You are asked to design a shaving mirror assuming that a person keeps it `10cm` from his face and views the magnified image of the face at the closest comfortable distance of `25cm`. The radius of curvature of the mirror would then be:

Options

A. `60cm`
B. `24cm`
C. `30cm`
D. `-24cm`

Explanation

To design a shaving mirror, we need to determine the radius of curvature of the mirror based on the given distances. The person keeps the mirror 10 cm from their face and views the magnified image at a comfortable distance of 25 cm. We will use the mirror formula and magnification formula to find the radius of curvature.

### Step 1: Identify the given values
- Object distance (u) = -10 cm (the object is in front of the mirror, so it is negative)
- Image distance (v) = +25 cm (the image is virtual and behind the mirror, so it is positive)

### Step 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.

### Step 3: Substitute the values into the mirror formula
Substituting \( u = -10 \) cm and \( v = 25 \) cm into the formula:
\[ \frac{1}{f} = \frac{1}{25} + \frac{1}{-10} \]

### Step 4: Calculate the right-hand side
Calculating the fractions:
\[ \frac{1}{25} = 0.04 \quad \text{and} \quad \frac{1}{-10} = -0.1 \]
Now, adding these values:
\[ \frac{1}{f} = 0.04 - 0.1 = -0.06 \]

### Step 5: Find the focal length
Taking the reciprocal to find \( f \):
\[ f = \frac{1}{-0.06} \approx -16.67 \text{ cm} \]

### Step 6: Relate the focal length to the radius of curvature
The relationship between the focal length \( f \) and the radius of curvature \( R \) is given by:
\[ f = \frac{R}{2} \]
Thus,
\[ R = 2f \]

### Step 7: Calculate the radius of curvature
Substituting the value of \( f \):
\[ R = 2 \times (-16.67) \approx -33.34 \text{ cm} \]

### Conclusion
The radius of curvature of the shaving mirror is approximately \( -33.34 \) cm. The negative sign indicates that it is a concave mirror, which is suitable for a shaving mirror.

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