Mathematical Ideas (13th Edition) - Standalone book
13th Edition
ISBN: 9780321977076
Author: Charles D. Miller, Vern E. Heeren, John Hornsby, Christopher Heeren
Publisher: PEARSON
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Chapter 12.2, Problem 54E
To determine
To calculate: The change in median if seven consecutive even whole numbers add up to 168.
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7. [10 marks]
Let G
=
(V,E) be a 3-connected graph. We prove that for every x, y, z Є V, there is a
cycle in G on which x, y, and z all lie.
(a) First prove that there are two internally disjoint xy-paths Po and P₁.
(b) If z is on either Po or P₁, then combining Po and P₁ produces a cycle on which
x, y, and z all lie. So assume that z is not on Po and not on P₁. Now prove that
there are three paths Qo, Q1, and Q2 such that:
⚫each Qi starts at z;
• each Qi ends at a vertex w; that is on Po or on P₁, where wo, w₁, and w₂ are
distinct;
the paths Qo, Q1, Q2 are disjoint from each other (except at the start vertex
2) and are disjoint from the paths Po and P₁ (except at the end vertices wo,
W1, and w₂).
(c) Use paths Po, P₁, Qo, Q1, and Q2 to prove that there is a cycle on which x, y, and
z all lie. (To do this, notice that two of the w; must be on the same Pj.)
6. [10 marks]
Let T be a tree with n ≥ 2 vertices and leaves. Let BL(T) denote the block graph of
T.
(a) How many vertices does BL(T) have?
(b) How many edges does BL(T) have?
Prove that your answers are correct.
4. [10 marks]
Find both a matching of maximum size and a vertex cover of minimum size in
the following bipartite graph. Prove that your answer is correct.
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Chapter 12 Solutions
Mathematical Ideas (13th Edition) - Standalone book
Ch. 12.1 - In Exercises I and 2, use the given data to do the...Ch. 12.1 - In Exercises 1 and 2, use the given data to do the...Ch. 12.1 - In Exercises 3-6, use the given data to do the...Ch. 12.1 - In Exercises 3-6, use the given data to do the...Ch. 12.1 - In Exercises 3-6, use the given data to do the...Ch. 12.1 - In Exercises 3-6, use the given data to do the...Ch. 12.1 - In each of Exercises 7-10, construct a...Ch. 12.1 - In each of Exercises 7-10, construct a...Ch. 12.1 - In each of Exercises 7-10, construct a...Ch. 12.1 - In each of Exercises 7-10, construct a...
Ch. 12.1 - Federal Revenue and Spending The graph shows U.S....Ch. 12.1 - Federal Revenue and Spending The graph shows U.S....Ch. 12.1 - Federal Revenue and Spending The graph shows U.S....Ch. 12.1 - Federal Revenue and Spending The graph shows U.S....Ch. 12.1 - Prob. 15ECh. 12.1 - Refer to the above graphs for Exercises 16-20....Ch. 12.1 - Prob. 17ECh. 12.1 - Refer to the above graphs for Exercises 16-20....Ch. 12.1 - Prob. 19ECh. 12.1 - Refer to the above graphs for Exercises 16-20....Ch. 12.1 - Reading a Circle Graph of Job-Training Sources The...Ch. 12.1 - Reading a Circle Graph of Job-Training Sources The...Ch. 12.1 - Prob. 23ECh. 12.1 - Correspondence between Education and Earnings Data...Ch. 12.1 - Prob. 25ECh. 12.1 - Net Worth of Retirement Savings Saura, wishing to...Ch. 12.1 - Prob. 27ECh. 12.1 - Net Worth of Retirement Savings Saura, wishing to...Ch. 12.1 - Prob. 29ECh. 12.1 - Sample Masses in a Geology Laboratory...Ch. 12.1 - Sample Masses in a Geology Laboratory...Ch. 12.1 - Record Temperatures According to the National...Ch. 12.1 - Prob. 33ECh. 12.1 - Frequencies and Probabilities of letter Occurrence...Ch. 12.1 - Prob. 35ECh. 12.1 - Prob. 36ECh. 12.1 - 37. Is the probability distribution in Exercise 36...Ch. 12.1 - Prob. 38ECh. 12.1 - Probabilities of Study Times Recall that the...Ch. 12.1 - Favorite Sports among Recreation Students The 40...Ch. 12.1 - Favorite Sports among Recreation Students The 40...Ch. 12.1 - Favorite Sports among Recreation Students The 40...Ch. 12.2 - For each list of data, calculate (a) the mean (to...Ch. 12.2 - For each list of data, calculate (a) the mean (to...Ch. 12.2 - For each list of data, calculate (a), the mean (to...Ch. 12.2 - For each list of data, calculate (a) the mean (to...Ch. 12.2 - For each list of data, calculate (a) the mean (to...Ch. 12.2 - For each list of data, calculate (a) the mean (to...Ch. 12.2 - For each list of data, calculate (a) the mean (to...Ch. 12.2 - For each list of data, calculate (a) the mean (to...Ch. 12.2 - Answer each question. Gymnasts' Scores An Olympic...Ch. 12.2 - Quiz Grades The average of 5 quiz grades is 10....Ch. 12.2 - For each category in Exercises 11-16, find (a) (he...Ch. 12.2 - For each category in Exercises 11-16, find (a) the...Ch. 12.2 - For each category in Exercises 11-16, find (a) the...Ch. 12.2 - For each category in Exercises 11-16, find (a) the...Ch. 12.2 - For each category in Exercises 11-16, find (a) the...Ch. 12.2 - For each category in Exercises 11-16, find (a) the...Ch. 12.2 - Leading U.S. Businesses The table shows the top...Ch. 12.2 - Leading U.S. Businesses The table shows the top...Ch. 12.2 - Measuring Elapsed Times While doing an experiment,...Ch. 12.2 - Measuring Elapsed Times White doing an experiment,...Ch. 12.2 - The student from Exercises 19 and 20,when...Ch. 12.2 - The student from Exercises 19 and 20,when...Ch. 12.2 - The student from Exercises 19 and 20,when...Ch. 12.2 - The student from Exercises 19 and 20, when...Ch. 12.2 - Scores on Management Examinations Thao earned the...Ch. 12.2 - Scores on Management Examinations Thao earned the...Ch. 12.2 - Scores on Management Examinations Thao earned the...Ch. 12.2 - For each of the following frequency distributions,...Ch. 12.2 - Prob. 29ECh. 12.2 - 30. Average Employee Salary A company has
15...Ch. 12.2 - Grade-Point Averages Find the grade-point average...Ch. 12.2 - Grade-Point Averages Find the grade-point average...Ch. 12.2 - Prob. 33ECh. 12.2 - Federal Budget Totals The table gives federal...Ch. 12.2 - Prob. 35ECh. 12.2 - World Cell Phone Use In 2012, just the top six...Ch. 12.2 - Prob. 37ECh. 12.2 - Crew, Passengers, and Hijackers on 3/11 Airliners...Ch. 12.2 - Prob. 39ECh. 12.2 - Olympic Medal Standings The top ten medal-winning...Ch. 12.2 - Prob. 41ECh. 12.2 - Olympic Medal Standings The top ten medal-winning...Ch. 12.2 - Prob. 43ECh. 12.2 - In Exercises 44 and 45, use the given...Ch. 12.2 - Prob. 45ECh. 12.2 - What does your box plot of Exercise 18 for...Ch. 12.2 - Prob. 47ECh. 12.2 - Prob. 48ECh. 12.2 - Prob. 49ECh. 12.2 - Prob. 50ECh. 12.2 - Prob. 51ECh. 12.2 - Prob. 52ECh. 12.2 - Prob. 53ECh. 12.2 - Prob. 54ECh. 12.2 - Prob. 55ECh. 12.2 - Prob. 56ECh. 12.2 - Prob. 57ECh. 12.2 - Prob. 58ECh. 12.2 - For Exercises 57-59, refer to the grouped...Ch. 12.2 - Prob. 60ECh. 12.3 - 1. If your calculator finds both the sample...Ch. 12.3 - 2. If your calculator finds only one kind of...Ch. 12.3 - Find (a) the range, and (b) the standard deviation...Ch. 12.3 - Find (a) the range, and (b) the standard deviation...Ch. 12.3 - Find (a) the range, and (b) the standard deviation...Ch. 12.3 - Find (a) the range, and (b) the standard deviation...Ch. 12.3 - Find (a) the range, and (b) the standard deviation...Ch. 12.3 - Find (a) the range, and (b) the standard deviation...Ch. 12.3 - Find (a) the range, and (b) the standard deviation...Ch. 12.3 - Find (a) the range, and (b) the standard deviation...Ch. 12.3 - Use Chebyshev’s theorem for Exercises 11-22. Give...Ch. 12.3 - Use Chebyshev's theorem for Exercises 11-22. Give...Ch. 12.3 - Use Chebyshev's theorem for Exercises 11-22. Give...Ch. 12.3 - Use Chebyshev's theorem for Exercises 11-22. Give...Ch. 12.3 - Prob. 15ECh. 12.3 - Use Chebyshev's theorem for Exercises 11-22. Give...Ch. 12.3 - Prob. 17ECh. 12.3 - Use Chebyshev's theorem for Exercises 11-22. Give...Ch. 12.3 - Use Chebyshev's theorem for Exercises 11-22. Give...Ch. 12.3 - Use Chebyshev's theorem for Exercises 11-22. Give...Ch. 12.3 - Prob. 21ECh. 12.3 - Use Chebyshev's theorem for Exercises 11-22. Give...Ch. 12.3 - Prob. 23ECh. 12.3 - Travel Accommodation CostsGabriel and Lucia took a...Ch. 12.3 - Travel Accommodation Costs Gabriel and Lucia took...Ch. 12.3 - Travel Accommodation Costs Gabriel and Lucia took...Ch. 12.3 - Travel Accommodation Costs Gabriel and Lucia took...Ch. 12.3 - Travel Accommodation Costs Gabriel and Lucia took...Ch. 12.3 - In Exercises 29 and 30, two samples are given. In...Ch. 12.3 - In Exercises 29 and 30, two samples are given. In...Ch. 12.3 - Utilize the following sample for Exercises...Ch. 12.3 - Utilize the following sample for Exercises 31-36....Ch. 12.3 - Prob. 33ECh. 12.3 - Utilize the following sample for Exercises...Ch. 12.3 - Prob. 35ECh. 12.3 - Utilize the following sample for Exercises...Ch. 12.3 - 37. Comparing Water Heater Lifetimes Two brands of...Ch. 12.3 - Lifetimes of Engine Control Modules Chin manages...Ch. 12.3 - Lifetimes of Engine Control Modules Chin manages...Ch. 12.3 - Prob. 40ECh. 12.3 - 41. In Section 12.2 we showed that the mean, as a...Ch. 12.3 - Prob. 42ECh. 12.3 - A Cereal-Marketing Survey A food distribution...Ch. 12.3 - A Cereal-Marketing Survey A food distribution...Ch. 12.3 - A Cereal-Marketing Survey A food distribution...Ch. 12.3 - Prob. 46ECh. 12.3 - Prob. 47ECh. 12.3 - Prob. 48ECh. 12.3 - Prob. 49ECh. 12.3 - Prob. 50ECh. 12.3 - Prob. 51ECh. 12.3 - Prob. 52ECh. 12.4 - Numbers of Restaurant Customers Refer to the...Ch. 12.4 - Numbers of Restaurant Customers Refer to the...Ch. 12.4 - Numbers of Restaurant Customers Refer to the...Ch. 12.4 - Numbers of Restaurant Customers Refer to the...Ch. 12.4 - In Exercises 5-8, make use of z-scores.
5....Ch. 12.4 - In Exercises 5-8, make use of z-scores.
6....Ch. 12.4 - In Exercises 5-8, make use of z-scores.
7....Ch. 12.4 - In Exercises 5-8, make use of z-scores. Relative...Ch. 12.4 - World's Largest Energy Producers and Consumers The...Ch. 12.4 - World's Largest Energy Producers and Consumers The...Ch. 12.4 - World's Largest Energy Producers and Consumers The...Ch. 12.4 - World's Largest Energy Producers and Consumers The...Ch. 12.4 - World's Largest Energy Producers and Consumers The...Ch. 12.4 - World's Largest Energy Producers and ConsumersThe...Ch. 12.4 - World's Largest Energy Producers and Consumers The...Ch. 12.4 - World's Largest Energy Producers and Consumers The...Ch. 12.4 - Determine which was relatively higher: Canada in...Ch. 12.4 - Construct box plots for both production and...Ch. 12.4 - Prob. 19ECh. 12.4 - 20. Comparing your two box plots of Exercise 18,...Ch. 12.4 - The text stated that for any distribution of data,...Ch. 12.4 - 22. Describe the basic difference between a...Ch. 12.4 - Prob. 23ECh. 12.4 - This chapter has introduced three major...Ch. 12.4 - Prob. 25ECh. 12.4 - Prob. 26ECh. 12.4 - Prob. 27ECh. 12.4 - Prob. 28ECh. 12.4 - Prob. 29ECh. 12.4 - Prob. 30ECh. 12.4 - Prob. 31ECh. 12.4 - Prob. 32ECh. 12.4 - In Exercises 32-35, answer yes or no and explain...Ch. 12.4 - In Exercises 32-35, answer yes or no and explain...Ch. 12.4 - In Exercises 32-35, answer yes or no and explain...Ch. 12.4 - Prob. 36ECh. 12.4 - Prob. 37ECh. 12.4 - Prob. 38ECh. 12.4 - Prob. 39ECh. 12.4 - Prob. 40ECh. 12.4 - Prob. 41ECh. 12.4 - Prob. 42ECh. 12.4 - Prob. 43ECh. 12.4 - Prob. 44ECh. 12.4 - Prob. 45ECh. 12.4 - Prob. 46ECh. 12.4 - Prob. 47ECh. 12.4 - Prob. 48ECh. 12.4 - Prob. 49ECh. 12.4 - Prob. 50ECh. 12.4 - Prob. 51ECh. 12.4 - Prob. 52ECh. 12.4 - Prob. 53ECh. 12.4 - Prob. 54ECh. 12.5 - Identify each variable quantity as discrete or...Ch. 12.5 - Identify each variable quantity as discrete or...Ch. 12.5 - Identify each variable quantity as discrete or...Ch. 12.5 - Prob. 4ECh. 12.5 - Identify each variable quantity as discrete or...Ch. 12.5 - Prob. 6ECh. 12.5 - Measuring the Mass of Ore Samples Suppose 100...Ch. 12.5 - Measuring the Mass of Ore Samples Suppose 100...Ch. 12.5 - Measuring the Mass of Ore Samples Suppose 100...Ch. 12.5 - Measuring the Mass of Ore Samples Suppose 100...Ch. 12.5 - Distribution of IQ Scores On standard IQ tests,...Ch. 12.5 - Distribution of IQ Scores On standard IQ tests,...Ch. 12.5 - Distribution of IQ Scores On standard IQ tests,...Ch. 12.5 - Distribution of IQ Scores On standard IQ tests,...Ch. 12.5 - Find the percentage of area under a normal curve...Ch. 12.5 - Find the percentage of area under a normal curve...Ch. 12.5 - Find the percentage of area under a normal curve...Ch. 12.5 - Find the percentage of area under a normal curve...Ch. 12.5 - Find the percentage of the total area under a...Ch. 12.5 - Find the percentage of the total area under a...Ch. 12.5 - Find the percentage of the total area under a...Ch. 12.5 - Find the percentage of the total area under a...Ch. 12.5 - Find a value of z such that each condition is met....Ch. 12.5 - Find a value of z such that each condition is met....Ch. 12.5 - Find a value of z such that each condition
25....Ch. 12.5 - Find a value of z such that each condition is...Ch. 12.5 - Weights of Peaches A fruit-packing company...Ch. 12.5 - Weights of Peaches A fruit-packing company...Ch. 12.5 - Weights of Peaches A fruit-packing company...Ch. 12.5 - Weights of Peaches A fruit-packing company...Ch. 12.5 - Weights of Peaches A fruit-packing company...Ch. 12.5 - Prob. 32ECh. 12.5 - Prob. 33ECh. 12.5 - Prob. 34ECh. 12.5 - IQs of Employees A large company employs workers...Ch. 12.5 - Prob. 36ECh. 12.5 - Net Weight of Cereal Boxes A certain dry cereal is...Ch. 12.5 - Prob. 38ECh. 12.5 - Prob. 39ECh. 12.5 - Net Weight of Cereal Boxes A certain dry cereal is...Ch. 12.5 - Prob. 41ECh. 12.5 - Prob. 42ECh. 12.5 - Prob. 43ECh. 12.5 - Prob. 44ECh. 12.5 - Assume the following distributions are all normal,...Ch. 12.5 - Prob. 46ECh. 12.5 - Prob. 47ECh. 12.5 - Prob. 48ECh. 12.5 - Prob. 49ECh. 12.5 - Prob. 50ECh. 12.5 - Prob. 51ECh. 12.5 - Prob. 52ECh. 12.5 - Prob. 53ECh. 12.5 - Prob. 54ECh. 12.5 - Prob. 55ECh. 12.5 - Prob. 56ECh. 12.5 - Prob. 57ECh. 12.5 - Prob. 58ECh. 12.5 - Prob. 59ECh. 12.5 - Prob. 60ECh. 12.5 - What percentage of the items lie within 1.25...Ch. 12.5 - Prob. 62ECh. 12 - Visits to Social Networking Sites The table shows...Ch. 12 - Cheaters Never Learn The table here shows the...Ch. 12 - Use this information for Exercises 3 and 4. For...Ch. 12 - Prob. 4TCh. 12 - College Endowment AssetsThe table shows the top...Ch. 12 - College Endowment Assets The table shows the top...Ch. 12 - College Endowment Assets The table shows the top...Ch. 12 - Prob. 8TCh. 12 - Pediatrics Patients Khalida worked the 22 weekdays...Ch. 12 - Prob. 10TCh. 12 - In Exercises 11-14, find the indicated measures...Ch. 12 - In Exercises 11-14, find the indicated measures...Ch. 12 - In Exercises 11-14, find the indicated measures...Ch. 12 - Prob. 14TCh. 12 - Exam Scores in a Criminal Justice Class The...Ch. 12 - Prob. 16TCh. 12 - Use the stem-and-leaf display below for Exercises...Ch. 12 - Prob. 18TCh. 12 - Use the stem-and-leaf display below for Exercises...Ch. 12 - Use the stem-and-leaf display below for...Ch. 12 - Use the stem-and-leaf display below for...Ch. 12 - Prob. 22TCh. 12 - Triple Jump ChampionsAll Summer Olympic...Ch. 12 - Prob. 24TCh. 12 - Season Statistics in Major League Baseball The...Ch. 12 - Season Statistics in Major League Baseball The...Ch. 12 - Season Statistics in Major League Baseball The...Ch. 12 - Prob. 28TCh. 12 - Remission Data for Cocaine and Marijuana Addiction...Ch. 12 - Remission Data for Cocaine and Marijuana Addiction...
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