Final answer:
The camera lens must be focused at a distance of 7.60 m to take a clear photo of the hummingbird's image in the plane mirror, accounting for the distances from the camera to the mirror and from the mirror to the image's apparent position behind it.
Step-by-step explanation:
To determine the distance for which the camera lens must be focused to take a clear photo of the hummingbird's image in a plane mirror, we need to assess the total path that light would travel from the hummingbird to the mirror and then from the mirror to the camera.
The hummingbird is 3.30 m from the mirror and 5,000 m to the right, but the 5,000 m distance is irrelevant here since it is not influencing the path of light in the direction of the mirror to the camera. Since plane mirrors create images that appear to be the same distance behind the mirror as the actual object is in front of it, the image's apparent position will also be 3.30 m behind the mirror.
The camera, being 4.30 m in front of the mirror, will need to focus on a point that is the sum of its distance to the mirror and the mirror's distance to the image. This means that the camera should focus at a distance of:
Distance to the image = Distance of camera from mirror + Distance of image behind mirror
Distance to the image = 4.30 m + 3.30 m
Distance to the image = 7.60 m
Therefore, for a clear photo, the camera lens needs to be focused at a distance of 7.60 m.