Final answer:
The object is 180cm to the left of the lens. The focal length of the lens is 180cm.
Step-by-step explanation:
To determine the distance of the object from the lens, we can use the magnification equation: m = -di/do, where m is the magnification, di is the image distance, and do is the object distance. Since the image is one-half the size of the object, the magnification is -1/2. We are given that the distance between the object and the image is 90cm, and since the image is formed to the right of the lens, the object distance must be negative. Therefore, using the magnification equation, we have -1/2 = (-90cm)/do. Solving for do, we find that the object distance is -180cm, or 180cm to the left of the lens.
The focal length of a lens can be determined using the lens formula: 1/f = 1/do + 1/di, where f is the focal length, do is the object distance, and di is the image distance. We are given that the object distance is -180cm and the image distance is 90cm. Substituting these values into the lens formula, we have 1/f = 1/(-180cm) + 1/90cm. Simplifying the expression, we find that 1/f = -1/180cm + 1/90cm. Adding the fractions, we get 1/f = 1/180cm. Solving for f, we find that the focal length of the lens is 180cm.