### Video Transcript

What is the focal length of a makeup mirror that produces a magnification of 1.50 when a personβs face is 12.0 centimeters away?

Letβs call the 1.50-magnification of the mirror capital π. And in this case, the object that creates an image is the personβs face, which is 12.0 centimeters from the mirror; weβll call that distance π sub π. We want to solve for the focal length of the mirror, which weβll call π.

We can start by recalling the equation for an objectβs magnification π. Magnification is equal to the image height divided by the object height, and itβs also equal to negative the image distance divided by the object distance. In our case, weβll want to use the fact that π is equal to negative π sub π over π sub π. Since weβre given both π and π sub π in the problem statement, we can solve for π sub π.

When we plug in using those given values, then we find that π sub π, the image distance, equals negative 18.0 centimeters. Now that we know π sub π, 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 weβve solved for π sub π and we know π sub π, and we want to find out π, the focal length of the mirror. We can rearrange this equation to solve for focal length π. First, we multiply all the terms by π, which cancels the focal length from the left-hand side.

Then we divide both sides by the terms in the parentheses, one over ππ plus one of ππ, which cancels that entire term from the right-hand side of our equation. Finally, we multiply the left-hand side by ππ times ππ divided by ππ times ππ, which gives us our final equation for the focal length π: focal length equals ππ times ππ divided by ππ plus ππ.

Weβre now ready to plug in with our values for those two variables. With those values entered in, weβre ready to calculate π, the focal length. And when we do, we find itβs equal to 0.360 meters. Thatβs the focal length of this concave makeup mirror.