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

To design the shaving mirror, we need to determine the radius of curvature. The person keeps the mirror 10 cm from his face, so the object distance (the face) from the mirror is 10 cm. Since the mirror is concave (as it magnifies the image), and using the sign convention where distances in front of the mirror are positive, the object distance $$u = +10$$ cm.

The person views the magnified image at the closest comfortable distance of 25 cm. This distance is from the eye to the image. Assuming the eye is close to the face (at the object position), the total distance from the eye to the image is the sum of the distance from the eye to the mirror (10 cm) and the distance from the mirror to the image. Since the image is virtual and formed behind the mirror, the distance from the mirror to the image is taken as positive in magnitude but negative in sign for the image distance $$v$$.

Let the magnitude of the image distance be $$q$$ cm. Then, the total distance is:

$$10 + q = 25$$

Solving for $$q$$:

$$q = 25 - 10 = 15 \text{ cm}$$

Since the image is virtual and behind the mirror, the image distance $$v = -q = -15$$ cm.

Now, we use the mirror formula:

$$\frac{1}{f} = \frac{1}{u} + \frac{1}{v}$$

where $$f$$ is the focal length. Substituting $$u = 10$$ cm and $$v = -15$$ cm:

$$\frac{1}{f} = \frac{1}{10} + \frac{1}{-15} = \frac{1}{10} - \frac{1}{15}$$

To subtract these fractions, find a common denominator. The least common multiple of 10 and 15 is 30:

$$\frac{1}{10} = \frac{3}{30}, \quad \frac{1}{15} = \frac{2}{30}$$

So,

$$\frac{1}{f} = \frac{3}{30} - \frac{2}{30} = \frac{1}{30}$$

Therefore, the focal length is:

$$f = 30 \text{ cm}$$

For a spherical mirror, the radius of curvature $$R$$ is twice the focal length:

$$R = 2f = 2 \times 30 = 60 \text{ cm}$$

Hence, the radius of curvature is 60 cm, which corresponds to option C.

So, the answer is Option C.