What is the focal length of a makeup mirror that produces a magnification of 1.50 when a person’s face is 12.0 centimeters away?
Let’s call the 1.50-magnification of the mirror capital 𝑀. And in this case, the object that creates an image is the person’s face, which is 12.0 centimeters from the mirror; we’ll call that distance 𝑑 sub 𝑜. We want to solve for the focal length of the mirror, which we’ll call 𝑓.
We can start by recalling the equation for an object’s magnification 𝑀. Magnification is equal to the image height divided by the object height, and it’s also equal to negative the image distance divided by the object distance. In our case, we’ll want to use the fact that 𝑀 is equal to negative 𝑑 sub 𝑖 over 𝑑 sub 𝑜. Since we’re given both 𝑀 and 𝑑 sub 𝑜 in the problem statement, we can solve for 𝑑 sub 𝑖.
When we plug in using those given values, then we find that 𝑑 sub 𝑖, the image distance, equals negative 18.0 centimeters. Now that we know 𝑑 sub 𝑖, we can refer to another equation sometimes called the lensmaker equation or the mirror equation. This equation connects the focal length of the mirror with the object and image distances.
When we apply it to our situation, we see that we’ve solved for 𝑑 sub 𝑖 and we know 𝑑 sub 𝑜, and we want to find out 𝑓, the focal length of the mirror. We can rearrange this equation to solve for focal length 𝑓. First, we multiply all the terms by 𝑓, which cancels the focal length from the left-hand side.
Then we divide both sides by the terms in the parentheses, one over 𝑑𝑜 plus one of 𝑑𝑖, which cancels that entire term from the right-hand side of our equation. Finally, we multiply the left-hand side by 𝑑𝑜 times 𝑑𝑖 divided by 𝑑𝑜 times 𝑑𝑖, which gives us our final equation for the focal length 𝑓: focal length equals 𝑑𝑜 times 𝑑𝑖 divided by 𝑑𝑖 plus 𝑑𝑜.
We’re now ready to plug in with our values for those two variables. With those values entered in, we’re ready to calculate 𝑓, the focal length. And when we do, we find it’s equal to 0.360 meters. That’s the focal length of this concave makeup mirror.