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 will follow these steps:

### Step 1: Understand the given data
- Focal length of the lens (f) = 5 cm = 0.05 m
- Height of the man (h_o) = 1.68 m
- Height of the film (h_i) = 24 mm = 0.024 m

### Step 2: Calculate the magnification (M)
Magnification (M) is defined as the ratio of the height of the image (h_i) to the height of the object (h_o):
\[ M = \frac{h_i}{h_o} \]
Substituting the values:
\[ M = \frac{0.024 \text{ m}}{1.68 \text{ m}} = \frac{24 \text{ mm}}{1680 \text{ mm}} = \frac{1}{70} \]

### Step 3: Use the magnification formula
The magnification can also be expressed in terms of object distance (U) and image distance (V):
\[ M = -\frac{V}{U} \]
From this, we can express V in terms of U:
\[ V = -M \cdot U \]
Substituting the value of M:
\[ V = -\left(-\frac{1}{70}\right) U = \frac{U}{70} \]

### Step 4: Apply the lens formula
The lens formula is given by:
\[ \frac{1}{f} = \frac{1}{V} - \frac{1}{U} \]
Substituting V from the previous step:
\[ \frac{1}{f} = \frac{1}{\frac{U}{70}} - \frac{1}{U} \]
This simplifies to:
\[ \frac{1}{f} = \frac{70}{U} - \frac{1}{U} \]
Combining the fractions:
\[ \frac{1}{f} = \frac{70 - 1}{U} = \frac{69}{U} \]

### Step 5: Solve for U
Rearranging gives:
\[ U = 69f \]
Substituting the focal length:
\[ U = 69 \times 0.05 \text{ m} = 3.45 \text{ m} \]

### Conclusion
The minimum distance of the man from the lens such that his complete image can be obtained is **3.45 m**.

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