You photograph a 2.0-m-tall person with a camera that has a \(5.0 \mathrm{cm}\) -focal length lens. The image on the film must be no more than \(2.0 \mathrm{cm}\) high. (a) What is the closest distance the person can stand to the lens? (b) For this distance, what should be the distance from the lens to the film?

Understanding how an image's size relates to its actual size is a crucial concept in optics. The magnification formula, which is expressed as
\( M = \frac{d_i}{d_o} \)
connects the image distance (\(d_i\)) to the object distance (\(d_o\)). In simpler terms, it tells us how much larger or smaller the image is compared to the object.
When \(M\) is greater than 1, the image is magnified; when \(M\) is less than 1, as in the exercise provided, the image is reduced in size.
For photographic purposes, such as the given problem, the magnification must be controlled to produce an image of a desired size. By rearranging the formula to \(d_o = \frac{d_i}{M}\), we can find out how close the object (the person) can be to the lens to achieve the necessary image size on the film.

The thin lens formula is a key equation for understanding how lenses form images of objects, formulated as:
\( \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \)
where \(f\) represents the focal length of the lens. This fundamental relation allows us to solve for one distance when the other and the focal length are known. It's instrumental in applications such as photography and optical instrument design.
In the context of our problem, knowing the object distance and focal length allowed us to calculate the required distance from the lens to the film (\(d_i\)). This is essential to ensure the film captures the image at the right size and clarity.
Additionally, the thin lens formula reinforces the interdependence between object and image distances, implying they must change concurrently to maintain focus with a given lens.

Calculation of the object distance is critical, especially when fine-tuning the desired size of a photographic image. As seen in the exercise, object distance (\(d_o\)) can be derived once we've determined the magnification and have knowledge of the image distance.
By isolating \(d_o\) in the magnification formula (\(d_o = \frac{d_i}{M}\)), we can find the specific distance at which a real object must be placed from the lens to create an image of the appropriate size. This understanding allows photographers to position their camera appropriately to capture a subject fully, as dictated by the limitations of their lens and the physical space available.
The step-by-step solution provided for the exercise represents a practical application of these calculations. Estimating the closest distance for the object not only maximizes the use of the lens' potential but also ensures that the desired image properties are achieved without the need for digital manipulation or cropping post-capture.