A photographic camera with a lens of focal length 5 cm is used for capturing images. The vertical length of the film used is 24 mm in which image of a 1.68 m tall man is to be captured. Find the minimum distance (in m) of the man from the lens such that his complete image can be obtained.

Answer

To solve the problem, we need to find the minimum distance of a 1.68 m tall man from a camera lens with a focal length of 5 cm, such that his complete image can be captured on a film of height 24 mm.

### Step-by-Step Solution:

1. **Convert Measurements to Consistent Units**:
- The height of the man (object height) \( h_o = 1.68 \, \text{m} = 1680 \, \text{mm} \).
- The height of the film (image height) \( h_i = 24 \, \text{mm} \).
- The focal length of the lens \( f = 5 \, \text{cm} = 50 \, \text{mm} \).

2. **Calculate the Magnification**:
The magnification \( m \) is given by the formula:
\[ m = \frac{h_i}{h_o} = \frac{24 \, \text{mm}}{1680 \, \text{mm}} = \frac{24}{1680} = \frac{1}{70} \]
Since the image is real and inverted, we take magnification as negative:
\[ m = -\frac{1}{70} \]

3. **Use the Magnification Formula**:
The magnification can also be expressed in terms of object distance \( u \) and image distance \( v \):
\[ m = -\frac{v}{u} \]
Rearranging gives:
\[ v = -mu \]

4. **Apply the Lens Formula**:
The lens formula is given by:
\[ \frac{1}{f} = \frac{1}{v} - \frac{1}{u} \]
Substituting \( v = -mu \) into the lens formula:
\[ \frac{1}{f} = \frac{1}{-mu} - \frac{1}{u} \]
This simplifies to:
\[ \frac{1}{f} = -\frac{1}{u}\left(m + 1\right) \]

5. **Substituting Known Values**:
Substituting \( f = 50 \, \text{mm} \) and \( m = -\frac{1}{70} \):
\[ \frac{1}{50} = -\frac{1}{u}\left(-\frac{1}{70} + 1\right) \]
Simplifying the expression in the parentheses:
\[ -\frac{1}{70} + 1 = 1 - \frac{1}{70} = \frac{70 - 1}{70} = \frac{69}{70} \]
Thus:
\[ \frac{1}{50} = \frac{1}{u} \cdot \frac{69}{70} \]

6. **Solve for \( u \)**:
Rearranging gives:
\[ u = \frac{69}{70} \cdot 50 \]
Calculating \( u \):
\[ u = \frac{69 \times 50}{70} = \frac{3450}{70} = 49.2857 \, \text{mm} \approx 49.29 \, \text{mm} \]

7. **Convert to Meters**:
Since the question asks for the distance in meters:
\[ u = 49.29 \, \text{mm} = 0.04929 \, \text{m} \]

### Final Answer:
The minimum distance of the man from the lens such that his complete image can be obtained is approximately **0.0493 m**.