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 solve the problem of designing a shaving mirror, we need to find the radius of curvature of the mirror based on the given distances. Here’s a step-by-step solution:

### Step 1: Identify the given values
- The distance of the object (the person's face) from the mirror, \( u = -10 \, \text{cm} \) (negative because the object is in front of the mirror).
- The closest comfortable distance for viewing the magnified image, \( v = -15 \, \text{cm} \) (since the image is virtual and on the same side as the object).

### 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 the values we have:
\[ \frac{1}{f} = \frac{1}{-15} + \frac{1}{-10} \]

### Step 4: Simplify the equation
To combine the fractions:
\[ \frac{1}{f} = -\frac{1}{15} - \frac{1}{10} \]
Finding a common denominator (which is 30):
\[ \frac{1}{f} = -\frac{2}{30} - \frac{3}{30} = -\frac{5}{30} \]
Thus,
\[ \frac{1}{f} = -\frac{1}{6} \]

### Step 5: Calculate the focal length
Taking the reciprocal gives:
\[ f = -6 \, \text{cm} \]

### Step 6: Find the radius of curvature
The radius of curvature \( R \) is related to the focal length by the formula:
\[ R = 2f \]
Substituting the value of \( f \):
\[ R = 2 \times (-6) = -12 \, \text{cm} \]

### Step 7: Conclusion
The radius of curvature in magnitude is:
\[ |R| = 12 \, \text{cm} \]
However, considering the sign convention, the radius of curvature is:
\[ R = -12 \, \text{cm} \]