Question

A convex spherical mirror, whose focal length has a magnitude of 15.8 cm, is to form an image 12.2 cm behind the mirror.

Answer 1

(a) we can solve the problem by using the mirror equation:
$\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}$
where 
f is the focal length of the mirror
$d_o$ is the distance of the object from the mirror
$d_i$ is the distance of the image from the mirror.

For the sign convention, f is taken as negative for a convex mirror and $d_i$ is taken as negative if the image is located behind the mirror, as in this problem. So we have 
$f=-15.8 cm$
$d_i=-12.2 cm$
and re-arranging the mirror equation we can find the distance of the object from the mirror:
$\frac{1}{d_o}= \frac{1}{f}- \frac{1}{d_i}= \frac{1}{-15.8 cm} - \frac{1}{-12.2 cm}$
from which we find
$d_o=53.5 cm$

(b) The magnitude is defined as the ratio between the size of the image and the size of the object, which is also equal to the negative of the ratio between the distance of the image and the distance of the object from the mirror:
$M= \frac{h_i}{h_o} =- \frac{d_i}{d_o}$
Using what we found at point (a), the magnification in this problem is
$M=- \frac{-12.2 cm}{53.5 cm}=0.23$
where the positive sign means the image is upright.