Answered

\begin{tabular}{|l|c|}
\hline \multicolumn{1}{|c|}{ Characteristic } & Value \\
\hline Focal length & 13 cm \\
\hline Distance of object from mirror & 8 cm \\
\hline Height of object & 4 cm \\
\hline
\end{tabular}

What is the distance of the image from the mirror? Round the answer to the nearest whole number.

A. [tex]$-5$[/tex] cm
B. [tex]$5$[/tex] cm
C. [tex]$-21$[/tex] cm
D. [tex]$21$[/tex] cm



Answer :

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.