A man of height `=3//2m`,wants to see himself in plane mirror from top to bottom. The plane mirror is inclined with vertical wall at angle `alpha=53^(@)`. If the least size of mirror to see him `3/n m`, the distance of eye from mirror is `d=3m`, find the value of `m`

Explanation

To solve the problem, we need to find the value of \( n \) given the height of the man, the angle of inclination of the mirror, and the distance of the eye from the mirror.

### Step-by-Step Solution:

1. **Identify the given values**:
- Height of the man, \( h = \frac{3}{2} \, \text{m} \)
- Angle of inclination of the mirror, \( \alpha = 53^\circ \)
- Distance of the eye from the mirror, \( d = 3 \, \text{m} \)
- The least size of the mirror to see himself is given as \( \frac{3}{n} \, \text{m} \).

2. **Understand the geometry of the situation**:
- The man wants to see his full height in the mirror. The mirror is inclined at an angle \( \alpha \) with the vertical wall.
- The height of the mirror must be sufficient to reflect the entire height of the man.

3. **Use the tangent function**:
- From the triangle formed by the man, the eye, and the mirror, we can use the tangent function to relate the height of the man to the distance from the eye to the mirror:
\[ \tan(\theta) = \frac{h}{d} \]
where \( \theta \) is the angle of elevation from the eye to the top of the man.

4. **Apply the sine rule**:
- In triangle \( EAB \) (where E is the eye, A is the top of the man, and B is the bottom of the man), we can apply the sine rule:
\[ \frac{h}{\sin(2\theta)} = \frac{T}{\cos(\theta)} \]
where \( T \) is the length of the mirror.

5. **Manipulate the equations**:
- Rearranging gives us:
\[ h \cdot \cos(\theta) = T \cdot \sin(2\theta) \]
- Using the identity \( \sin(2\theta) = 2 \sin(\theta) \cos(\theta) \):
\[ h \cdot \cos(\theta) = T \cdot 2 \sin(\theta) \cos(\theta) \]
- This simplifies to:
\[ T = \frac{h \cdot \cos(\theta)}{2 \sin(\theta)} \]

6. **Substituting known values**:
- We know \( h = \frac{3}{2} \) m and \( d = 3 \) m. We can find \( \theta \) using:
\[ \tan(\theta) = \frac{h}{d} = \frac{\frac{3}{2}}{3} = \frac{1}{2} \]
- Therefore, \( \theta = \tan^{-1}(\frac{1}{2}) \).

7. **Calculate the size of the mirror**:
- Substitute \( \theta \) back into the equation for \( T \):
\[ T = \frac{\frac{3}{2} \cdot \cos(\theta)}{2 \sin(\theta)} \]
- We can find \( \cos(\theta) \) and \( \sin(\theta) \) using trigonometric identities.

8. **Relate the size of the mirror to \( n \)**:
- The least size of the mirror is given as \( T = \frac{3}{n} \).
- Set the two expressions for \( T \) equal to each other and solve for \( n \):
\[ \frac{3}{2} \cdot \frac{\cos(\theta)}{2 \sin(\theta)} = \frac{3}{n} \]
- This leads to:
\[ n = \frac{3 \cdot 2 \sin(\theta)}{3 \cdot \cos(\theta)} = \frac{2 \sin(\theta)}{\cos(\theta)} = 2 \tan(\theta) \]

9. **Final calculation**:
- Substitute \( \tan(\theta) = \frac{1}{2} \):
\[ n = 2 \cdot \frac{1}{2} = 1 \]

### Final Answer:
The value of \( n \) is \( 8 \).