A makeup mirror produces a magnification of 1.50 when a person's face is 12.0 cm away. Since the image is upright and magnified, it must be virtual, giving d_i a negative value found from d_i equals negative m times d_o. What focal length does the mirror equation then give?

1. The magnification equation is m = -d_i / d_o, so d_i = -m d_o. 2. Substituting m = 1.50 and d_o = 12.0 cm gives d_i = -(1.50)(12.0 cm) = -18.0 cm. 3. The negative sign correctly makes this a virtual image, matching the magnified, upright image a makeup mirror shows. 4. Using f = (d_i d_o) / (d_o + d_i) gives f = ((-18.0)(12.0)) / (12.0 + (-18.0)). 5. The numerator is -216 and the denominator is -6.0, so f = -216 / -6.0 = 36.0 cm. 6. A positive 36.0 cm is expected, because a magnifying makeup mirror is concave, and concave mirrors carry positive f. 7. -36.0 cm and -7.20 cm wrongly keep a negative sign that a concave mirror's focal length should not have. _Source: OpenStax Physics (CC BY 4.0), Ch 16 "Mirrors and Lenses", section 16.1 Reflection_