Final answer:
The radius of curvature of the mirror is 90 cm.
Step-by-step explanation:
To find the radius of curvature of the concave spherical shaving mirror, we can use the formula:
f = R/2
where f is the focal length and R is the radius of curvature. Since the image is 1.5 times as large as the man's face, the magnification (m) is 1.5. Using the magnification formula:
m = -d_i / d_o = h_i / h_o = -1.5
we can determine the image distance (d_i) and object distance (d_o) relative to the mirror. Given that the face is 30 cm in front of the mirror, the object distance (d_o) is -30 cm (negative because it is in front of the mirror) and the image distance (d_i) is unknown. Solving for d_i, we get:
d_i = (-1.5)(-30) = 45 cm
Since the object distance and image distance are on the same side of the mirror, the focal length (f) is positive. Using the mirror equation:
1/f = 1/d_i + 1/d_o
we can substitute the known values:
1/f = 1/45 + 1/(-30)
Simplifying, we find:
f = 45 cm
Finally, using the formula f = R/2, we can determine the radius of curvature (R):
R = 2f = 2(45) = 90 cm