86 CHAPTER 2 Sets 12. Appetizers Survey Da Tulio’s Restaurant hired Dennis to determine what kind of appetizers customers liked. He surveyed 100 people, with the following results: 78 liked shrimp cocktail, 56 liked mozzarella sticks, and 35 liked both shrimp cocktail and mozzarella sticks. Every person interviewed liked one or the other or both kinds of appetizers. Does this result seem correct? Explain your answer. No; only 99 of 100 interviewed are accounted for. 13. Discovering an Error A New York State Thruway agent sampled cars going from New York into Pennsylvania. The report indicated that of the 85 cars sampled, 35 cars were red. 53 cars were sport-utility vehicles. 43 cars had two or more passengers. 27 cars were red sport-utility vehicles. 25 cars were red and had two or more passengers. 20 cars were sport-utility vehicles with two or more passengers. 15 cars were red sport-utility vehicles and had two or more passengers. After reading the agent's report, the supervisor explains that a mistake has been made. Explain how the supervisor knew that the agent’s report contained an error. In a Venn diagram, regions II, IV, and V contain a total of 37 red cars. This total is greater than the 35 red cars, as given in the exercise. Challenge Problems/Group Activities 14. Parks A survey of 300 parks showed the following. 15 had only camping. 20 had only hiking trails. 35 had only picnicking. 185 had camping. 140 had camping and hiking trails. 125 had camping and picnicking. 210 had hiking trails. Determine the number of parks that a) had at least one of these features. 290 b) had all three features. 95 c) did not have any of these features. 10 d) had exactly two of these features. 125 15. Surveying Farmers Data from a survey of 500 farmers in a midwestern state showed the following. 125 grew only wheat. 110 grew only corn. 90 grew only oats. 200 grew wheat. 60 grew wheat and corn. 50 grew wheat and oats. 180 grew corn. Determine the number of farmers who a) grew at least one of the three. 410 b) grew all three. 35 c) did not grow any of the three. 90 d) grew exactly two of the three. 50 16. Family Reunion When the Montesano family members discussed where their annual reunion should take place, they determined that of all the family members, 8 would not go to a park. 7 would not go to a beach. 11 would not go to the family cottage. 3 would go to neither a park nor a beach. 4 would go to neither a beach nor the family cottage. 6 would go to neither a park nor the family cottage. 2 would not go to a park or a beach or to the family cottage. 1 would go to all three places. What is the total number of family members? 16 Recreational Mathematics 17. Number of Elements A universal set U consists of 12 elements. If sets A B, , and C are proper subsets of U and = = = = nU nAB nAC nBC ( ) 12, ( ) ( ) ( ) 6, > > > = n A B C ( ) 4, > > and = n A B C ( ) 10, < < determine a) n A B ( ) < 10 b) ′ n A C ( ) < 10 c) ′ n A B ( ) > 6 Shutterstock
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