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Math 105 Workbook Exploring Mathematics

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Math 105 Workbook Exploring Mathematics Math 105 Workbook Exploring Mathematics Douglas R. Anderson, Professor Fall 2018: MWF 11:50-1:00, ISC 101 Acknowledgment First we would like to thank all of our former Math 105 students. Their successes, struggles, and suggestions have shaped how we teach this course in many important ways. We also want to thank our departmental colleagues and several Concordia math- ematics majors for many fruitful discussions and resources on the content of this course and the makeup of this workbook. Some of the topics, examples, and exercises in this workbook are drawn from other works. Most significantly, we thank Samantha Briggs, Ellen Kramer, and Dr. Jessie Lenarz for their work in Exploring Mathematics, as well as other Cobber mathemat- ics professors. We have also used: • Taxicab Geometry: An Adventure in Non-Euclidean Geometry by Eugene F. Krause, • Excursions in Modern Mathematics, Sixth Edition, by Peter Tannenbaum. • Introductory Graph Theory by Gary Chartrand, • The Heart of Mathematics: An invitation to effective thinking by Edward B. Burger and Michael Starbird, • Applied Finite Mathematics by Edmond C. Tomastik. Finally, we want to thank (in advance) you, our current students. Your suggestions for this course and this workbook are always encouraged, either in person or over e-mail. Both the course and workbook are works in progress that will continue to improve each semester with your help. Let's have a great semester this fall exploring mathematics together and fulfilling Concordia's math requirement in 2018. Skol Cobbs! i ii Contents 1 Taxicab Geometry 3 1.1 Taxicab Distance . .3 Homework . .8 1.2 Taxicab Circles . .9 Homework . 13 1.3 Taxicab Applications . 15 Homework . 21 1.4 Taxicab Minimizing Regions . 23 Homework . 25 1.5 Taxicab Midsets . 26 Homework . 31 1.6 Taxicab Lines . 32 Homework . 38 1.7 Chapter Projects . 41 1.8 Chapter Review . 42 2 Counting and Probability 55 2.1 Introduction to Counting . 55 Homework . 62 2.2 Combinations . 64 Homework . 69 2.3 Introduction to Probability . 71 iii Homework . 77 2.4 Complements, Unions, and Intersections . 79 Homework . 84 2.5 Conditional Probability . 86 Homework . 90 2.6 Chapter Projects . 93 2.7 Chapter Review . 94 3 Graph Theory 103 3.1 Introduction to Graph Theory . 103 Homework . 107 3.2 Paths and Circuits . 109 Homework . 114 3.3 Subgraphs and Trees . 118 Homework . 122 3.4 Graph Colorings . 126 Homework . 131 3.5 Planar Graphs . 133 Homework . 136 3.6 Possible Project: Directed Graphs . 138 Homework . 143 3.7 Chapter Projects . 146 3.8 Chapter Review . 147 4 Consumer Mathematics 155 4.1 Percentages and Simple Interest . 155 Homework . 159 4.2 Compound Interest . 161 Homework . 166 4.3 Effective Annual Yield . 168 iv 1 Homework . 170 4.4 Ordinary Annuities . 173 Homework . 176 4.5 Mortgages . 177 Homework . 180 4.6 Chapter Projects . 182 4.7 Chapter Review . 183 5 Voting Theory 189 5.1 Voting Systems . 189 Homework . 199 5.2 Voting Paradoxes and Problems . 201 Homework . 208 5.3 Weighted Voting Systems . 211 Homework . 215 5.4 Banzhaf Power Index . 216 Homework . 221 5.5 Voting Theory Homework Set . 222 5.6 Possible Project: Antagonists . 224 5.7 Chapter Projects . 226 5.8 Chapter Review . 228 A Projects 235 B Syllabus 237 2 Chapter 1 Taxicab Geometry 1.1 Taxicab Distance 1. Suppose, in the city shown below, that we want to ride in a taxicab along city streets from the corner of 8th Street and 10th Avenue to the corner of 3rd Street and 13th Avenue. (a) How many blocks does it take to make such a trip? (b) Does every route in the city grid from the corner of 8th Street and 10th Avenue to the corner of 3rd Street and 13th Avenue take the same dis- tance? (c) Does every route in the city grid from the corner of 8th Street and 10th Avenue to the corner of 3rd Street and 13th Avenue that continues to make progress at every point take the same distance? 3 4 CHAPTER 1. TAXICAB GEOMETRY Definition: The taxicab distance between two points is the shortest possible distance between the two points where we are only allowed to move horizontally or vertically. Examples: 1.1. TAXICAB DISTANCE 5 2. (a) Graph the points A = (1; 3), B = (1; −2), C = (−3; −1), and D = (0; 3). (b) Now find the following distances in both Euclidean and taxicab geome- tries. Give a decimal approximation to 2 decimal places. Euclidean distance Taxicab distance from A to B from B to C from C to D (c) If you know the Euclidean distance between two points, does that tell you what the taxicab distance is? Why or why not? (d) If you know the taxicab distance between two points, does that tell you what the Euclidean distance is? Why or why not? 6 CHAPTER 1. TAXICAB GEOMETRY 3. (a) Consider the points in the following graph: C D B A E Calculate the following distances in both Euclidean and taxicab geome- tries. Give a decimal approximation to 2 decimal places. Euclidean distance Taxicab distance from A to B from A to C from A to D from A to E (b) Is the Euclidean distance between two points always less than or equal to the taxicab distance? If so, explain why. If not, give an example where the Euclidean distance is greater than the taxicab distance. 1.1. TAXICAB DISTANCE 7 4. One night the 911 dispatcher for Taxicab City receives a report of an accident at X = (−1; 4). There are two police cars in the area, car C at (2; 1) and car D at (−1; −1). Which car should be sent to the scene of the accident to arrive most quickly? (Since the cars must drive on the streets, we use taxicab geometry to measure distances.) 1 3 8 1 5. Find the taxicab distance between A = ( 3 ; 2 ) and B = ( 3 ; − 2 ). 8 CHAPTER 1. TAXICAB GEOMETRY Homework Throughout this chapter, all taxicab pictures should be completed on graph paper. 1. (a) On a single large graph, plot the following points: A = (5; 4) B = (1; 2) C = (4; −3) D = (−1; 5) E = (−5; −4) F = (1; −2) (b) Find the Euclidean distance between A and B. (c) Find the taxicab distance between A and B. (d) Find the Euclidean distance between B and F . (e) Find the taxicab distance between B and F . (f) Find the Euclidean distance between F and C. (g) Find the taxicab distance between F and C. 2. Let C = (1; 0). (a) Find five different points that are a taxicab distance of 5 from C, but are not a Euclidean distance of 5 from C. (b) Graph all of the points that are a taxicab distance of 5 from C, including those that are a Euclidean distance of 5 from C. Be sure to include all possible points, not just the ones with integer coordinates. (c) Come up with a mathematically appropriate name for the answer to part 2b. 3. Let A = (−1; 1) and B = (3; 3). (a) Find a point C so that the taxicab distance between A and C is the same as the taxicab distance between B and C. (b) Find a different point D so that the taxicab distance between A and D equals the taxicab distance between B and D. (c) Graph all of the points P where the taxicab distance between A and P and the taxicab distance between B and P is the same. Be sure to include all possible points, not just the ones with integer coordinates. 1.2. TAXICAB CIRCLES 9 1.2 Taxicab Circles Definition: The taxicab circle centered at a point C with a radius of r (where r is a number, r ≥ 0) is all of the points that are a taxicab distance of r from C. 1. Draw the taxicab circle of radius 5 around the point P = (3; 4). 10 CHAPTER 1. TAXICAB GEOMETRY 2. Draw the taxicab circle of radius 6 around the point Q = (2; −1). 3. (a) On a single graph, draw taxicab circles around the point R = (1; 2) of radii 1, 2, 3, and 4. (b) What are the taxicab perimeters (circumferences) of the circles? Do you see a pattern? (c) How many grid squares are inside of each of the circles? Do you see a pattern? 1.2. TAXICAB CIRCLES 11 4. Describe a quick technique for drawing a taxicab circle of radius r around a point P . 5. Tyrion Lannister has fled King's Landing and now works in Taxicab City for the 3M plant, located at M = (1; 2). He goes out to eat for lunch once a week, and out of company loyalty, he likes to walk exactly 3 blocks from the plant to do so. Where in the city are restaurants at which Tyrion can eat? Draw their locations on the graph. M 12 CHAPTER 1. TAXICAB GEOMETRY 6. A developing company wants to construct an apartment building in Taxicab City within six blocks of the mall at M = (−2; 1) and within four blocks of the tennis courts at T = (3; 3). Shade in the area of the graph that suits the builder's requirements. 1.2. TAXICAB CIRCLES 13 Homework 1. Graph all of the points that are a taxicab distance of exactly 4 from the point A = (−2; −1). 2. Put your answers to these questions on separate graphs. (a) Graph the taxicab circle that is centered at (1; 4) with a radius of 3. 5 (b) Graph the taxicab circle that is centered at (−1; −3) with a radius of 2 .
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