Understanding Pinhole Cameras: The Height of the Man Behind the Image

Have you ever wondered how a pinhole camera can form an image that is smaller than the actual object? This classic problem involves using the properties of similar triangles to determine the height of an object from its image. In this article, we will explore how to calculate the height of a man based on the dimensions of his image in a pinhole camera setup.

The Pinhole Camera Principle

A pinhole camera is a simple optical device that uses a tiny hole as an aperture to create an inverted, sharp image on a screen. The principle relies on the properties of similar triangles. When light passes through the pinhole, it creates an inverted image on the opposite side of the camera. The height of the object and the height of its image are directly proportional to their respective distances from the pinhole.

Setting Up the Problem

We are given the following information:

The height of the pinhole camera is 15 cm. The height of the image of the man is 3 cm. The man stands 9 meters away from the camera.

The first step is to set up the equation that relates the height of the image, the height of the object, and their respective distances from the pinhole. The equation is:

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Solving for the Height of the Man

Let's substitute the known values into the equation:

(frac{3 text{ cm}}{h_{object}} frac{15 text{ cm}}{900 text{ cm}})

Cross-multiplying gives:

(3 text{ cm} times 900 text{ cm} 15 text{ cm} times h_{object})

Calculating the left side:

2700 text{ cm}^2 15 text{ cm} times h_{object})

Now, divide both sides by 15 cm:

(h_{object} frac{2700 text{ cm}^2}{15 text{ cm}} 180 text{ cm})

Therefore, the height of the man is approximately 180 cm, or 1.8 meters.

Conclusion

Using the properties of similar triangles, we can accurately determine the height of an object from its image in a pinhole camera setup. The key is to correctly identify the relevant ratios and perform the necessary calculations. With a little bit of math, anyone can solve seemingly complex optical problems like this one.

Keywords: pinhole camera, similar triangles, image height calculation