Final answer:
The lens is a diverging lens with a negative focal length, indicating the image is virtual.
Step-by-step explanation:
The scenario described involves an object being imaged on the same side as the object itself and closer to the lens. This suggests that we are dealing with a diverging lens, as converging lenses produce real images on the opposite side. To determine the focal length and the nature of the image, we can use the lensmaker's equation 1/f = 1/do + 1/di, where f is the focal length, do is the object distance, and di is the image distance.
Given that do = -37.5 cm (the negative sign indicates the object is on the same side as the incoming light) and di = -8.20 cm (again, negative since the image is on the same side as the object), we can calculate the focal length as follows:
1/f = 1/(-37.5) + 1/(-8.20)
Upon solving, we find the focal length f is negative, confirming the lens is indeed a diverging lens. Additionally, since the image is formed on the same side of the lens as the object, the image is virtual.