An object is placed at a distance of 15 cm from a convex mirror and images is formed at a distance of 5 cm from the mirror. Calculate the radius of curvature of the mirror.

Answer

To solve the problem, we will use the mirror formula and the relationship between the focal length and the radius of curvature for a convex mirror.

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Object distance (u) = -15 cm (The object distance is negative for a convex mirror)
- Image distance (v) = +5 cm (The image distance is positive for a convex mirror)

2. **Use the Mirror Formula:**
The mirror formula is given by:
\[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \]
Where \( f \) is the focal length of the mirror.

3. **Substitute the Values into the Mirror Formula:**
\[ \frac{1}{f} = \frac{1}{5} + \frac{1}{-15} \]

4. **Calculate the Right Side:**
- Find a common denominator for the fractions:
\[ \frac{1}{5} = \frac{3}{15} \quad \text{and} \quad \frac{1}{-15} = -\frac{1}{15} \]
- Therefore,
\[ \frac{1}{f} = \frac{3}{15} - \frac{1}{15} = \frac{2}{15} \]

5. **Find the Focal Length (f):**
- Taking the reciprocal gives:
\[ f = \frac{15}{2} \text{ cm} = 7.5 \text{ cm} \]

6. **Calculate the Radius of Curvature (R):**
The radius of curvature \( R \) is related to the focal length by the formula:
\[ R = 2f \]
- Substituting the value of \( f \):
\[ R = 2 \times \frac{15}{2} = 15 \text{ cm} \]

### Final Answer:
The radius of curvature of the convex mirror is **15 cm**.

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