To solve this problem, let’s go step-by-step through the physics principles involved, using the given information about the concave mirror.
### Given:
- Focal length, [tex]\( f = 51 \)[/tex] cm
- The image is inverted and 4 times smaller than the object, implying the magnification, [tex]\( m = -\frac{1}{4} \)[/tex]
### Step 1: Use the Magnification Formula
The magnification formula for mirrors is given by:
[tex]\[ m = -\frac{v}{u} \][/tex]
Where:
- [tex]\( v \)[/tex] is the image distance
- [tex]\( u \)[/tex] is the object distance
Given [tex]\( m = -\frac{1}{4} \)[/tex], we can rearrange this formula to express [tex]\( v \)[/tex] in terms of [tex]\( u \)[/tex]:
[tex]\[ -\frac{1}{4} = -\frac{v}{u} \Rightarrow v = \frac{u}{4} \][/tex]
### Step 2: Use the Mirror Formula
The mirror formula is given by:
[tex]\[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \][/tex]
### Step 3: Substitute [tex]\( v \)[/tex] into the Mirror Formula
From the magnification formula, we have [tex]\( v = \frac{u}{4} \)[/tex]. Substitute this into the mirror equation:
[tex]\[ \frac{1}{51} = \frac{1}{\frac{u}{4}} + \frac{1}{u} \][/tex]
[tex]\[ \frac{1}{51} = \frac{4}{u} + \frac{1}{u} \][/tex]
[tex]\[ \frac{1}{51} = \frac{4 + 1}{u} \][/tex]
[tex]\[ \frac{1}{51} = \frac{5}{u} \][/tex]
### Step 4: Solve for [tex]\( u \)[/tex]
Rearranging the equation to solve for [tex]\( u \)[/tex] gives:
[tex]\[ u = 51 \times 5 \][/tex]
[tex]\[ u = 255 \][/tex]
So, the object distance [tex]\( u \)[/tex] is 255 cm.
### Step 5: Calculate the Image Distance [tex]\( v \)[/tex]
Now, we use [tex]\( v = \frac{u}{4} \)[/tex] to find the image distance:
[tex]\[ v = \frac{255}{4} \][/tex]
[tex]\[ v = 63.75 \][/tex]
So, the image distance [tex]\( v \)[/tex] is 63.75 cm.
### Conclusion:
The position of the object is 255 cm from the mirror, and the position of the resulting image is 63.75 cm from the mirror.
