Answer:
focal length = 30 cm
Step-by-step explanation:
You want the focal length of a convex mirror when moving the mirror an additional 15 cm away from the object causes its image size to decrease from 1/2 to 2/5 the size of the object.
Mirror equation
The relation between focal length (f), object distance (do) and image distance (di) is ...
1/f = 1/do +1/di
The ratio of heights is the same as the ratio of distances:
hi/ho =di/do
Application
If we let do represent the original object distance, then (1/2)do = di, the original image distance. This gives one equation we need:
1/f = 1/do +1/(1/2do) = 3/do
Moving the mirror increases the object distance to (do+15). Now, the equation becomes ...
1/f = 1/(do+15) +1/(2/5(do+15)) = 7/(2(do+15))
Solution
Substituting for 1/f, we have ...
3/do = 7/(2(do +15))
6(do +15) = 7do . . . . . . . . . cross multiply
90 = do . . . . . . . . . . . . subtract 6·do
Then the focal length is ...
1/f = 3/do
f = do/3 = 90/3 = 30
The focal length of the mirror is 30 cm.