Reflection and Mirrors Legacy Problem #22 Guided Solution

An effective problem solver reads the problem carefully and identifies the given quantities, representing them in terms of the variables within common physics equations, then identifies the unknown quantities, and before ever picking up a calculator, begins to plot out a strategy as to how to get from the known quantities to the unknown quantities. Here in this problem, we read about two characteristics of an object. First of all, it's 4.9 centimeters tall, and that's HO, HO equals 4.9 centimeters. And it's positioned 14.8 centimeters from the mirror, and that's DO, DO is equal to 14.8 centimeters. We're asked to find a radius of curvature that would be able to produce an upright image, which is 7.2 centimeters tall. So the third unknown, or third known quantity in this problem is HI, HI equals 7.2 centimeters. That's the image height. And it's a positive image height, since it's described as an upright image. Now in order to calculate the radius of curvature, I understand that I must first find the focal length. In order to find the focal length, I first must know the object distance and the image distance. Now the object distance is easy, it's given as 14.8 centimeters, but I don't have the image distance, and the only way to do so is to use the relationship that the HI to HO relationship, the HI to HO ratio, is equal to the negative di-do ratio. And so, I could say 7.2 centimeters divided by 4.9 centimeters, that's the HI-HO ratio, is equal to the negative of di divided by 14.8 centimeters. Now I can rearrange that equation in order to solve for di, and when I do, I get negative 21.7469 centimeters. That's my first step. And my second step, you might recall, was to find the focal length using the MIR equation. So I say 1 over F equals 1 over do plus 1 over di. And I substitute 14.8 in for do, and negative 21.7469 centimeters in for di. Evaluating the right side of the equation gives me 0.021584, and taking the reciprocal of that gives me the focal length, and it comes out to be 46.3304 centimeters. Now the radius of curvature is twice the focal length, and so knowing the focal length, I can double it and get the radius of curvature, and that comes to 92.6609, and I can round that to three significant digits, 92.7 centimeters.