A slide projector has to project a 35 mm slide `(35 mm xx 23 mm)` on a 2 m xx 2m screen at a distance of 10 m from the lens. What should be the focal length of the lens in the projector ?

Explanation

To find the focal length of the lens in the projector, we can follow these steps:

### Step 1: Understand the problem
We need to project a 35 mm x 23 mm slide onto a 2 m x 2 m screen, with the lens positioned 10 m away from the screen. We need to determine the focal length of the lens.

### Step 2: Convert units
Convert the dimensions of the slide and the screen from meters to millimeters for consistency:
- Slide dimensions: 35 mm x 23 mm
- Screen dimensions: 2 m x 2 m = 2000 mm x 2000 mm

### Step 3: Calculate magnification
The magnification (M) is given by the formula:
\[ M = \frac{V}{U} \]
Where:
- \( V \) = image distance (distance from the lens to the screen)
- \( U \) = object distance (distance from the lens to the slide)

For the width of the slide:
- \( U = 35 \) mm
- \( V = 2000 \) mm

Calculating magnification for the width:
\[ M = \frac{2000}{35} \approx 57.14 \]

For the height of the slide:
- \( U = 23 \) mm
- \( V = 2000 \) mm

Calculating magnification for the height:
\[ M = \frac{2000}{23} \approx 86.96 \]

### Step 4: Choose the minimum magnification
Since we want to project the slide as a square, we will consider the minimum magnification, which is approximately 57.14.

### Step 5: Relate object distance to image distance
Using the magnification formula:
\[ M = \frac{V}{U} \]
We can rearrange it to find \( U \):
\[ U = \frac{V}{M} \]
Substituting \( V = 10 \) m (which is 10000 mm) and \( M \approx 57.14 \):
\[ U = \frac{10000}{57.14} \approx 175.0 \text{ mm} \]

### Step 6: Apply the lens formula
The lens formula is given by:
\[ \frac{1}{f} = \frac{1}{V} - \frac{1}{U} \]
Substituting \( V = 10000 \) mm and \( U = -175.0 \) mm (the object distance is taken as negative):
\[ \frac{1}{f} = \frac{1}{10000} - \frac{1}{-175} \]
\[ \frac{1}{f} = \frac{1}{10000} + \frac{1}{175} \]

### Step 7: Calculate the focal length
Finding a common denominator and calculating:
\[ \frac{1}{f} = \frac{1}{10000} + \frac{1}{175} \]
Calculating \( \frac{1}{175} \) in terms of 10000:
\[ \frac{1}{175} = \frac{10000}{175 \times 10000} \approx \frac{57.14}{10000} \]
So:
\[ \frac{1}{f} = \frac{1 + 57.14}{10000} = \frac{58.14}{10000} \]
Thus:
\[ f \approx \frac{10000}{58.14} \approx 172.0 \text{ mm} \]

### Step 8: Convert to centimeters
Converting to centimeters:
\[ f \approx 17.2 \text{ cm} \]

### Final Answer
The focal length of the lens in the projector should be approximately **17.2 cm**.