Convex mirror, reduced image, find radius of curvature

[awertag]

Homework Statement

A child holds a candy bar 16.5 cm in front of the convex side-view mirror of an automobile. The image height is reduced by one-half. What is the radius of curvature of the mirror?
1Your answer is incorrect. cm

Homework Equations

1/f = 1/do + 1/di

M=-di/do
M=hi/ho

f=focal point
do=object distance
di=image distance
M=magnification
hi/ho=height image/object
2f=radius

The Attempt at a Solution

ive done it a bunch of times and i got different answers incliuding 1.33, 11, and 33 cm. All were incorrect.
Basically I set M=1/2 and hi/ho=-di/do and plugged in what i knew (made up numbers height to satisfy the 1/2 ratio) and then solved for di. Then i used the first equation to find the focal point, and multiplied that by two for the radius.

I cannot seem to get the right answer. What am I doing wrong??

Thanks so much,
aweg.

 

[anathema]

Hello aweg,

It seems like you are on the right track with your approach. However, there may be some slight errors in your calculations. I will walk you through the correct steps to solve this problem.

First, let's define our variables:

f = focal point (unknown)
do = object distance (16.5 cm)
di = image distance (unknown)
M = magnification (1/2)
hi = image height (unknown)
ho = object height (unknown)

Next, we can use the magnification formula to relate the image and object distances:

M = -di/do

Since we know that M = 1/2, we can substitute that in and solve for di:

1/2 = -di/16.5
di = -8.25 cm

Note that the negative sign indicates that the image is inverted.

Now, we can use the first equation you listed to find the focal point:

1/f = 1/do + 1/di
1/f = 1/16.5 + 1/-8.25
1/f = -1/8.25
f = -8.25 cm

Again, the negative sign indicates that the focal point is located behind the mirror.

Finally, we can use the formula 2f = radius to find the radius of curvature:

2f = 2(-8.25) = -16.5 cm

So, the radius of curvature of the mirror is -16.5 cm. Note that the negative sign indicates that the mirror is convex, as stated in the problem.

I hope this helps clarify where you may have gone wrong in your calculations. Let me know if you have any further questions. Keep up the good work!