A tourist takes a picture of a mountain 14 km away using a camera that has a lens with a focal l

step 1 A tourist takes a picture of a mountain 14 kilometers away using a camera that has a lens with a

A tourist takes a picture of a mountain 14 km away using a...

A tourist takes a picture of a mountain $14 \mathrm{km}$ away using a camera that has a lens with a focal length of $50 \mathrm{mm}$. She then takes a second picture when she is only $5.0 \mathrm{km}$ away. What is the ratio of the height of the mountain's image on the camera's image sensor for the second picture to its height on the image sensor for the first picture?

Key Concepts

Proportional Relationships in Imaging

Proportional reasoning in imaging involves understanding that the size of the image is directly related to the magnification factor, which in turn depends on the focal length and the object distance. This concept allows for the comparison of image scales when the distance to the object changes, by revealing that the image size on the sensor is inversely proportional to the object distance for far objects.

Focal Length

Focal length is a key parameter of a lens that indicates the distance over which initially collimated (parallel) rays are brought to a focus. It is fundamental in determining the field of view and the scale of the image on the sensor. In practical applications, a fixed focal length helps to maintain a consistent proportional relation between object size and image size, when objects are at varying distances.

Thin Lens Equation

The thin lens equation relates the focal length of a lens to the object distance and the image distance by the formula 1/f = 1/d_o + 1/d_i. This relationship is central to understanding how lenses focus light to form images, as it provides the mathematical basis for determining the image location when an object is at a certain distance.

Lens Magnification

Magnification is defined as the ratio of the size of the image produced by the lens to the actual size of the object. In the context of a thin lens, it can be calculated by the formula m = -d_i/d_o, where the negative sign indicates image inversion. For objects that are very far compared to the focal length, the magnification approximates to f/d_o, demonstrating how image size scales inversely with object distance.

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