Solutions Manual
A First Course in Mathematical
Modeling
FIFTH EDITION
Frank R. Giordano
William P. Fox
Steven B. Horton
Not For Sale
Australia • Brazil • Japan • Korea • Mexico • Singapore • Spain • United Kingdom • United States
, CONTENTS
PART I: ORGANIZING A MODELING COURSE 1
PART II: TEACHING SUGGESTIONS 7
1 Modeling Change 7
2 The Modeling Process,
Proportionality, and Geometric Similarity 16
3 Model Fitting 20
4 Experimental Modeling 25
5 Simulation Modeling 29
6 Discrete Probabilistic Modeling 38
7 Optimization of Discrete Models 39
8 Modeling Using Graph Theory 41
9 Modeling with Decision Theory 43
10 Game Theory 44
11 Modeling with a Differential Equation 46
12 Modeling with Systems of Differential Equations 49
13 Optimization of Continuous Models 52
14 Dimensional Analysis and Similitude 54
15 Graphs as Functions as Model 59
PART III: SAMPLE PROBLEM SOLUTIONS 62
1 Modeling Change 62
2 The Modeling Process,
Proportionality, and Geometric Similarity 70
3 Model Fitting 88
4 Experimental Modeling 103
5 Simulation Modeling 111
6 Discrete Probabilistic Modeling 121
7 Optimization of Discrete Models 124
Not For Sale
iii
© 2014 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed
with a certain product or service or otherwise on a password-protected website for classroom use.
, 8 Modeling Using Graph Theory 143
9 Modeling with Decision Theory 147
10 Game Theory 158
11 Modeling with a Differential Equation 167
12 Modeling with Systems of Differential Equations 189
© Cengage Learning. All rights reserved. No distribution allowed without express authorization.
13 Optimization of Continuous Modeling 209
14 Dimensional Analysis and Similitude 226
15 Graphs of Functions as Models 246
iv
© 2014 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed
with a certain product or service or otherwise on a password-protected website for classroom use.
, PART I
ORGANIZING A MODELING COURSE
ISSUES
Organizing a modeling course is a significant, educational challenge. A number of resource
materials must be gathered: an appropriate text, supplemental references both for students as
well as the instructor, sources and scenarios for student projects, and, possibly, computer
software. Furthermore, there are crucial pedagogical issues that must be resolved in designing
the course, including the following:
1. Course objectives
2. Course prerequisites
3. Course content
4. Number and type of student projects
5. Individual versus group projects
6. The role of computation
7. Grading considerations
8. Opportunities for follow-on modeling courses.
We believe that a textbook can only serve as a base for a modeling course, which must then
be tailored to meet the specific needs of students, as well as overall objectives in the
curriculum. Moreover, a modeling course needs to be flexible and dynamic to allow for each
individual instructor to take advantage of his or her particular mathematical expertise,
experiences, and modeling preferences.
OBJECTIVES
The overall goal of our course is to provide a thorough introduction to the entire modeling
process while affording students the opportunity to practice:
1. Creative and Empirical Model Construction: Given a real-world scenario, the student
must identify a problem, make assumptions and collect data, propose a model, test the
assumptions, refine the model as necessary, fit the model to data if appropriate, and analyze
the underlying mathematical structure of the model in order to appraise the sensitivity of
the conclusions in relation to the assumptions. Furthermore, the student should be able to
generalize the construction to related scenarios.
2. Model Analysis: Given a model, the student must work backward to uncover the implicit
underlying assumptions, assess critically how well the assumptions reflect the scenario at
hand, and estimate the sensitivity of the conclusions when the assumptions are not
precisely met.
3. Model Research: The student investigates an area of interest to gain knowledge,
understanding, and an ability to use what already has been created or discovered. Model
research provides for determining the ‘state of the art’ in a subject area.
Not For Sale
© 2014 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed
with a certain product or service or otherwise on a password-protected website for classroom use.
A First Course in Mathematical
Modeling
FIFTH EDITION
Frank R. Giordano
William P. Fox
Steven B. Horton
Not For Sale
Australia • Brazil • Japan • Korea • Mexico • Singapore • Spain • United Kingdom • United States
, CONTENTS
PART I: ORGANIZING A MODELING COURSE 1
PART II: TEACHING SUGGESTIONS 7
1 Modeling Change 7
2 The Modeling Process,
Proportionality, and Geometric Similarity 16
3 Model Fitting 20
4 Experimental Modeling 25
5 Simulation Modeling 29
6 Discrete Probabilistic Modeling 38
7 Optimization of Discrete Models 39
8 Modeling Using Graph Theory 41
9 Modeling with Decision Theory 43
10 Game Theory 44
11 Modeling with a Differential Equation 46
12 Modeling with Systems of Differential Equations 49
13 Optimization of Continuous Models 52
14 Dimensional Analysis and Similitude 54
15 Graphs as Functions as Model 59
PART III: SAMPLE PROBLEM SOLUTIONS 62
1 Modeling Change 62
2 The Modeling Process,
Proportionality, and Geometric Similarity 70
3 Model Fitting 88
4 Experimental Modeling 103
5 Simulation Modeling 111
6 Discrete Probabilistic Modeling 121
7 Optimization of Discrete Models 124
Not For Sale
iii
© 2014 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed
with a certain product or service or otherwise on a password-protected website for classroom use.
, 8 Modeling Using Graph Theory 143
9 Modeling with Decision Theory 147
10 Game Theory 158
11 Modeling with a Differential Equation 167
12 Modeling with Systems of Differential Equations 189
© Cengage Learning. All rights reserved. No distribution allowed without express authorization.
13 Optimization of Continuous Modeling 209
14 Dimensional Analysis and Similitude 226
15 Graphs of Functions as Models 246
iv
© 2014 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed
with a certain product or service or otherwise on a password-protected website for classroom use.
, PART I
ORGANIZING A MODELING COURSE
ISSUES
Organizing a modeling course is a significant, educational challenge. A number of resource
materials must be gathered: an appropriate text, supplemental references both for students as
well as the instructor, sources and scenarios for student projects, and, possibly, computer
software. Furthermore, there are crucial pedagogical issues that must be resolved in designing
the course, including the following:
1. Course objectives
2. Course prerequisites
3. Course content
4. Number and type of student projects
5. Individual versus group projects
6. The role of computation
7. Grading considerations
8. Opportunities for follow-on modeling courses.
We believe that a textbook can only serve as a base for a modeling course, which must then
be tailored to meet the specific needs of students, as well as overall objectives in the
curriculum. Moreover, a modeling course needs to be flexible and dynamic to allow for each
individual instructor to take advantage of his or her particular mathematical expertise,
experiences, and modeling preferences.
OBJECTIVES
The overall goal of our course is to provide a thorough introduction to the entire modeling
process while affording students the opportunity to practice:
1. Creative and Empirical Model Construction: Given a real-world scenario, the student
must identify a problem, make assumptions and collect data, propose a model, test the
assumptions, refine the model as necessary, fit the model to data if appropriate, and analyze
the underlying mathematical structure of the model in order to appraise the sensitivity of
the conclusions in relation to the assumptions. Furthermore, the student should be able to
generalize the construction to related scenarios.
2. Model Analysis: Given a model, the student must work backward to uncover the implicit
underlying assumptions, assess critically how well the assumptions reflect the scenario at
hand, and estimate the sensitivity of the conclusions when the assumptions are not
precisely met.
3. Model Research: The student investigates an area of interest to gain knowledge,
understanding, and an ability to use what already has been created or discovered. Model
research provides for determining the ‘state of the art’ in a subject area.
Not For Sale
© 2014 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed
with a certain product or service or otherwise on a password-protected website for classroom use.
