Mathematics All Around (6th Edition)
6th Edition
ISBN: 9780134434681
Author: Tom Pirnot
Publisher: PEARSON
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Chapter 14.2, Problem 31E
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Assume that a company is considering purchasing a machine for $50,000 that will have a five-year useful life and a $5,000 salvage value. The
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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.)
Chapter 14 Solutions
Mathematics All Around (6th Edition)
Ch. 14.1 - Sharpening Your Skills. In Exercises 1 and 2,...Ch. 14.1 - Sharpening Your Skills. In Exercises 1 and 2,...Ch. 14.1 - Sharpening Your Skills. In Exercises 3 and 4,...Ch. 14.1 - Sharpening Your Skills. In Exercises 3 and 4,...Ch. 14.1 - Sharpening Your Skills. In Exercises 5 and 6,...Ch. 14.1 - Sharpening Your Skills. In Exercises 5 and 6,...Ch. 14.1 - Prob. 7ECh. 14.1 - Sharpening Your Skills. In Exercises 7 and 8,...Ch. 14.1 - Prob. 9ECh. 14.1 - Sharpening Your Skills. In Exercises 9 and 10,...
Ch. 14.1 - Prob. 11ECh. 14.1 - Prob. 12ECh. 14.1 - Applying What Youve Learned. Customers at...Ch. 14.1 - Applying What Youve Learned. Scheduling a rec...Ch. 14.1 - Applying What Youve Learned. Data similar to that...Ch. 14.1 - Applying What Youve Learned. Data similar to that...Ch. 14.1 - Comparing wage data. The following bar graphs...Ch. 14.1 - Applying What Youve Lerarned. Comparing wage data....Ch. 14.1 - Applying What Youve Learned. Comparing wage data....Ch. 14.1 - Applying What Youve Learned. Comparing wage data....Ch. 14.1 - Applying What Youve Learned. Comparing wage data....Ch. 14.1 - Prob. 22ECh. 14.1 - Prob. 23ECh. 14.1 - Prob. 24ECh. 14.1 - Prob. 25ECh. 14.1 - For Exercises 23 26, use the four graphs of the...Ch. 14.1 - Applying What Youve Learned. Comparing training...Ch. 14.1 - Applying What Youve Learned. Comparing weight-loss...Ch. 14.1 - Applying What Youve Learned. Super Bowl scores....Ch. 14.1 - Applying What Youve Learned. Law School Aptitude...Ch. 14.1 - In Exercises 31 34, first group the data using an...Ch. 14.1 - Applying What Youve Learned. In Exercises 31 34,...Ch. 14.1 - In Exercises 31 34, first group the data using an...Ch. 14.1 - Applying What Youve Learned. In Exercises 31 34,...Ch. 14.1 - Communicating Mathematics What is the difference...Ch. 14.1 - Communicating Mathematics If you are using a bar...Ch. 14.1 - Communicating Mathematics What is the difference...Ch. 14.1 - Communicating Mathematics What do you see as an...Ch. 14.1 - Challenge Yourself How might you present three...Ch. 14.1 - Challenge Yourself 42. The following table is an...Ch. 14.2 - Prob. 1ECh. 14.2 - Prob. 2ECh. 14.2 - Sharpening Your Skills Find the mean, median, and...Ch. 14.2 - Prob. 4ECh. 14.2 - Prob. 5ECh. 14.2 - Prob. 6ECh. 14.2 - Prob. 7ECh. 14.2 - Prob. 8ECh. 14.2 - Prob. 9ECh. 14.2 - Prob. 10ECh. 14.2 - Prob. 11ECh. 14.2 - Prob. 12ECh. 14.2 - Prob. 13ECh. 14.2 - Prob. 14ECh. 14.2 - Prob. 15ECh. 14.2 - Prob. 16ECh. 14.2 - In Exercises 1720, a give the five-number summary...Ch. 14.2 - Prob. 18ECh. 14.2 - In Exercises 1720, a give the five-number summary...Ch. 14.2 - In Exercises 1720, a give the five-number summary...Ch. 14.2 - In Exercises 2128, find the mean, median, and mode...Ch. 14.2 - In Exercises 2128, find the mean, median, and mode...Ch. 14.2 - In Exercises 2128, find the mean, median, and mode...Ch. 14.2 - In Exercises 2128, find the mean, median, and mode...Ch. 14.2 - In Exercises 2128, find the mean, median, and mode...Ch. 14.2 - In Exercises 2128, find the mean, median, and mode...Ch. 14.2 - In Exercises 2128, find the mean, median, and mode...Ch. 14.2 - In Exercises 2128, find the mean, median, and mode...Ch. 14.2 - Many colleges assign numerical points to grades as...Ch. 14.2 - Prob. 30ECh. 14.2 - Exam scores. Izzy had an 84 and an 86 on his first...Ch. 14.2 - Prob. 32ECh. 14.2 - Prob. 33ECh. 14.2 - Prob. 34ECh. 14.2 - Exam scores. Assume that in your History of Film...Ch. 14.2 - Prob. 36ECh. 14.2 - Prob. 37ECh. 14.2 - Prob. 38ECh. 14.2 - Prob. 39ECh. 14.2 - Example 6 Using Stem-and-Leaf Home Run Records...Ch. 14.2 - Prob. 41ECh. 14.2 - Prob. 42ECh. 14.2 - Prob. 43ECh. 14.2 - Prob. 44ECh. 14.2 - Prob. 45ECh. 14.2 - Prob. 46ECh. 14.2 - Prob. 47ECh. 14.2 - A college placement office has made a comparative...Ch. 14.2 - Prob. 49ECh. 14.2 - Prob. 50ECh. 14.2 - Give three separate real-life examples in which...Ch. 14.2 - Which measure of central tendency do you think is...Ch. 14.2 - Manipulating data. Choose some set of data...Ch. 14.2 - Analyzing misleading graphs. Find a graph in the...Ch. 14.2 - Prob. 55ECh. 14.2 - Prob. 56ECh. 14.2 - Prob. 57ECh. 14.2 - Prob. 58ECh. 14.2 - Prob. 59ECh. 14.2 - Prob. 60ECh. 14.2 - Prob. 61ECh. 14.2 - Prob. 62ECh. 14.2 - Prob. 63ECh. 14.2 - Prob. 64ECh. 14.2 - Prob. 65ECh. 14.3 - Sharpening Your Skills Find the range, mean, and...Ch. 14.3 - Sharpening Your Skills Find the range, mean, and...Ch. 14.3 - Sharpening Your Skills Find the range, mean, and...Ch. 14.3 - Sharpening Your Skills Find the range, mean, and...Ch. 14.3 - Sharpening Your Skills Find the range, mean, and...Ch. 14.3 - Sharpening Your Skills Find the range, mean, and...Ch. 14.3 - Sharpening Your Skills Find the range, mean, and...Ch. 14.3 - Sharpening Your Skills Find the range, mean, and...Ch. 14.3 - Prob. 9ECh. 14.3 - Prob. 10ECh. 14.3 - Prob. 11ECh. 14.3 - Prob. 12ECh. 14.3 - Prob. 13ECh. 14.3 - Prob. 14ECh. 14.3 - Prob. 15ECh. 14.3 - Prob. 16ECh. 14.3 - Prob. 17ECh. 14.3 - Prob. 18ECh. 14.3 - Prob. 19ECh. 14.3 - Applying What Youve Learned Summarizing age data....Ch. 14.3 - Prob. 21ECh. 14.3 - Prob. 22ECh. 14.3 - Prob. 23ECh. 14.3 - Prob. 24ECh. 14.3 - Prob. 25ECh. 14.3 - Prob. 26ECh. 14.3 - Prob. 27ECh. 14.3 - Prob. 28ECh. 14.3 - Prob. 29ECh. 14.3 - Prob. 30ECh. 14.3 - In Exercises 31 and 32, we present information on...Ch. 14.3 - In Exercises 31 and 32, we present information on...Ch. 14.3 - Human resources. As the manager of the human...Ch. 14.3 - Prob. 34ECh. 14.3 - Prob. 35ECh. 14.3 - Prob. 36ECh. 14.3 - Prob. 37ECh. 14.3 - Prob. 38ECh. 14.3 - Laptop batteries. A particular brand of laptop was...Ch. 14.3 - Study times. A group of students were sampled with...Ch. 14.3 - Prob. 41ECh. 14.3 - Prob. 42ECh. 14.3 - Prob. 43ECh. 14.3 - Use the following graphs for Exercise 44 a. Which...Ch. 14.3 - Prob. 45ECh. 14.3 - Prob. 46ECh. 14.3 - Prob. 47ECh. 14.3 - Prob. 48ECh. 14.4 - Assume that the distribution in Exercises 1 6 has...Ch. 14.4 - Assume that the distribution in Exercises 1 6 has...Ch. 14.4 - Assume that the distribution in Exercises 1 6 has...Ch. 14.4 - Assume that the distribution in Exercises 1 6 has...Ch. 14.4 - Assume that the distribution in Exercises 1 6 has...Ch. 14.4 - Assume that the distribution in Exercises 1 6 has...Ch. 14.4 - Assume that the distribution in Exercises 7 12...Ch. 14.4 - Assume that the distribution in Exercises 7 12...Ch. 14.4 - Assume that the distribution in Exercises 7 12...Ch. 14.4 - Assume that the distribution in Exercises 7 12...Ch. 14.4 - Assume that the distribution in Exercises 7 12...Ch. 14.4 - Assume that the distribution in Exercises 7 12...Ch. 14.4 - Use the following graph of the standard normal...Ch. 14.4 - Use the following graph of the standard normal...Ch. 14.4 - Use Table 14.19 to find the percent of the area...Ch. 14.4 - Use Table 14.19 to find the percent of the area...Ch. 14.4 - Use Table 14.19 to find the percent of the area...Ch. 14.4 - Prob. 18ECh. 14.4 - Prob. 19ECh. 14.4 - Prob. 20ECh. 14.4 - Prob. 21ECh. 14.4 - Prob. 22ECh. 14.4 - Use Table 14.19 to find the percent of the area...Ch. 14.4 - Prob. 24ECh. 14.4 - Use Table 14.19 to find the percent of the area...Ch. 14.4 - Use Table 14.19 to find the percent of the area...Ch. 14.4 - Use Table 14.19 to find the percent of the area...Ch. 14.4 - Prob. 28ECh. 14.4 - Use Table 14.19 to find the percent of the area...Ch. 14.4 - Prob. 30ECh. 14.4 - Use Table 14.19 to find the percent of the area...Ch. 14.4 - Prob. 32ECh. 14.4 - Use Table 14.19 to find the percent of the area...Ch. 14.4 - Prob. 34ECh. 14.4 - Prob. 35ECh. 14.4 - In Exercises 35 - 40, we give you a mean, a...Ch. 14.4 - Prob. 37ECh. 14.4 - Prob. 38ECh. 14.4 - Prob. 39ECh. 14.4 - Prob. 40ECh. 14.4 - Prob. 41ECh. 14.4 - In Exercises 41 - 46, we give you a mean, a...Ch. 14.4 - Prob. 43ECh. 14.4 - Prob. 44ECh. 14.4 - Prob. 45ECh. 14.4 - Prob. 46ECh. 14.4 - Prob. 47ECh. 14.4 - Prob. 48ECh. 14.4 - Prob. 49ECh. 14.4 - Prob. 50ECh. 14.4 - Prob. 51ECh. 14.4 - Prob. 52ECh. 14.4 - Prob. 53ECh. 14.4 - Prob. 54ECh. 14.4 - Prob. 55ECh. 14.4 - Prob. 56ECh. 14.4 - Prob. 57ECh. 14.4 - Prob. 58ECh. 14.4 - Prob. 59ECh. 14.4 - Prob. 60ECh. 14.4 - Prob. 61ECh. 14.4 - Prob. 62ECh. 14.4 - Prob. 63ECh. 14.4 - Weight lifting.In a power lifting competition the...Ch. 14.4 - Strength of cables.A certain type of cable has a...Ch. 14.4 - Prob. 66ECh. 14.4 - Prob. 67ECh. 14.4 - Prob. 68ECh. 14.4 - Prob. 69ECh. 14.4 - Prob. 70ECh. 14.4 - Prob. 71ECh. 14.4 - Prob. 72ECh. 14.4 - Prob. 73ECh. 14.4 - Prob. 74ECh. 14.4 - Analyzing the SATs. Assume that the math SAT...Ch. 14.4 - Prob. 76ECh. 14.4 - Prob. 77ECh. 14.4 - Prob. 78ECh. 14.4 - Prob. 79ECh. 14.4 - Prob. 80ECh. 14.4 - Prob. 81ECh. 14.4 - Prob. 82ECh. 14.4 - Prob. 83ECh. 14.4 - Prob. 84ECh. 14.4 - Prob. 85ECh. 14.4 - Prob. 86ECh. 14.4 - If a distribution has a mean of 40 and a standard...Ch. 14.4 - Prob. 88ECh. 14.4 - Prob. 89ECh. 14.4 - Prob. 90ECh. 14.5 - In Exercises 1 and 2, state what kind of...Ch. 14.5 - In Exercises 1 and 2, state what kind of...Ch. 14.5 - For Exercises 3 6, do the following: a. Plot the...Ch. 14.5 - Prob. 4ECh. 14.5 - Prob. 5ECh. 14.5 - Prob. 6ECh. 14.5 - Prob. 7ECh. 14.5 - Prob. 8ECh. 14.5 - Prob. 9ECh. 14.5 - Prob. 10ECh. 14.5 - Prob. 11ECh. 14.5 - Prob. 12ECh. 14.5 - In Exercises 11 14, find the line of best fit for...Ch. 14.5 - Prob. 14ECh. 14.5 - Prob. 15ECh. 14.5 - Prob. 16ECh. 14.5 - Prob. 17ECh. 14.5 - Prob. 18ECh. 14.5 - In Exercises 19 22, find the line of best fit for...Ch. 14.5 - In Exercises 19 22, find the line of best fit for...Ch. 14.5 - In Exercises 19 22, find the line of best fit for...Ch. 14.5 - In Exercises 19 22, find the line of best fit for...Ch. 14.5 - In Example 3, we found the absolute value of the...Ch. 14.5 - Prob. 24ECh. 14.5 - Prob. 25ECh. 14.5 - Prob. 26ECh. 14.5 - Prob. 27ECh. 14.CR - Prob. 1CRCh. 14.CR - Prob. 2CRCh. 14.CR - Prob. 3CRCh. 14.CR - Prob. 4CRCh. 14.CR - The lists represent the ages of actors M and...Ch. 14.CR - Prob. 6CRCh. 14.CR - Prob. 7CRCh. 14.CR - Prob. 8CRCh. 14.CR - Prob. 9CRCh. 14.CR - Prob. 10CRCh. 14.CR - Prob. 11CRCh. 14.CR - Prob. 12CRCh. 14.CR - Prob. 13CRCh. 14.CR - Prob. 14CRCh. 14.CR - Prob. 15CRCh. 14.CR - Prob. 16CRCh. 14.CR - Prob. 17CRCh. 14.CR - State what kind of correlation, if any, is...Ch. 14.CR - Prob. 19CRCh. 14.CT - Prob. 1CTCh. 14.CT - Prob. 2CTCh. 14.CT - Prob. 3CTCh. 14.CT - Prob. 4CTCh. 14.CT - Prob. 5CTCh. 14.CT - Prob. 6CTCh. 14.CT - Prob. 7CTCh. 14.CT - Prob. 8CTCh. 14.CT - Prob. 9CTCh. 14.CT - Prob. 10CTCh. 14.CT - Prob. 11CTCh. 14.CT - Prob. 12CTCh. 14.CT - Prob. 13CTCh. 14.CT - Prob. 14CTCh. 14.CT - Prob. 15CTCh. 14.CT - Prob. 16CTCh. 14.CT - Prob. 17CTCh. 14.CT - Prob. 18CT
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- 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.arrow_forward4. [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. ย ພarrow_forward5. [10 marks] Let G = (V,E) be a graph, and let X C V be a set of vertices. Prove that if |S||N(S)\X for every SCX, then G contains a matching M that matches every vertex of X (i.e., such that every x X is an end of an edge in M).arrow_forward
- Q/show that 2" +4 has a removable discontinuity at Z=2i Z(≥2-21)arrow_forwardRefer to page 100 for problems on graph theory and linear algebra. Instructions: • Analyze the adjacency matrix of a given graph to find its eigenvalues and eigenvectors. • Interpret the eigenvalues in the context of graph properties like connectivity or clustering. Discuss applications of spectral graph theory in network analysis. Link: [https://drive.google.com/file/d/1wKSrun-GlxirS3IZ9qoHazb9tC440 AZF/view?usp=sharing]arrow_forwardRefer to page 110 for problems on optimization. Instructions: Given a loss function, analyze its critical points to identify minima and maxima. • Discuss the role of gradient descent in finding the optimal solution. . Compare convex and non-convex functions and their implications for optimization. Link: [https://drive.google.com/file/d/1wKSrun-GlxirS31Z9qo Hazb9tC440 AZF/view?usp=sharing]arrow_forward
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- Only 100% sure experts solve it correct complete solutions okarrow_forwardGive an example of a graph with at least 3 vertices that has exactly 2 automorphisms(one of which is necessarily the identity automorphism). Prove that your example iscorrect.arrow_forward3. [10 marks] Let Go (Vo, Eo) and G₁ = (V1, E1) be two graphs that ⚫ have at least 2 vertices each, ⚫are disjoint (i.e., Von V₁ = 0), ⚫ and are both Eulerian. Consider connecting Go and G₁ by adding a set of new edges F, where each new edge has one end in Vo and the other end in V₁. (a) Is it possible to add a set of edges F of the form (x, y) with x € Vo and y = V₁ so that the resulting graph (VUV₁, Eo UE₁ UF) is Eulerian? (b) If so, what is the size of the smallest possible F? Prove that your answers are correct.arrow_forward
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