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What is the focal length of the mirror, the length of the image, and the magnification?

When a butterfly of body length 4.2 cm is 10 cm from a concave mirror, its image is 15 cm behind the mirror.

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Richard P. | Fairfax County Tutor for HS Math and ScienceFairfax County Tutor for HS Math and Sci...
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The key to this type of question is the lens equation which is usually written as:
 
1/p + 1/q  =  1/f    where   p is the object distance , q is the image distance and f is the focal length.
 
In this problem  p = 10 cm,   q =  - 15  (negative because the image is a virtual image behind the mirror).
 
So  1/10 - 1/15 = 1/f   .   Solving for f yields   f =  30 cm.   The magnification , M, is   =- q/p  = 1.5, so the image of the butterfly will be upright  = 6.3 cm.