Final answer:
The focal length of the makeup mirror with a magnification of 1.5 and an object distance of 11.5 cm is 0.345 meters or -34.5 centimeters, indicating a concave mirror.
Step-by-step explanation:
To determine the focal length of a makeup mirror that produces a magnification of 1.5, we can use the mirror equation and magnification equation for curved mirrors. The object distance (do) is given as 11.5 cm (or 12.0 cm in similar problems), and we need to convert this into meters by dividing by 100, do = 0.115 m. The magnification (m) is given as 1.5, and since it is positive, this implies a virtual image. The mirror equation is given by 1/f = 1/do + 1/di and the magnification equation is m = -di/do, where di is the image distance and f is the focal length of the mirror.
Since m = 1.5 = -di/do, we can rearrange to find di = -1.5 × do. Substituting do = 0.115 m, we get di = -1.5 × 0.115 m = -0.1725 m. Using the mirror equation 1/f = 1/do + 1/di, we get 1/f = 1/0.115 + 1/ (-0.1725), which gives us the focal length, f.
Calculating this we get f = 1 / ((1/0.115) - (1/0.1725)) = 1 / (8.6957 - 5.7971) = 1 / 2.8986 = 0.345 m.
Therefore, the focal length of the mirror is 0.345 m or, when converted to centimeters, -34.5 cm (the negative sign indicates a concave mirror).