A concave mirror for face viewing has focal length of `0.4m`. The distance at which you hold the mirror from your face in oreder to see your image upright with a magnification of 5 is :

Options

A. `0.24 m`
B. `1.60 m`
C. `0.32 m`
D. `0.16 m`

Answer

To solve the problem, we need to find the distance at which you should hold a concave mirror from your face in order to see your image upright with a magnification of 5. The focal length of the concave mirror is given as \( f = -0.4 \, \text{m} \) (negative because it is a concave mirror).

### Step-by-Step Solution:

1. **Understand the Magnification Formula**:
The magnification \( m \) for mirrors is given by the formula:
\[ m = \frac{h'}{h} = -\frac{v}{u} \]
where \( h' \) is the height of the image, \( h \) is the height of the object, \( v \) is the image distance, and \( u \) is the object distance. Since we want the image to be upright, the magnification will be positive. Thus, we have:
\[ m = +5 \]

2. **Relate Magnification to Object and Image Distances**:
From the magnification formula, we can express \( v \) in terms of \( u \):
\[ 5 = -\frac{v}{u} \implies v = -5u \]

3. **Use the Mirror Formula**:
The mirror formula relates the object distance \( u \), the image distance \( v \), and the focal length \( f \):
\[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \]
Substituting \( f = -0.4 \, \text{m} \) and \( v = -5u \):
\[ \frac{1}{-0.4} = \frac{1}{-5u} + \frac{1}{u} \]

4. **Simplify the Equation**:
Rewrite the equation:
\[ -\frac{5}{2} = \frac{-1 + 5}{5u} \implies -\frac{5}{2} = \frac{4}{5u} \]
Cross-multiplying gives:
\[ -5 \cdot 5u = 8 \implies -25u = 8 \implies u = -\frac{8}{25} \, \text{m} \]

5. **Calculate the Object Distance**:
The value of \( u \) is negative, indicating that the object is real and located on the same side as the incoming light. Thus:
\[ u = -0.32 \, \text{m} \]
The distance at which you hold the mirror from your face is:
\[ |u| = 0.32 \, \text{m} \]

### Final Answer:
You should hold the mirror at a distance of **0.32 meters** from your face.