2.5 Set Applications and Survey Data Analysis 83 U 22 A N D I 29 16 IV II III VI VII 11 8 5 18 16 VIII V Figure 2.28 Example 3 Travel Packages Liberty Travel surveyed 125 potential customers. The following data were obtained. 68 wished to travel to Alaska (A). 47 wished to travel to Dollywood (D). 53 wished to travel to New Orleans (N). 23 wished to travel to Alaska and Dollywood. 26 wished to travel to Dollywood and New Orleans. 34 wished to travel to Alaska and New Orleans. 18 wished to travel to all three destinations. Use a Venn diagram to answer the following questions. How many of those surveyed a) did not wish to travel to any of these destinations? b) wished to travel only to Alaska? c) wished to travel to Dollywood and New Orleans, but not to Alaska? d) wished to travel to Dollywood or New Orleans, but not to Alaska? e) wished to travel to exactly one of these destinations? Solution The Venn diagram is constructed using the procedures we outlined in Example 2. The diagram is illustrated in Fig. 2.28. We suggest you construct the diagram by yourself now and check your diagram with Fig. 2.28. a) Twenty-two potential customers did not wish to travel to any of these destinations (see region VIII). b) Twenty-nine potential customers wished to travel only to Alaska (see region I). c) Those potential customers in region VI wished to travel to Dollywood and New Orleans, but not to Alaska. Therefore, eight customers satisfied the criteria. d) The word or in this type of problem means one or the other or both. All the potential customers in regions II, III, IV, V, VI, and VII wished to travel to Dollywood or New Orleans. Those in regions II, IV, and V also wished to travel to Alaska. The potential customers who wished to travel to Dollywood or New Orleans, but not to Alaska, are determined by adding the numbers in regions III, VI, and VII. There are + + = 16 8 11 35 potential customers who satisfy the criteria. e) Those potential customers in regions I, III, and VII wished to travel to exactly one of the destinations. Therefore, + + = 29 16 11 56 customers wished to travel to exactly one of these destinations. 7 Instructor Resources for Section 2.5 in MyLab Math • Objective-Level Videos 2.5 • Animation: Application of Venn Diagrams • PowerPoint Lecture Slides 2.5 • MyLab Exercises and Assignments 2.5 Exercises Problem Solving In Exercises 1–15, draw a Venn diagram to obtain the answers. * 1. Study Locations At Lansing Community College, a survey was taken to determine where students studied on campus. Of the 250 students surveyed, it was determined that 177 studied in the library. 138 studied in the cafeteria. 80 studied in both the library and cafeteria. Of those students surveyed, a) how many studied only in the library? 97 b) how many studied only in the cafeteria? 58 c) how many did not study in either the library or cafeteria? 15 SECTION 2.5 *See Instructor Answer Appendix Now try Exercise 7 m Dollywood Chad Robertson Media/ Shutterstock SeventyFour/Shutterstock
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