Final answer:
To find the radius of curvature of the convex mirror, the mirror equation 1/f = 1/do + 1/di is used, since the images in both mirrors coincide, which suggests the object and image distances from the convex mirror are equal. Solving for the focal length, we find it to be -20 cm, meaning the radius of curvature is -40 cm (40 cm as a positive value). However, this answer is not listed among the options provided, suggesting a possible mistake in the question or answer choices.
Step-by-step explanation:
The question asks for the radius of curvature of a convex mirror given that an image coincides with the same produced by a plane mirror placed between an object and the convex mirror. In this situation, the radius of curvature is twice the focal length of the mirror. Since the plane mirror creates an image at the same distance behind the mirror as the object is in front of it, and the images coincide for both mirrors, the distance from the convex mirror to the object can be used to determine the image distance for the convex mirror. Given that the object is 40 cm from the convex mirror and the plane mirror is at 32 cm, the remaining 8 cm is the image distance behind the plane mirror, which is the same as the object distance from the convex mirror. Using the mirror 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 can find the focal length and thus the radius of curvature (R = 2f).
The mirror equation for the convex mirror becomes 1/f = 1/do + 1/di, which is 1/f = 1/40 + 1/(-8), leading to 1/f = 1/40 - 1/8 = -1/20. Thus, f = -20 cm (a negative focal length denotes a convex mirror). The radius of curvature of the convex mirror is therefore R = 2f, which equals 2 * (-20) = -40 cm. However, as the radius of curvature is typically stated as a positive value, the absolute value is taken, resulting in 40 cm, which is not an option in the provided choices, indicating a potential issue with the question or the choices given.