Final answer:
The focal length of a concave mirror where an object is placed at 40 cm and creates an equally sized image is 20 cm, assuming that the object is located at the center of curvature of the mirror.
Step-by-step explanation:
To calculate the focal length of a concave mirror where an object is placed at a distance of 40 cm and forms an image of equal size, we use the mirror equation: 1/f = 1/do + 1/di, where f is the focal length, do is the object distance, and di is the image distance. When the object and image are of equal size, the magnification (m) is -1, because magnification is equal to -di/do for mirrors, and the negative sign indicates an inverted image. Given that the object distance (do) is 40 cm and magnification (m) is -1, we can infer that the image distance (di) is also -40 cm (negative because the image formed by a concave mirror is real and inverted). Substituting the values into the mirror equation we get 1/f = 1/40 + 1/(-40), which simplifies to 1/f = 0. Therefore, the focal length (f) of the mirror is infinity, which theoretically cannot happen because an image of the same size as the object can only be formed at the center of curvature, which is at a finite distance. However, if we assume the question implies that the object is at the center of curvature, then the center of curvature and the focal point are related by f = C/2, where C is the center of curvature. Therefore, in this scenario, with an object distance of 40 cm, which is equal to the center of curvature, the focal length (f) would be 20 cm.