Reflection and Mirrors Legacy Problem #17 Guided Solution

This question pertains to a convex mirror that produces an image of a shoplifter. The shoplifter is located a distance of 4.8 meters from the mirror. That's the doh value of the shoplifter. The objects of DO, or object distance, equal 4.8 meters. The next part of this problem is the critical part. It says her image is magnified by a factor of one half. That means that the high per hole ratio, image height to object height, is equal to one half. Now the one half could be either a positive or a negative, so you have to think this part through. The fact that it's a convex mirror means that the image is going to be an upright image. I know that from my understanding of convex mirrors. If you're uncertain, just get out a spoon and look at yourself in the convex side. You'll notice your image is upright. So the fact that high is positive means that the high per hole ratio has got to be positive. And so high per hole equals positive one half. Now the high per hole ratio is equal to the negative dye per doh ratio, where dye is image distance and doh is object distance. So if I know object distance and I know the ratio of eye to hole, I can actually calculate the image distance, and the question asks me to find the focal length. So I'm thinking that the only equation with focal length in it is the mirror equation, which relates focal length to object distance, doh, and image distance, dye. So if I can just find dye, I'll be able to calculate the focal length. So I'm going to just say that the positive one half is equal to the negative dye per doh ratio, where the doh is 4.8. So positive one half equals negative dye divided by 4.8. Solving for dye, I end up getting dye equal negative 2.4 meters. Now take a deep breath. You're just about ready to calculate the focal length. You know that doh, object distance, is 4.8 meters. You know that dye, image distance, is negative 2.4 meters. And you just say one over f equal one over 4.8 plus one over negative 2.4. Evaluate the right side of that equation and you get negative 0.20833. Take the reciprocal of it and that gives you the focal length. It comes out to be negative 4.8 meters. We would expect a negative answer here because convex mirrors have negative focal lengths.