Final answer:
Using the information provided and the mirror equations, the radius of curvature for the concave mirror would be -66.67 cm. The negative sign indicates that the mirror is concave, as expected.
Step-by-step explanation:
The concept to solve this question revolves around the mirror equation for spherical mirrors, which states 1/f = 1/v + 1/u. Here, f is the focal length, v is the image distance, and u is the object distance. The given information reveals that the image is at the same distance from the mirror as the object, but on the same side. This implies that the image distance v is -25 cm (as per conventions in mirror equations).
The magnification (m) provided is 2, which equals -v/u for a mirror. Since v = -u in this case, m = -(-u/u) = 1, which contradicts our given magnification. However, the concave mirror can give an upright and magnified image when the object is placed within its focal point, causing a virtual image. This means our presumption that the image is real (and hence v is negative) is false. With a magnification of 2 (a positive m means the image is upright), v is, therefore, 2u or -50 cm.
Plugging these values into the mirror equation, we have 1/f = 1/(-50 cm) + 1/(-25 cm). After solving, we find that f equals -33.33 cm. The radius of curvature (R) is twice the focal length, R = 2f or R = -66.67 cm. The negative sign indicates that it is a concave mirror.
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