Final answer:
To find the radius of curvature of a makeup mirror with a magnification of 3.50, the magnification formula and mirror equation are used. The image distance is solved for first, then the focal length, from which the radius of curvature is determined as twice the focal length.
Step-by-step explanation:
To determine the radius of curvature of the mirror, we can use the magnification formula for mirrors, which is magnification (m) = -image distance (di) / object distance (do). Since we are given that the magnification is 3.50 when the eye (object) is 20.0 cm from the mirror, we can rearrange this formula to solve for di.
First, note that the image has the same orientation as the object, which means the mirror is convex, and the image formed is virtual and upright. Then, using the magnification formula:
3.50 = -di / 20.0 cm
Rearranging this to solve for di gives di = -3.50 \(\times\) 20.0 cm = -70.0 cm.
Next, we apply the mirror equation which links object distance (do), image distance (di), and the focal length (f):
1/f = 1/do + 1/di
Plugging in do = 20.0 cm and di = -70.0 cm, and solving for f:
1/f = 1/20.0 cm + 1/(-70.0 cm)
This yields the focal length f. Then, the radius of curvature (R) is twice the focal length of a spherical mirror:
R = 2f
Using the absolute value of f ensures R is positive. We conclude that the radius of curvature of the convex makeup mirror is the calculated R, expressed in meters.