Question

A photograph of a landscape is captured by a drone camera at a height of 18 km. The size of the camera film is 2 cm \( \times \) 2 cm and the area of the landscape photographed is 400 km\(^2\). The focal length of the lens in the drone camera is:

• A. \( 0.9 \, \text{cm} \)
• B. \( 2.8 \, \text{cm} \)
• C. \( 2.5 \, \text{cm} \)
• D. \( 1.8 \, \text{cm} \)

Hint:

To solve problems involving photographs and cameras, use the relationship between the image size, the actual size of the object, and the distance (height) from the object. This is a direct application of similar triangles.

Answer 1

The Correct Option is A

We use the formula for a camera to relate the size of the image, the size of the landscape, the height of the camera, and the focal length: \[ \frac{\text{Size of image}}{\text{Size of landscape}} = \frac{\text{Focal length}}{\text{Height of camera}}. \] Here, the size of the image is \( 2 \times 2 \) cm (so the area is 4 cm\(^2\)), the size of the landscape is 400 km\(^2\), and the height of the camera is 18 km. Substituting these values into the equation and solving for the focal length, we find that the focal length is \( 0.9 \, \text{cm} \). 
Final Answer: \( 0.9 \, \text{cm} \).

Answer 2

Step 1: Understand the problem setup.
We are given the following information:
- The height of the drone camera is \( 18 \, \text{km} \),
- The size of the camera film is \( 2 \, \text{cm} \times 2 \, \text{cm} \),
- The area of the landscape photographed is \( 400 \, \text{km}^2 \).
We are asked to find the focal length of the lens in the drone camera.

Step 2: Use the concept of similar triangles.
The image formed on the camera film is a scaled-down version of the landscape. The relationship between the size of the image, the focal length, and the height of the drone can be modeled using similar triangles.
Let:
- \( A_{\text{landscape}} \) be the area of the landscape photographed,
- \( A_{\text{image}} \) be the area of the camera film,
- \( h \) be the height of the camera, and
- \( f \) be the focal length of the lens.
The areas of the landscape and the image are related as:
\[ \frac{A_{\text{image}}}{A_{\text{landscape}}} = \left( \frac{f}{h} \right)^2. \] Substitute the given values:
- \( A_{\text{image}} = 2 \, \text{cm} \times 2 \, \text{cm} = 4 \, \text{cm}^2 \),
- \( A_{\text{landscape}} = 400 \, \text{km}^2 \).
First, convert the landscape area into \( \text{cm}^2 \). Since \( 1 \, \text{km} = 10^5 \, \text{cm} \), we have:
\[ A_{\text{landscape}} = 400 \, \text{km}^2 = 400 \times (10^5)^2 = 400 \times 10^{10} \, \text{cm}^2. \] Now, use the equation to find the focal length \( f \):
\[ \frac{4}{400 \times 10^{10}} = \left( \frac{f}{18 \times 10^5} \right)^2. \] Simplifying the equation:
\[ \frac{4}{400 \times 10^{10}} = \frac{f^2}{(18 \times 10^5)^2}. \] Solve for \( f^2 \):
\[ f^2 = \frac{4 \times (18 \times 10^5)^2}{400 \times 10^{10}}. \] Now, calculating the value of \( f \), we get:
\[ f = 0.9 \, \text{cm}. \]

Step 3: Final answer.
The focal length of the lens is \( 0.9 \, \text{cm} \).
\[ \boxed{0.9 \, \text{cm}}.\]