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The lens formula is a fundamental concept in optics and is used to relate the object distance, image distance, and the focal length of a lens. It is expressed as: \[ \frac{1}{u} + \frac{1}{v} = \frac{1}{f} \]Where:- \( u \) is the object distance (the distance between the object and the lens).- \( v \) is the image distance (the distance between the image and the lens).- \( f \) is the focal length of the lens.To solve most exercises involving lenses, the lens formula allows us to calculate any one of the three parameters if the other two are known. For example, in the case of the camera optics exercise, we rearrange the formula to find the image distance given the focal length and the object distance. Remember, it's crucial to keep all measurements in the same unit when using the formula, typically converting everything to meters or millimeters as needed.

Image distance is the distance from the lens to the image that is formed. This concept is key when determining whether an image will focus correctly on a sensor. Using the lens formula, we can isolate the image distance \( v \) with the equation:\[ v = \frac{1}{\left(\frac{1}{f} - \frac{1}{u}\right)} \]In our exercise, the image distance tells us how far the lens needs to be from the sensor to focus the image correctly. Calculating \( v \) involves substituting the given focal length and object distance into the equation while ensuring they are in the same units.For a real-world application, as in a camera, the lens adjusts its position from the sensor to achieve this focused state.

The focal length \( f \) of a lens is an inherent property describing how strongly the lens converges or diverges light. It is measured from the lens to the focal point, where parallel rays of light meet after passing through the lens.In the camera scenario, the focal length of 85 mm determines the lens's capability to focus light on the sensor from objects at various distances. A shorter focal length implies a wider field of view, while a longer focal length provides a magnified view but a narrower field of view.Understanding focal length is crucial when setting up a camera to ensure that you capture the desired frame and focus on distant or near objects with clarity.

The magnification formula relates the height of the image to the height of the object, as well as the image and object distances:\[ m = \frac{h'}{h} = -\frac{v}{u} \]- \( m \) is the magnification.- \( h' \) is the height of the image.- \( h \) is the height of the object.- The negative sign indicates the image is inverted.In solving the problem, this formula helps determine whether the image of the friend will fit on the sensor. By calculating the image height using:\[ h' = m \times h \]It helps us assess if the image dimensions are suitable for the sensor's size, considering both the height and width constraints.