With a 4 cm high object, positioned 12 cm away from a 4 cm focal length concave mirror, what is the height of the image?

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Question restatement: A 4 cm high object is placed 12 cm from a concave mirror with focal length 4 cm. Find the height of the image. Answer options: a) 3 cm, b) 16 cm, c) 1 cm, d) 2 cm.

Option a: 3 cm. This would imply a magnification of hi/ho = 3/4 = 0.75. However, for a concave mirror with do = 12 cm and f = 4 cm, the image distance di comes from 1/f = 1/do + 1/di, giving di = 6 cm and magnification m = -di/do = -0.5. The resulting image height hi = m*ho = -2 cm, not +3 cm. The sign and magnitude do not align with the actual geometry, so this is incorrect.

Option b: 16 cm. This would require a large magnification of hi/ho = 16/4 = 4, or a very large image distance, which is not compatible with the mirror’s geometry for the given object distance. Using the mirror equation, di = 6 cm and m = -0.5 yield hi = -2 cm. Therefore 16 cm is not achievable in this setup.

Option c: 1 cm. This would correspond to a magnification of hi/ho = 1/4 = 0.25 in magnitude, implying di = 3 cm if ho = 4 cm. But the proper calculation from 1/f = 1/do + 1/di with f = 4 cm and do = 12 cm gives di = 6 cm, leading to hi = -2 cm. Hence 1 cm is not consistent with the actual image distance and magnification.

Option d: 2 cm. This is the magnitude of the image height, with hi = m*ho and m = -di/do = -6/12 = -0.5, so hi = (-0.5)*(4 cm) = -2 cm. The negative sign indicates the image is inverted; if the question reports height by magnitude, 2 cm is the correct magnitude. This option matches the correct calculation for di = 6 cm and the resulting image height.

Summary: The correct computations give di = 6 cm and hi = -2 cm, so the image height in magnitude is 2 cm with inversion. Among the options, only d) 2 cm aligns with this result, while the others conflict with the derived image distance or sign, making them incorrect.

With a 4 cm high object, positioned 12 cm away from a 4 cm focal length concave mirror, what is the height of the image?

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