A movie camera with a (single) lens of focal length `75mm` takes a picture of a person standing `27m` away. If the person is `180cm`, tall what is the height of the image on the film ?

Answer

To solve the problem of finding the height of the image on the film from a movie camera with a lens of focal length `75mm`, we will follow these steps:

### Step 1: Convert Units
Convert the focal length and the distance of the person from meters to the same unit.
- Focal length, \( f = 75 \text{ mm} = 0.075 \text{ m} \)
- Distance of the person from the camera, \( u = 27 \text{ m} \)
- Height of the person, \( H_o = 180 \text{ cm} = 1.8 \text{ m} \)

### Step 2: Apply the Lens Formula
The lens formula relates the object distance \( u \), the image distance \( v \), and the focal length \( f \):
\[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \]
Rearranging gives us:
\[ \frac{1}{v} = \frac{1}{f} - \frac{1}{u} \]

### Step 3: Substitute Values into the Lens Formula
Substituting the known values into the lens formula:
\[ \frac{1}{v} = \frac{1}{0.075} - \frac{1}{-27} \]
Calculating each term:
\[ \frac{1}{0.075} = 13.33 \quad \text{and} \quad \frac{1}{-27} = -0.037 \]
Thus:
\[ \frac{1}{v} = 13.33 + 0.037 = 13.367 \]
Now, find \( v \):
\[ v = \frac{1}{13.367} \approx 0.0747 \text{ m} \]

### Step 4: Calculate Magnification
The magnification \( m \) is given by:
\[ m = \frac{H_i}{H_o} = \frac{v}{u} \]
Substituting the values:
\[ m = \frac{0.0747}{-27} \]
Calculating \( m \):
\[ m \approx -0.00276 \]

### Step 5: Find the Height of the Image
Using the magnification to find the height of the image \( H_i \):
\[ H_i = m \cdot H_o \]
Substituting the values:
\[ H_i = -0.00276 \cdot 1.8 \approx -0.00497 \text{ m} \approx -0.497 \text{ cm} \]

### Step 6: Interpret the Result
The negative sign indicates that the image is inverted. Therefore, the height of the image on the film is approximately \( 0.5 \text{ cm} \).

### Final Answer
The height of the image on the film is approximately \( 0.5 \text{ cm} \).

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