Solution Manual for A First Course in Mathematical Modeling, 5th Edition

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’s Manual with Solutionsn • Koin •A First Course in MathematicalModelingFIFTH EDITIONFrank R. GiordanoNaval Postgraduate SchoolWilliam P. FoxNaval Postgraduate SchoolSteven B. HortonUnited States Military AcademySOLUTIONSGUIDE

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CONTENTSPART I: ORGANIZING A MODELING COURSE1PART II: TEACHING SUGGESTIONS71Modeling Change72The Modeling Process,Proportionality, and Geometric Similarity163Model Fitting204Experimental Modeling255Simulation Modeling296Discrete Probabilistic Modeling387Optimization of Discrete Models398Modeling Using Graph Theory419Modeling with Decision Theory4310Game Theory4411Modeling with a Differential Equation4612Modeling with Systems of Differential Equations4913Optimization of Continuous Models5214Dimensional Analysis and Similitude5415Graphs as Functions as Model59PART III: SAMPLE PROBLEM SOLUTIONS621Modeling Change622The Modeling Process703Model Fitting4Experimental Modeling1035Simulation Modeling1116Discrete Probabilistic Modeling1217Optimization of Discrete Models124Proportionality, and Geometric Similarity,88

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11Modeling with a Differential Equation16712Modeling with Systems of Differential Equations18913Optimization of Continuous Modeling20914Dimensional Analysis and Similitude22615Graphs of Functions as Models2iv8Modelingsing Graph Theory1439Modeling with Decision Theory14710Game Theory158U46

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PART IORGANIZING A MODELING COURSEISSUESOrganizing a modeling course is a significant, educational challenge. A number of resourcematerials must be gathered: an appropriate text, supplemental references both for students aswell as the instructor, sources and scenarios for student projects, and, possibly, computersoftware. Furthermore, there are crucial pedagogical issues that must be resolved in designingthe course, including the following:1. Course objectives2. Course prerequisites3. Course content4. Number and type of student projects5. Individual versus group projects6. The role of computation7. Grading considerations8. Opportunities for follow-on modeling courses.We believe that a textbook can only serve as a base for a modeling course, which must thenbe tailored to meet the specific needs of students, as well as overall objectives in thecurriculum. Moreover, a modeling course needs to be flexible and dynamic to allow for eachindividual instructor to take advantage of his or her particular mathematical expertise,experiences, and modeling preferences.OBJECTIVESThe overall goal of our course is to provide a thorough introduction to the entire modelingprocess while affording students the opportunity to practice:1.Creative and Empirical Model Construction:Given a real-world scenario, the studentmust identify a problem, make assumptions and collect data, propose a model, test theassumptions, refine the model as necessary, fit the model to data if appropriate, and analyzethe underlying mathematical structure of the model in order to appraise the sensitivity ofthe conclusions in relation to the assumptions. Furthermore, the student should be able togeneralize the construction to related scenarios.2.Model AnalysisGiven a model, the student must work backward to uncover the implicitunderlying assumptions, assess critically how well the assumptions reflect the scenario athand, and estimate the sensitivity of the conclusions when the assumptions are notprecisely 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. Modelresearch prnin‘art’:1

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can be organized in support of amodeling course.COURSE PREREQUISITESThere are strong arguments to require such courses as advanced calculus, linear algebra,differential equations, probability, numerical analysis, and optimization as prerequisites to anintroductory modeling course. Certainly the level and sophistication of the mathematics thatstudents are capable of using increases significantly as more advanced courses are added totheir programs. However, our desire is to gain the modeling experience as early as possible inthe student’s career. Although some unfamiliar mathematical ideas are taught as part of themodeling process, the emphasis is on using mathematics already known to the student aftercompleting high school. As outlined in the preface to the text, a course can be constructedrequiring only high school mathematics as a prerequisite. Some sections do require anintroductory calculus course as a prerequisite or corequisite, as detailed in the preface to thetext. In our modeling courses, we emphasize teaching students how to use mathematics theyalready know in a context of significant applications with which they can readily identify. Thisapproach stimulates student interest in mathematics and motivates them to study moreadvanced topics such as those mentioned above. Moreover, our students are eager to seemeaningful applications of the mathematics they have learned.To accomplish our goals, we provide students with a diversity of scenarios for practicingall three of these facets of modeling. In addition, normal and routine exercises are assigned totest the student’s understanding of the instructional material we present. Thus, our textbookprovides expository material and a framework around which a diversity of additional materialsCOURSE CONTENTMany modeling courses select from an inventory of specific model types which can beadapted to a variety of situations. Certainly model selection is a valid step in theproblem-solving process and it is important that students learn to use what already has beencreated. However, our experience is that undergraduate students seldom comprehend theassumptions inherent in type models. We want our students to realize the necessity of makingassumptions, the need to determine the appropriateness of the assumptions, and the importanceof investigating how sensitive the conclusions are to the assumptions. Consequently, while wedo discuss how to fit type models in the text, we have chosen to emphasize model construction,leaving the study of type models for more advanced courses.We feel that model construction promotes student creativity, demonstrates the artisticnature of model building, and develops an appreciation for how mathematics can be usedeffectively in various settings. Since the student needs practice in the first several steps of theproblem-solving process–identifying the problem, making assumptions, determininginterrelationships between the variables and submodels–it is tempting to compose an entirecourse of creative model construction. However, there are serious difficulties with such acourse. Typically, students are very anxious, at least initially, because they don’t know how tobegin the modeling process. After all, they have probably never before attacked an open-endedsemester is dedicated to creative model construction. Moreover, a course consisting entirely ofproblem. When they are successful in constructing a model, they usually find the procedureenjoyable and exhilarating. Nevertheless, they tend to “burn out” if an entire quarter or2

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creative model construction cannot address other important aspects of modeling, likeexperimentation and simulation. Furthermore, there are difficulties with such a course for theinstructor. Preparation of the course requires enormous effort in researching and generatingscenarios to be modeled. Grading is difficult and tends to be subjective because each studentapproaches each project in a different way, yet students need constant feedback on their work.These difficulties then contribute to the anxieties of the student who is overly concerned aboutbeing graded in class under a time constraint in an area perceived to be relatively subjective.Under these conditions, success of the course becomes highly instructor dependent andcircumstantial.For the above reasons, we have chosen to design a course consisting of a mixture of bothcreativeandempiricalmodeling projects, along with projects in model analysis and modelresearch. We begin by interactively constructing graphical models in class to engage studentsimmediately in model analysis, which is relatively familiar to them. The transition to creativemodel construction commences with the students learning to make assumptions about areal-world behavior and by providing them with data to check their assumptions (initially withsimple proportionality arguments). We then apply the modeling process to constructinteractively in class relatively simple submodels and models in a variety of settings coveringmany disciplines. Students begin to see that situations arise where it may be very difficult orimpossible to construct an analytic model, yet predictive capabilities are highly desirable. Thisperception motivates the study of empirical modeling. Students find empirical modelconstruction more procedural and reproducible than creative model construction, and we findthey welcome the mixture. In the text, some scenarios are modeled several ways creatively,and several ways empirically, so students can experience the alternatives that may be available.STUDENT PROJECTSWe have developed our modeling courses to promote progressive development in thestudent’s modeling capabilities. To achieve our objectives, we require each student to completeat least 6significantproblem assignments or projects to be handed-in for a grade. Each studentis assigned a mixture of problems/projects in creative and empirical model construction, modelanalysis, and model research. We purposely select problems/projects which address scenariosfor which there are no unique solutions. Several of the projects includerealdata that thestudent is either given or canreadilycollect.If the course is taught early in the student’s program we recommend a combination ofindividual and group projects. Individual work is essential if the student is to develop adequatemodeling skills. By way of contrast, group projects are exhilarating and allow for theexperience of the synergistic effect that takes place in a ‘brainstorming’ session. For example,during the instruction of Chapter 2 on proportionality, we give a choice of 5 or 6 differentscenarios for which students individually create a model. During the instruction of the nextchapter on model fitting, teams are organized to test, refine, and fit these models to data. Theresulting group model is typically substantially superior to any of the individual models. Asimilar procedure is followed during the instruction of the simulation chapter where studentsare required to develop models individually followed by a team effort to refine the models andimplement them on the computer.3

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Finally, we choose projects from a wide variety of disciplines including biology,economics and the physical sciences. We make certain that these projects require littleoverhead to be paid by the student in order to understand, and successfully attack, the problem.We also allow students to choose a project that can be turned in at the end of the course.Students are free to choose from a diversity of possibilities such as completing UMAPmodules, researching a model studied earlier, developing a model in a scenario of interest tothem, or analyzing a model presented in another subject they are studying. This chosen projectgives the student an opportunity to pursue a subject in some detail. Typically, our studentschoose a project of interest from one of the Project sections in the text. We require thatstudents complete one of the following for their selected projects:1.Model ConstructionDevelop a scenario of interest to the student and develop a modelto address the situation posed. For example, the student may pose a relevant economicquestion and construct a graphical model to explain the behavior being studied.2.Model Analysis:Analyze a model of interest by identifying the underlying assumptionsand discussing their applicability. Determine the mathematical conclusions of the modeland the sensitivity of the conclusions to the assumptions made. Finally, interpret themathematical conclusions for real-world scenarios.3.Model ResearchResearch a model studied in class to determine the ‘state of the art’ inthat area. For example, many UMAP modules are available that treat scenarios discussed inclass in greater detail. The student may also study a section of the text not covered in classand complete the problems from the corresponding problem set.THE ROLE OF COMPUTATIONWe emphasize that computing and programming capabilities are not required for ourmodeling courses. However, beginning with Chapter One, Modeling Change, computationdoes play a role of increasing importance. The use of computers can significantly enhance amodeling course: in a demonstrational mode to facilitate student understanding of a concept(such as a graphical solution of a linear program) or in a computational mode to reduce thetediousness in carrying out certain numerical procedures. We have found the integration of acomputer in a supportive role adds significantly to student interest and to the realism of ourmodeling course.We provide our students with packaged software for making scatterplots, fitting modelsaccording to various criteria, solving linear programs by the Simplex method, constructingdivided difference tables and cubic spline models, solving initial value problems by variousmethods, and graphing functions.A TYPICAL COURSEIn this edition, we have provided several options for organizing a course. We have detailedthe prerequisites in Figure 2 of the Preface. We incorporate lecture/discussion lessons, hourexams, and computer workshop sessions as part of our modeling courses. Additionally, time isprovided for students to complete modeling projects. Detailed discussions of the variouslessons and suggestions on how many lessons to devote to each section appear in the TeachingSuggestions portion of this Manual, arranged according to the chapters of our text.::4

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GRADING CONSIDERATIONSThe difficulty in grading a modeling course has dissuaded many instructors from teachingmodeling, and discouraged many students from taking the course as well. Given the objectivesof our course, both creative and technical skills are to be acquired by the student. We havefound it best to have the creative requirements accomplished outside the classroom, withoutthe added pressure of a fixed, short time constraint. We use the classroom exams strictly fortesting techniques we expect the student to master. These classroom exams test theunderstanding of basic concepts or straightforward applications of particular techniques, suchas fitting a cubic spline (see the section on Sample Tests in this Manual). The more interpretiveapplications of the modeling techniques are treated in problems and projects assigned for workoutside the classroom.The classroom exams are similar to those in other mathematics courses and present noadded difficulty in grading. For most of the student projects we are able to establish a gradingscale that assigns weights to various parts of each problem. Homework is collected often,principally for purposes of feedback but also for a grade. (Some homework is collected but notgraded, to assist students in allowing their minds to roam without fear of being ‘wrong’ whileat the same time providing some guidance and feedback.)THE USE OF UMAP MODULESWe have found material provided by The Consortium For Mathematics and ItsApplications (COMAP) to be outstanding and particularly well suited to the course wepropose. COMAP started under a grant from the National Science Foundation and has as itsgoal the production of instructional materials to introduce applications of mathematics into theundergraduate curriculum.The individual modules may be used in a variety of ways. First, they may be used asinstructional material to support several lessons. (We have incorporated several modules in thetext in precisely this manner.) In this mode a student completes the self-study module byworking through its exercises (the detailed solutions provided with the module can be removedconveniently before it is issued). Another option is to put together a block of instructioncovering, for example, linear programming or difference equations, using as instructionalmaterial one or more UMAP modules suggested in the projects sections of the text. Themodules also provide excellent sources for ‘model research’ since they cover a wide variety ofapplications of mathematics in many fields. In this mode, a student is given an appropriatemodule to be researched and is asked to complete and report on the module. Finally, themodules are excellent resources for scenarios for which students can practice modelconstruction. In this mode the teacher writes a scenario for a student project based on anapplication addressed in a particular module and uses the module as background material,perhaps having the student complete the module at a later date. Information may be obtainedby contacting COMAP Inc., Suite 3B, 175 Middlesex Turnpike, Bedford, MA 01730,800-722-6627.5

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FOLLOW-ON MODELING COURSESMany of our students express an interest in taking an additional modeling course. While wehave tailored many courses to the individual needs of the students’ programs, a very successfuland popular course is one which combines studies from various UMAP modules selected to fita student’s particular interests, coupled with significant advanced student projects. Theproportion of time devoted to the modules versus the student projects varies, depending on thenature of the projects that are available.We are fortunate to have real-world projects readily available at our schools andappropriate to assign for advanced student work. There are, however, several other excellentsources which can be used as background material for student projects The comprehensivefour-volume seriesModules in Applied Mathematics, edited by William F. Lucas andpublished by Springer-Verlag, provides important and realistic applications of mathematicsappropriate for undergraduates. The volumes treat differential equations models, political andrelated models, discrete and system models, and life science models. Another book,CaseStudies in Mathematical Modelling,edited by D. J. James and J. J. McDonald and published byHalsted Press, provides case studies explicitly designed to facilitate the development ofmathematical models. Finally, scenarios with an industrial flavor are contained in the excellentmodular series edited by J. L. Agnew and M. S. Keener of Oklahoma State University.Frank R. GiordanoWilliam P. FoxSteven B. HortonMaurice D. Weir6

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PART IITEACHING SUGGESTIONSCHAPTER ONEModeling ChangeSUGGESTED SYLLABUSClass HoursTopicText1Modeling Change with Difference EquationsSect 1.11Approximating Change with Difference EquationsSect 1.21Solutions to Dynamical SystemsSect 1.31Systems of Difference EquationsSect 1.4OBJECTIVESThe major objectives of this chapter are:1.To build and solve models involving change that takes place in discrete intervals.2.To prepare the students for modeling change taking place continuously in Chapters 11and 12.3.To introduce numerical solutions by iterating difference equations.4.To extend the modeling process by modeling interactive systems early.DISCUSSIONWe have found that freshman students model dynamical systems quite naturally. Since theycan iterate the systems they build given an initial value, they gain intuition by graphing theirresults and analyzing the long-term behavior. Further, by experimenting with different initialvalues, they begin to appreciate the sensitivity of the model’s conclusions to the initialconditions. Finally, the experience of this chapter prepares them for modeling with differentialequations and systems of differential equations, which students generally find more difficult.We strongly suggest beginning with behavior that can be modeled exactly, such as theaccumulation of money in a savings account. Moving to annuities or mortgages retains studentinterest while introducing more sophisticated models. Since the students can readily enumeratethese sequences even before building the model, they gain confidence in their work. Once thestudents are confident with exactly modeling behavior, we move to approximating change withdiscrete systems. Only an elementary notion of proportionality is needed for the scenarios inthis chapter. (The concepts of proportionality and geometric similarity are studied in moredetail in Chapter Two). A powerful advantage of studying discrete systems early is that theresulting models can be iterated if starting values are known. We have found that students haveno difficulty iterating the systems. Finally, introducing systems of difference equations permitsthe students to model interesting systems with rather elementary mathematics. Many of thescenarios intrositand7

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A spreadsheet is of great use to iterate and graph the models the students build. We havefound that even if the students have not used a spreadsheet before, they learn the fewcommands they need for this chapter quickly. They do not have difficulties extending theirknowledge of spreadsheets to handle systems of difference equations in Section 1.4. We addeda new modeling example and problems onSIRmodels.We add a few additional examples for nonlinear DDS here for the instructors use. We buildnonlinear discrete dynamical systems to describe the change in behavior of the quantities westudy. We also will study systems of DDS to describe the changes in various systems that acttogether in some way or ways. We define a nonlinear DDS– If the function ofaninvolvespowers ofan(likean2, or a functional relationship (likeanan−1), we will say that the discretedynamical system is nonlinear. A sequence is a function whose domain is the set ofnon-negative integers (n0, 1, 2, . . . ). We will restrict our model solution to the numerical andgraphical solutions. Analytical solutions may be studied in more advanced mathematicscourses.Example 1.Growth of a Yeast CultureWe often model population growth by assuming that the change in population is directlyproportional to the current size of the given population. This produces a simple, first orderDDS similar to those seen earlier. It might appear reasonable at first examination, but thelong-term behavior of growth without bound is disturbing. Why would growth without boundof a yeast culture in a jar (or controlled space) be alarming?There are certain factors that affect population growth. Things include resources (food,oxygen, space, etc.) These resources can support some maximum population. As this number isapproached, the change (or growth rate) should decrease and the population should neverexceed its resource supported amount.Problem IdentificationPredict the growth of yeast in a controlled environment as afunction of the resources available and the current population.Assumptions and VariablesWe assume that the population size is best described by the weight of the biomass ofthe culture. We defineynas the population size of the yeast culture after periodn. There existsa maximum carrying capacity,M, that is sustainable by the resources available. The yeastculture is growing under the conditions established.Model:yn1ynkyn M−ynwhereynis the population size after periodnnis the time period measured in hourskis the growth (or decay) constant of proportionalityMis the carrying capacity of our systemIn our experiment, we first plotynversusnand find a stable equilibrium value ofapproximately 665. Next, we plotyn1−ynversusyn665−ynto find the slope,k; it isapproximately 0. 00082 , Withk0. 00082 and the carrying capacity in biomass is 665. Thisanalysis is currently in the textbook. The final model isyn1yn0. 00082yn 665−yn::8

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Again, this is nonlinear because of theyn2term obtained in the expansion. The solution isiterated via technology (there is no closed form analytical solution for this equation) from aninitial condition, biomass, of 9. 6:Plot of DDS from growth of a yeast culture.The model shows stability in that the population (biomass) of the yeast culture approaches665 asngets large. Thus, the population is eventually stable at approximately 665 units.Example 2.Spread of a Contagious DiseaseThere are one thousand students in a college dormitory, and some students have beendiagnosed with meningitis, a highly contagious disease. The health center wants to build amodel to determine how fast the disease will spread.Problem Identification:Predict the number of students affected with meningitis as afunction of time.Assumptions and Variables:Letmnbe the number of students affected withmeningitis afterndays. We assume all students are susceptible to the disease. The possibleinteractions of infected and susceptible students are proportional to their product (as aninteraction term).The model is,mn1mnkmn1000−mn9

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Two students returned from spring break with meningitis. The rate of spreading per day ischaracterized byk0. 0090. It is assumed that a vaccine can be in place and studentsvaccinated within 1-2 weeks.mn1mn0. 00090mn1000mnWe iterated with technology:We see that without treatment everyone will eventually get the disease.Plot of DDS for the spread of a disease.−Interpretation:The results show that most students will be affected within 2 weeks. Sinceonly about 10% will be affected within one week, every effort must be made to get thevaccination at the school and get the students vaccinated within one week.We also add an additional example for systems here for the instructor’s use as needed. Thisis a military model concerning insurgencies.10

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ExampleModeling Military Interactions and InsurgenciesInsurgent forces have a strong foothold in the city of Urbania, a major metropolis in thecenter of the country of Ibestan. Intelligence estimates they currently have a force of about1000 fighters. The local police force has approximately 1300 officers, many of which have hadno formal training in law enforcement methods or modern tactics for addressing insurgentactivity. Based on data collected over the past year, approximately 8% of insurgents switchsides and join the police each week, whereas about 11% of police switch sides and join theinsurgents. Intelligence also estimates that around 120 new insurgents arrive from theneighboring country of Moronka each week. Recruiting efforts in Ibestan yield about 85 newpolice recruits each week as well. In armed conflict with insurgent forces, the local police areable to capture or kill approximately 10% of the insurgent force each week on average whilelosing about 3% of their force.Problem Statement: Build a mathematical model of this insurgency. Determine theequilibrium state (if it exists) for this DDS.We define the variablesPnthe number of police in the system after time period n.Inthe number of insurgents in the system after time period n.n0, 1, 2, 3,…weeksModel:Pn1Pn−0. 03Pn−0. 11Pn0. 08In85,P01300In1In0. 11Pn−0. 08In−0. 01In120,I01000PnIn13001000128310831275.021149.191273.4521202.5881276.3761246.2021282.381282.2871290.4291312.5371299.7721338.2281309.8621360.3221320.3071379.5491330.8281396.4641341.2291411.4911351.3771424.9581361.181437.1171370.5851448.1661379.5561458.261388.0791467.5251396.151476.0591403.7741483.9451410.9611491.251417.7261498.0313.11

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1424.0871504.3351430.0621510.2041435.6691515.6741440.931520.7771445.8621525.539Plot of InsurgentsInover time.12

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