Question

For a camera with focal length \( 150\ \text{mm} \) and a \( 30\ \text{cm} \times 30\ \text{cm} \) format size, what height above ground (in meters) is necessary for a vertical photograph to cover an area of \( 9\ \text{km}^2 \)?

• A. 1500
• B. 150
• C. 15
• D. 15000

Hint:

Use the vertical-photograph scale relation S = f/H together with S = photo side / ground side, after converting the 9 km² area to a ground square side.

Answer 1

The Correct Option is A

Step 1: Write down the given data and the scale relation for a vertical photograph.
The focal length of the camera is \( f = 150\ \text{mm} = 0.15\ \text{m} \). The photo format is square with side \( a = 30\ \text{cm} = 0.3\ \text{m} \). The photo scale of a vertical aerial photograph is \( S = \dfrac{f}{H} \), where \( H \) is the flying height above the ground. The same scale also equals the ratio of any photo distance to the corresponding ground distance, \( S = \dfrac{\text{photo distance}}{\text{ground distance}} \).
Step 2: Find the ground distance corresponding to the photo format.
The photograph covers a square area of \( 9\ \text{km}^2 \) on the ground. Since the format is square, the ground coverage is also square, so the side of the ground square is \( \sqrt{9\ \text{km}^2} = 3\ \text{km} = 3000\ \text{m} \).
Step 3: Compute the photo scale from the format side and the ground side.
\[ S = \frac{a}{\text{ground side}} = \frac{0.3\ \text{m}}{3000\ \text{m}} = \frac{1}{10000} \]
Step 4: Use the scale relation to find the flying height.
\[ S = \frac{f}{H} \implies H = \frac{f}{S} = 0.15\ \text{m} \times 10000 = 1500\ \text{m} \]
Step 5: Check why the other options are wrong.
Options (B) 150 m and (C) 15 m come from mishandling the format side or the km-to-m conversion by a factor of 10 or 100, and option (D) 15000 m results from inverting the scale ratio. Only \( H = 1500\ \text{m} \) is consistent with \( S = f/H \) and the 9 km² ground coverage.\[ \boxed{H = 1500\ \text{m}} \]