The Heart of Mathematics: An Invitation to Effective Thinking
4th Edition
ISBN: 9781118156599
Author: Edward B. Burger, Michael Starbird
Publisher: Wiley, John & Sons, Incorporated
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Textbook Question
Chapter 6.4, Problem 16MS
Hamilton Study. Look at the graph you drew to model part of your campus in section 6.4, Mindscape 8. Is there a Hamiltonian circuit in your graph? Why or why not?
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There are three options for investing $1150. The first earns 10% compounded annually, the second earns 10% compounded quarterly, and the third earns 10% compounded continuously. Find equations that model each investment growth and
use a graphing utility to graph each model in the same viewing window over a 20-year period. Use the graph to determine which investment yields the highest return after 20 years. What are the differences in earnings among the three
investment?
STEP 1: The formula for compound interest is
A =
nt
= P(1 + − − ) n²,
where n is the number of compoundings per year, t is the number of years, r is the interest rate, P is the principal, and A is the amount (balance) after t years. For continuous compounding, the formula reduces to
A = Pert
Find r and n for each model, and use these values to write A in terms of t for each case.
Annual Model
r=0.10
A = Y(t) = 1150 (1.10)*
n = 1
Quarterly Model
r = 0.10
n = 4
A = Q(t) = 1150(1.025) 4t
Continuous Model
r=0.10
A = C(t) =…
The following ordered data list shows the data speeds for cell phones used by a
telephone company at an airport:
A. Calculate the Measures of Central Tendency from the ungrouped data list.
B. Group the data in an appropriate frequency table.
C. Calculate the Measures of Central Tendency using the table in point B.
D. Are there differences in the measurements obtained in A and C? Why (give at
least one justified reason)?
I leave the answers to A and B to resolve the remaining two.
0.8
1.4
1.8
1.9
3.2
3.6
4.5
4.5
4.6
6.2
6.5
7.7
7.9
9.9
10.2
10.3
10.9
11.1
11.1
11.6
11.8
12.0
13.1
13.5
13.7
14.1
14.2
14.7
15.0
15.1
15.5
15.8
16.0
17.5
18.2
20.2
21.1
21.5
22.2
22.4
23.1
24.5
25.7
28.5
34.6
38.5
43.0
55.6
71.3
77.8
A. Measures of Central Tendency
We are to calculate:
Mean, Median, Mode
The data (already ordered) is:
0.8, 1.4, 1.8, 1.9, 3.2, 3.6, 4.5, 4.5, 4.6, 6.2, 6.5, 7.7, 7.9, 9.9, 10.2, 10.3, 10.9,
11.1, 11.1, 11.6,
11.8, 12.0, 13.1, 13.5, 13.7, 14.1, 14.2, 14.7, 15.0, 15.1, 15.5,…
A tournament is a complete directed graph, for each pair of vertices x, y either (x, y) is an arc or
(y, x) is an arc. One can think of this as a round robin tournament, where the vertices represent
teams, each pair plays exactly once, with the direction of the arc indicating which team wins.
(a) Prove that every tournament has a direct Hamiltonian path. That is a labeling of the teams
V1, V2,..., Un so that vi beats Vi+1. That is a labeling so that team 1 beats team 2, team 2
beats team 3, etc.
(b) A digraph is strongly connected if there is a directed path from any vertex to any other
vertex. Equivalently, there is no partition of the teams into groups A, B so that every team
in A beats every team in B. Prove that every strongly connected tournament has a directed
Hamiltonian cycle. Use this to show that for any team there is an ordering as in part (a) for
which the given team is first.
(c) A king in a tournament is a vertex such that there is a direct path of length at most 2 to
any…
Chapter 6 Solutions
The Heart of Mathematics: An Invitation to Effective Thinking
Ch. 6.1 - Map maker, map maker make me a graph. Represent...Ch. 6.1 - Unabridged list. Represent cach landmass from...Ch. 6.1 - Will the walk work? Does your graph from...Ch. 6.1 - Walk around the house. Is it possibel to traverse...Ch. 6.1 - Walk the line. Does this graph above have an Euler...Ch. 6.1 - Walkabout. Does this graph have an Euler circuit?...Ch. 6.1 - Linking the loops. In this map, the following...Ch. 6.1 - Scenic drive. (S) Here is a map of Rockystone...Ch. 6.1 - Under-edged. (H) Does this graph have an Euler...Ch. 6.1 - No man is an island. The country of Pelago...
Ch. 6.1 - Path-o-rama. For each graph below, determine if...Ch. 6.1 - Walk around the block. Create a graph of the...Ch. 6.1 - Walking the dogs. Your dogs, Abbey and Bear, love...Ch. 6.1 - Delivery query. The next time you see a postal...Ch. 6.1 - Snow job. (ExH) Shown here is a map of the tiny...Ch. 6.1 - Special delivery. (ExH) Julia is the letter...Ch. 6.1 - Draw this old house. Suppose you wanted to trace...Ch. 6.1 - Path of no return. Consider this map showing a...Ch. 6.1 - Without a trace. Is it possibel to trace out...Ch. 6.1 - New Euler. In the three previous Mindscapes, you...Ch. 6.1 - New edge—new circuit. Look at the graph for...Ch. 6.1 - New edge—new path. Review your work for...Ch. 6.1 - Path to proof. Suppose you have a connected graph...Ch. 6.1 - No Euler no how. Look at graph (a) for Mindscape...Ch. 6.1 - Degree day. (S) For cach graph below, determine...Ch. 6.1 - degrees of proof. Review your work for Mindscape...Ch. 6.1 - Degrees in sequence. Can you draw a graph that has...Ch. 6.1 - Even Steven. Review your work in Mindscape 28 to...Ch. 6.1 - Little League lesson. (H) You are in charge of...Ch. 6.1 - With a group of folks. In a small group, discuss...Ch. 6.1 - Power beyond the mathematics. Provide several...Ch. 6.1 - Here we celebrate the power of algebra as a...Ch. 6.1 - Here we celebrate the power of algebra as a...Ch. 6.1 - Here we celebrate the power of algebra as a...Ch. 6.1 - Here we celebrate the power of algebra as a...Ch. 6.1 - Here we celebrate the power of algebra as a...Ch. 6.2 - What a character! What expression gives the Euler...Ch. 6.2 - Count, then verify. What are the values of V, E,...Ch. 6.2 - Sneeze, then verify. Look at an unopened tissue...Ch. 6.2 - Blow, then verify. Inflate a ballon and use a...Ch. 6.2 - Add one. Find the values V, E, and F for the graph...Ch. 6.2 - Bowling. What is the Euler Characteristic of the...Ch. 6.2 - Making change. We begin with the graph pictured at...Ch. 6.2 - Making a point. Take a connected graph and add a...Ch. 6.2 - On the edge (H). Is it possible to add an edge to...Ch. 6.2 - Soap films. Consider the following sequence of...Ch. 6.2 - Dualing. What is the relationship between the...Ch. 6.2 - Prob. 12MSCh. 6.2 - Lots of separation. Suppose we are told that a...Ch. 6.2 - Prob. 14MSCh. 6.2 - Psychic readings. Someone is thinking of a...Ch. 6.2 - Prob. 16MSCh. 6.2 - Prob. 17MSCh. 6.2 - Circular reasoning. Create a connected graph as...Ch. 6.2 - Prob. 19MSCh. 6.2 - More circles. Consider the sphere described in...Ch. 6.2 - In the rough (S). Count the number of facets,...Ch. 6.2 - Cutting corners (H). The following collection of...Ch. 6.2 - Stellar. The following collection of pictures...Ch. 6.2 - A torus graph (ExH). The Euler Characteristic...Ch. 6.2 - Regular unfolding. Each graph below represents...Ch. 6.2 - A tale of two graphs. Suppose we draw a graph that...Ch. 6.2 - Two graph conjectures (S). Can you conjecture a...Ch. 6.2 - Lots of graphs conjecture. Can you conjecture a...Ch. 6.2 - Torus count. Three hollowed, triangular prisms...Ch. 6.2 - Torus two count (H). Carefully count the number of...Ch. 6.2 - Torus many count. Using the preceding calculations...Ch. 6.2 - Prob. 32MSCh. 6.2 - Tell the truth. Someone said that she made a...Ch. 6.2 - No sphere. Suppose we have a sphere built out of...Ch. 6.2 - Soccer ball. A soccer ball is made of pentagons...Ch. 6.2 - Klein bottle. Using the diagram here for building...Ch. 6.2 - Not many neighbors. Show that every map has at...Ch. 6.2 - Infinite edges. Suppose we consider a conn ected...Ch. 6.2 - Here we celebrate the power of algebra as a...Ch. 6.2 - Prob. 44MSCh. 6.2 - Prob. 45MSCh. 6.2 - Here we celebrate the power of algebra as a...Ch. 6.2 - Here we celebrate the power of algebra as a...Ch. 6.3 - Dont be cross. Here is a drawing of a graph with...Ch. 6.3 - De Plane! De Plane! (S) Is the graph given in...Ch. 6.3 - Countdown (H). For the graph drawing shown, count...Ch. 6.3 - Prob. 4MSCh. 6.3 - Criss-Cross. Is it possible to redraw the graph...Ch. 6.3 - Dont cross in the edge. Each of the graphs drawn...Ch. 6.3 - Hot crossed buns. Each of the graphs drawn below...Ch. 6.3 - Prob. 8MSCh. 6.3 - Spider on a mirror. Is it possible to redraw the...Ch. 6.3 - One more vertex. The graph here is drawn to show...Ch. 6.3 - Yet one more vertex (H). The graph shown is drawn...Ch. 6.3 - Familiar freckles. Is it possible to redraw the...Ch. 6.3 - Remind you of anyone you know? Is it possible to...Ch. 6.3 - Final countdown. For this graph drawing, count the...Ch. 6.3 - Euler check-up. Use your answer to the previous...Ch. 6.3 - Euler second opinion. For the graph drawing shown...Ch. 6.3 - Prob. 17MSCh. 6.3 - Prob. 18MSCh. 6.3 - A colorful museum. This figure shows the floor...Ch. 6.3 - Limit of 5. Start drawing a planar graph. Keep...Ch. 6.3 - Starring the hexagon. Is it possible to redraw...Ch. 6.3 - Prob. 22MSCh. 6.3 - Prob. 23MSCh. 6.3 - Getting greedy. (H) Suppose you are asked to color...Ch. 6.3 - Stingy rather than greedy. By coloring the...Ch. 6.3 - Getting more colorful. Graphs dont have to be...Ch. 6.3 - Prob. 27MSCh. 6.3 - Prob. 28MSCh. 6.3 - Chromatically applied. There are eight radio...Ch. 6.3 - Prob. 30MSCh. 6.3 - Personal perspectives. Write a short essay...Ch. 6.3 - Here we celebrate the power of algebra as a...Ch. 6.3 - Here we celebrate the power of algebra as a...Ch. 6.3 - Prob. 37MSCh. 6.3 - Here we celebrate the power of algebra as a...Ch. 6.3 - Here we celebrate the power of algebra as a...Ch. 6.4 - Up close and personal. Create a graph to model...Ch. 6.4 - Network lookout. Find an examle of a network...Ch. 6.4 - Prob. 3MSCh. 6.4 - Hamiltonian holiday (S). You are interning for a...Ch. 6.4 - Home style. Create a graph to model the rooms in...Ch. 6.4 - Six degrees or less. Suppose this graph is a model...Ch. 6.4 - Degrees of you. Find ten willing friends or...Ch. 6.4 - Campus shortcut. Find a map of your campus and...Ch. 6.4 - Arborist lesson. Which of the graphs below are...Ch. 6.4 - Prob. 10MSCh. 6.4 - Prob. 11MSCh. 6.4 - Prob. 12MSCh. 6.4 - Prob. 13MSCh. 6.4 - Prob. 14MSCh. 6.4 - Prob. 15MSCh. 6.4 - Hamilton Study. Look at the graph you drew to...Ch. 6.4 - Business trip redux. Look back in the section and...Ch. 6.4 - Handling Hamiltons. For each graph below, find a...Ch. 6.4 - Road trip. You are checking out gradua te programs...Ch. 6.4 - Back to Hatties trip. Look back in this section...Ch. 6.4 - Solve the Icosian Game. Find a Hamiltonian circuit...Ch. 6.4 - Hunt for Hamilton (S). A large island country has...Ch. 6.4 - Has no Hamilton. Give some characteristics that...Ch. 6.4 - Cubing Hamilton (ExH). Can you find a Hamihonian...Ch. 6.4 - Hamiltonian path. A Hamiltonian path is a path in...Ch. 6.4 - Sorry, no path. Give some characteristics that...Ch. 6.4 - Prob. 27MSCh. 6.4 - Prob. 28MSCh. 6.4 - Prob. 29MSCh. 6.4 - Prob. 30MSCh. 6.4 - Edge count. Look at all the trees you drew in the...Ch. 6.4 - Personal perspecthes. Write a short essay...Ch. 6.4 - Prob. 33MSCh. 6.4 - Prob. 34MSCh. 6.4 - Dollars and cents. Your spanning tree has three...Ch. 6.4 - Adding up. Your spanning tree has four edges with...Ch. 6.4 - Prob. 38MSCh. 6.4 - Vertex search (H). Your graph has a Hamiltonian...Ch. 6.4 - Binary gossip tree. You told a secret to two of...
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