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for All Practical Purposes: Mathematical Literacy in Today's World Tenth Edition Test Bank

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  1. Page 1
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    Chapter 1: Urban Services Free - Response Page 30 Format: Short Answer 1. On a graph that represents six cities and the roads between them, the valence of vertex A is 4. What does this mean in real - world terms? Ans: If the valence of vertex A is four, it means there are four roads that lead to town A. Format: Short Answer 2. A graph that represents six cities and the roads among them is connected. What does this mean in real - world terms? Ans: Any pair of two cities has at least one connecting path of roads . Format: Short Answer 3. Describe the two conditions for a graph to have an Euler circuit. Ans: The graph must be connected and ha ve all valences even. Format: Short Answer 4 . Draw a graph representing four cities, A, B, C, and D, with a road that connects each pair of cities given: AB, AC, BC, BD, CD Ans: Format: Short Answer 5 . Draw a graph with vertices A, B, C, and D in which the valence of vertices A and D is 3 and the valence of vertices B and C is 2. Ans: Answer s may vary. One solution is:

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    Chapter 1 Free - Response Page 31 Format: Short Answer 6. Draw a graph with vertices A, B, and C in which A has valence 4 and B and C have valence 5. Ans: Format: Short Answer 7. Draw a graph that has four vertices, with valences 1, 2, 3, and 4. Ans: Format: Short Answer 8. Consider the path represented by the sequence of numbered edges on the graph below. Explain why the path is NOT an Euler circuit. Ans: The path is not an Euler circuit because it does not start and stop at the same vertex.

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    Chapter 1 Free - Response Page 32 Format: Short Answer 9. Consider the path represented by the sequence of numbered edges on the graph below. Explain why the path is NOT an Euler circuit. Ans: The path is not an Euler circuit because it does not cover every edge of the graph. Format: Short Answer 10. Draw a graph with eight vertices, with the valence of each vertex being even, that does NOT have an Euler circuit. Ans: Answers may vary. One solution is: Format: Short Answer 11. Explain why the graph shown below does NOT have an Euler circuit. Ans: The graph does not have an Euler circuit because the valences of vertex B and vertex D are odd.

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    Chapter 1 Free - Response Page 33 Format: Short Answer 12. Explain why the graph shown below does NOT have an Euler circuit. Ans: The graph does not have an Euler circuit because it is not connected. Use the following to answer Questions 13 – 16: Identify an Euler circuit on the following graphs by numbering the sequence of edges in the order traveled. Reference: 1 - 1 Format: Short Answer 13. Ans: Answers may vary. One solution is:

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    Chapter 1 Free - Response Page 34 Reference: 1 - 1 Format: Short Answer 14. Ans: Answers may vary. One solution is: Reference: 1 - 1 Format: Short Answer 15. Ans: Answers may vary. One solution is:

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    Chapter 1 Free - Response Page 35 Reference: 1 - 1 Format: Short Answer 16. Ans: Answers may vary. One solution is: Use the following to answer Questions 17 – 22: Add wig gly edges to find an efficient e ulerization of the following graphs. Reference: 1 - 2 Format: Short Answer 17. Ans: Answers may vary. One solution is:

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    Chapter 1 Free - Response Page 36 Reference: 1 - 2 Format: Short Answer 18. Ans: Answers may vary. One solution is: Reference: 1 - 2 Format: Short Answer 19. Ans: Answers may vary. One solution is:

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    Chapter 1 Free - Response Page 37 Reference: 1 - 2 Format: Short Answer 20. Ans: Answers may vary. One solution is: Reference: 1 - 2 Format: Short Answer 21 . Ans: Answers may vary. One solution is:

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    Chapter 1 Free - Response Page 38 Reference: 1 - 2 Format: Short Answer 22. Ans: Answers may vary. One solution is:

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    Chapter 1 Free - Response Page 39 Format: Short Answer 23. The map below gives the territory of a parking control officer. The dots represent meters that must be checked. Draw the graph that would be useful for finding an efficient route. Ans:

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    Chapter 1 Free - Response Page 40 Format: Short Answer 24. For the street network shown below, draw a graph that would be useful for routing a garbage truck. Assume all streets are two - way and that passing once down the street is sufficient for collecting trash from both sides. Ans:

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    Chapter 1 Free - Response Page 41 Format: Short Answer 25. Find an Euler circuit on the graph on the left and use it to find a circuit on the graph on the right that reuses one edge. Ans: Answers may vary. One solution is: Format: Short Answer 26. Find an Euler circuit on the graph on the left and use it to find a circuit on the graph on the right that reuses four edges. Ans: Answers may vary. One solution is:

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    Chapter 1 Free - Response Page 42 Format: Short Answer 27. Find an Euler circuit on the graph on the left and use it to find a circuit on the graph on the right that reuses three edges. Ans: Answers may vary. One solution is: Format: Short Answer 28. Find an Euler circuit on the graph on the left and use it to find a circuit on the graph on the right that reuses three edges. Ans: Answers may vary. One solution is:

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    Chapter 1 Free - Response Page 43 Format: Short Answer 29. Why would a city street department want its snowplow operator’s path to follow an Euler circuit if possible? Ans: If the snowplow operator followed an Euler circuit, he would end his route at the city garage where he started. Also, no time or gas would be wasted traveling down streets that were already plowed. Format: Short Answer 30. Give three real - world applications in which a worker would want to find an Euler circuit on a street network. Ans: Answers may vary. Some situations include: Plowing snow Painting lines down the center of streets Collecting trash Checking gutters on the corners of all streets Checking parking meters along the edge of all streets Delivering mail to all houses on all streets Format: Short Answer 31. Draw a connected graph with seven vertices where each vertex has valence 4. Ans: Answers will vary. One solution is:

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    Chapter 1 Free - Response Page 44 Format: Short Answer 32. Draw a graph with six vertices where the valences are 1, 2, 2, 3, 4, and 4. Ans: Answers will vary. One solution is: Format: Short Answer 33. Find an eulerization with nine added edges for a 3  6 – block rectangular street network. Ans: Format: Short Answer 34. Draw a graph where every vertex has a valence of at least 2 but removing a single edge disconnects the graph. Ans: Answers will vary. One solution is:

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