In convex mirrors the focus is virtual and the focal distance is negative. This is how the reflected rays diverge and only their extensions are cut at a point on the main axis, resulting in a virtual image of the real object .
The Mirror equation is:
[tex]\frac{1}{f}=\frac{1}{u}+\frac{1}{v}[/tex] (1)
Where:
[tex]f=-6cm[/tex] is the focal distance
[tex]u=12cm[/tex] is the distance between the object and the mirror
[tex]v[/tex] is the distance between the image and the mirror
We already know the values of [tex]f[/tex] and [tex]u[/tex], let's find [tex]v[/tex] from (1):
[tex]v=\frac{u.f}{u-f}[/tex] (2)
[tex]v=\frac{(12cm)(-6cm)}{12cm-(-6cm)}[/tex]
[tex]v=-4cm[/tex] (3)
On the other hand, the magnification [tex]m[/tex] of the image is given by the following equations:
[tex]m=-\frac{v}{u}[/tex] (4)
[tex]m=\frac{h_{i}}{h_{o}}[/tex] (5)
Where:
[tex]h_{i}[/tex] is the image height
[tex]h_{o}=15cm[/tex] is the object height
Now, if we want to find the image height, we firstlu have to find [tex]m[/tex] from (4), substitute it on (5) and find [tex]h_{i}[/tex]:
Substituting (3) in (4):
[tex]m=-\frac{-4cm}{12cm}[/tex]
[tex]m=\frac{1}{3}[/tex] (6)
Substituting (6) in (5):
[tex]\frac{1}{3}=\frac{h_{i}}{15cm}[/tex]
[tex]h_{i}=\frac{15cm}{3}[/tex]
Finally we obtain the value of the height of the image produced by the mirror:
[tex]h_{i}=5cm[/tex]
