Let's start by using the mirror equation:
[tex] \frac{1}{f}= \frac{1}{d}+ \frac{1}{q} [/tex]
where f=10 cm is the focal lenght of the mirror, d=38 cm is the distance of the object from the mirror, while q is the distance of the image from the mirror.
For the sign convention, f is taken as positive for a concave mirror. Therefore, we can solve the equation for q:
[tex] \frac{1}{q}= \frac{1}{f}- \frac{1}{d} = \frac{1}{10 cm}- \frac{1}{38 cm} [/tex]
from which we find q=13.6 cm.
The fact that q is positive means that the image is real, so it is on the same side of the object, with respect to the mirror.
Then we can also find the size of the image with respect to the original object. The magnification is given by
[tex]M=- \frac{q}{d} =- \frac{13.6}{38}=-0.36 [/tex]
The negative sign means that the image is inverted, and the size of the image is 0.36 times the size of the object.
