To determine the distance of the image from the mirror, we will use the mirror formula:
[tex]\[
\frac{1}{F} = \frac{1}{d_o} + \frac{1}{d_i}
\][/tex]
where:
- [tex]\( F \)[/tex] is the focal length of the mirror,
- [tex]\( d_o \)[/tex] is the distance of the object from the mirror, and
- [tex]\( d_i \)[/tex] is the distance of the image from the mirror.
The given values are:
- Focal length, [tex]\( F = 13 \)[/tex] cm
- Distance of the object from the mirror, [tex]\( d_o = 8 \)[/tex] cm
We need to find the image distance, [tex]\( d_i \)[/tex].
First, let's rearrange the mirror formula to solve for [tex]\( \frac{1}{d_i} \)[/tex]:
[tex]\[
\frac{1}{d_i} = \frac{1}{F} - \frac{1}{d_o}
\][/tex]
Substitute the known values into the equation:
[tex]\[
\frac{1}{d_i} = \frac{1}{13} - \frac{1}{8}
\][/tex]
Next, we calculate the right-hand side of the equation to find [tex]\( \frac{1}{d_i} \)[/tex]:
[tex]\[
\frac{1}{d_i} = \frac{8 - 13}{104} = \frac{-5}{104}
\][/tex]
Thus, [tex]\( d_i \)[/tex] is given by the reciprocal of [tex]\( \frac{1}{d_i} \)[/tex]:
[tex]\[
d_i = \frac{104}{-5} = -20.8
\][/tex]
Rounding this to the nearest whole number, we obtain [tex]\( d_i = -21 \)[/tex] cm.
Therefore, the distance of the image from the mirror is:
[tex]\[
\boxed{-21 \, \text{cm}}
\][/tex]
The correct option is [tex]\( -21 \)[/tex] cm.
