Certainly! Let's solve this problem step-by-step.
We are given:
- Focal length [tex]\( f \)[/tex] of the concave mirror = 30 cm
- Object distance [tex]\( u \)[/tex] = 120 cm
Since we are dealing with a concave mirror, the focal length [tex]\( f \)[/tex] is considered negative, so [tex]\( f = -30 \)[/tex] cm. The object distance for concave mirrors is also considered negative, so [tex]\( u = -120 \)[/tex] cm.
We need to find the image distance [tex]\( v \)[/tex]. To do this, we'll use the mirror formula:
[tex]\[
\frac{1}{f} = \frac{1}{v} + \frac{1}{u}
\][/tex]
Rearranging the mirror formula to solve for [tex]\( v \)[/tex]:
[tex]\[
\frac{1}{v} = \frac{1}{f} - \frac{1}{u}
\][/tex]
Substitute the given values into the equation:
[tex]\[
\frac{1}{v} = \frac{1}{-30} - \frac{1}{-120}
\][/tex]
Simplify the right-hand side of the equation:
[tex]\[
\frac{1}{v} = -\frac{1}{30} + \frac{1}{120}
\][/tex]
To combine these fractions, find a common denominator, which is 120:
[tex]\[
\frac{1}{v} = -\frac{4}{120} + \frac{1}{120}
\][/tex]
[tex]\[
\frac{1}{v} = \frac{-4 + 1}{120}
\][/tex]
[tex]\[
\frac{1}{v} = \frac{-3}{120}
\][/tex]
Simplifying the fraction:
[tex]\[
\frac{1}{v} = -\frac{1}{40}
\][/tex]
To find [tex]\( v \)[/tex]:
[tex]\[
v = -40 \text{ cm}
\][/tex]
So, the distance of the image from the mirror is [tex]\( -40 \text{ cm} \)[/tex].
The negative sign indicates that the image is formed on the same side as the object, which is typical for a real image formed by a concave mirror.
Thus, the image distance is [tex]\( 40 \text{ cm} \)[/tex] in front of the mirror.
