text{m} \).

### Step 2: Calculate the focal length of the mirror
The focal length \( f \) of a concave mirror is given by the formula:
\[ f = \frac{R}{2} \]
Substituting the value of \( R \):
\[ f = \frac{15}{2} = 7.5 \, \text{m} \]

### Step 3: Convert the angle from degrees to radians
To use the angle in calculations, we convert \( \theta \) from degrees to radians:
\[ \theta \text{ (in radians)} = \frac{\theta \text{ (in degrees)} \times \pi}{180} = \frac{0.5 \times \pi}{180} = \frac{\pi}{360} \, \text{radians} \]

### Step 4: Relate the angle to the diameter of the image
Using the small angle approximation, we can relate the angle \( \theta \) to the diameter \( D \) of the image formed at the focus:
\[ \theta = \frac{D}{f} \]
Thus, we can express \( D \) as:
\[ D = \theta \cdot f \]

### Step 5: Substitute the values into the equation
Substituting the values of \( \theta \) and \( f \):
\[ D = \left(\frac{\pi}{360}\right) \cdot 7.5 \]

### Step 6: Calculate the diameter
Now, we calculate \( D \):
\[ D = \frac{\pi \cdot 7.5}{360} \approx \frac{23.5619}{360} \approx 0.0654 \, \text{m} \]

### Step 7: Convert the diameter from meters to centimeters
Since the question asks for the diameter in centimeters:
\[ D = 0.0654 \, \text{m} \times 100 = 6.54 \, \text{cm} \]

### Step 8: Final answer
Thus, the diameter of the image of the sun formed by the concave mirror is approximately:
\[ \text{Diameter} \approx 6.54 \, \text{cm} \]

### Conclusion
The correct answer is approximately \( 6.55 \, \text{cm} \).

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