Reflection and Mirrors Legacy Problem #15 Guided Solution

A physics student is able to blend good understanding of concepts in mathematics with effective problem-solving habits and good thinking in order to solve difficult problems like this one. The effective habits involve reading the problem carefully, identifying what's known and what the unknowns are, then plotting out a strategy as to how to get to unknown values. Now in this question what we have is a mirror sculpture which has on one side a convex mirror and on the other side a concave mirror. The convex side is said to approach, it's said to face Fifth Avenue in New York City and a taxi is located 38 meters from the convex side of the sculpture and its image is one-fifth the size of the taxi itself. So I can take that statement that the image is one-fifth the size of the taxi and I can write the following. The height of the image HI is or equal one divided by five times the size of the taxi or times HO. That is again high equal one-fifth whole. Now if high equal one-fifth whole then the high per whole ratio is one-fifth and it's a positive one-fifth because convex mirrors produce upright images and upright images always have positive image heights. So I now know that the magnification is one divided by five and the magnification is equal to the negative of the die-dough ratio where the dough here is 38 meters. So I can say positive one-fifth equal negative die over 38. I can cross multiply and get 38 equal negative five die and I can divide through by negative five and I can get the value of die that is the image distance for the taxi. It ends up being negative 7.6 meters. Now the negative indicates that the taxi's image is located on the opposite side of the mirror. That is it's a virtual image. So I have die equal negative 7.6 meters and now what I wish to calculate is the focal line. So I go one over F equal one over 38 minus one over negative 7.6. When I solve for F or when I solve for one over F I get negative .10526 and taking the reciprocal of that number gives me the value of F. Comes out to be negative 9.5 meters and that's the answer to part A. Take a deep breath, you've got more to go. On part B you're given that the Rockefeller Center is located on the concave side of the Smear Sculpture. It's height HO equal 260 meters and it's distance DO equal 95 meters from the concave mirror. I'm to assume that the focal length of the concave side is the same as the magnitude of the focal length of the convex side and as such the F of the mirror is equal to positive 9.5 meters. So here are three quantities that are known, HO equal 260, DO equal 95, and F equal 9.5 meters. What I'm asked to calculate is the die value and the high value. To get the die value I need to use the mirror equation. One over die equal one over F minus one over DO. Substituting in 9.5 for F and 95 for DO I end up getting that one over die equal .094737 and that's the reciprocal of the answer. If I take the reciprocal of that number I get the answer. It comes out to be 10.555 meters. I can round that to two significant digits such as 11 meters. Now that I've got the die value I can calculate the high value using the relationship that the high per HO ratio is equal to the negative of the die per DO ratio. Now I can rearrange that to say high equal the negative of HO times die divided by DO. Substituting in values of die of 10.555 meters and of HO which is 260 meters and of DO which is 95 meters I can solve for the high value. It comes out to be negative 28.888 meters which I can round to negative 29 meters. The negative indicating that the image is an inverted image and that's the answer to this part B.