Question 23.4: The Face in the Mirror GOAL Find a focal length from a magni...
The Face in the Mirror
GOAL Find a focal length from a magnification and an object distance.
PROBLEM When a woman stands with her face 40.0 cm from a cosmetic mirror, the upright image is twice as tall as her face. What is the focal length of the mirror?
STRATEGY To find f in this example, we must first find q, the image distance. Because the problem states that the image is upright, the magnification must be positive (in this case, M = +2), and because M = -q/p, we can determine q.
Obtain q from the magnification equation:
M=-\,{\frac{q}{{p}}}=2
q=-2p=-2(40.0\;\mathrm{cm})=-80.0\;\mathrm{cm}
Because q is negative, the image is on the opposite side of the mirror and hence is virtual. Substitute q and p into the mirror equation and solve for f:
{\frac{1}{40.0\,\mathrm{cm}}}-{\frac{1}{80.0\,\mathrm{cm}}}={\frac{1}{f}}
f = 80.0 cm
REMARKS The positive sign for the focal length tells us that the mirror is concave, a fact we already knew because the mirror magnified the object. (A convex mirror would have produced a smaller image.)