Final answer:
To determine the radius of curvature of the concave mirror, the mirror and magnification equations are used with given object distance and magnification. Once the focal length is found, the radius of curvature is twice the focal length, resulting in a radius of 40 cm.
Step-by-step explanation:
The question pertains to the properties of a concave mirror and its ability to form images of objects placed in front of it. In the scenario described, a man's face is 30 cm in front of a concave mirror, and the image formed is erect and magnified two times. Using the mirror equation ∑1/f = 1/do + 1/di and the magnification equation m = -di/do, where f is the focal length, do is the object distance (30 cm), di is the image distance, and m is the magnification (-2 for an erect image), we can solve for the focal length and then find the radius of curvature R, with the relationship R = 2f. Since the magnification is negative for a real image and positive for a virtual image, and the image is larger (positive) and erect, we know that the magnification is +2. Thus, we can set up the equation as 1/f = 1/do + 1/(2do), which simplifies to 1/f = 3/(2do). Substituting the object distance, we get 1/f = 3/(2(30 cm)) = 1/20 cm¹. Therefore, f = 20 cm. With the focal length, the radius of curvature is R = 2f = 2(20 cm) = 40 cm.