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CHAPTER 1 Modeling Change1

Introduction1

Example 1:Testing for Proportionality2

1.1 Modeling Change with Difference Equations4

Example 1:A Savings Certificate5

Example 2:Mortgaging a Home6

1.2 Approximating Change with Difference Equations9

Example 1:Growth of a Yeast Culture9

Example 2:Growth of a Yeast Culture Revisited10

Example 3:Spread of a Contagious Disease12

Example 4:Decay of Digoxin in the Bloodstream13

Example 5:Heating of a Cooled Object14

1.3 Solutions to Dynamical Systems18

Example 1:A Savings Certificate Revisited18

Example 2:Sewage Treatment21

Example 3:Prescription for Digoxin25

Example 4:An Investment Annuity26

Example 5:A Checking Account28

Example 6:An Investment Annuity Revisited30

Example 1:A Car Rental Company35

1.4 Systems of Difference Equations35

Example 2:The Battle of Trafalgar38

Example 3:Competitive Hunter Model—Spotted Owls and Hawks41

Example 4:Voting Tendencies of the Political Parties44

CHAPTER 2 The Modeling Process,Proportionality,and Geometric Similarity52

Introduction52

2.1 Mathematical Models54

Example 1:Vehicular Stopping Distance59

2.2 Modeling Using Proportionality65

Example 1:Kepler's Third Law67

2.3 Modeling Using Geometric Similarity75

Example 1:Raindrops from a Motionless Cloud77

Example 2:Modeling a Bass Fishing Derby79

2.4 Automobile Gasoline Mileage88

2.5 Body Weight and Height,Strength and Agility91

CHAPTER 3 Model Fitting97

Introduction97

3.1 Fitting Models to Data Graphically101

3.2 Analytic Methods of Model Fitting107

3.3 Applying the Least-SquaresCriterion114

3.4 Choosing a Best Model119

Example 1:Vehicular Stopping Distance122

CHAPTER 4 Experimental Modeling126

Introduction126

4.1 Harvesting in the Chesapeake Bay and Other One-Term Models127

Example 1:Harvesting Bluefish130

Example 2:Harvesting Blue Crabs131

4.2 High-Order Polynomial Models138

Example 1:Elapsed Time of a Tape Recorder140

4.3 Smoothing:Low-Order Polynomial Models146

Example 1:Elapsed Time of a Tape Recorder Revisited147

Example 2:Elapsed Time of a Tape Recorder Revisited Again151

Example 3:Vehicle Stopping Distance153

Example 4:Growth of a Yeast Culture155

4.4 Cubic Spline Models159

Example 1:Vehicle Stopping Distance Revisited167

CHAPTER 5 Simulation Modeling175

Introduction175

5.1 Simulating Deterministic Behavior:Area Under a Curve177

5.2 Generating Random Numbers182

5.3 Simulating Probabilistic Behavior186

5.4 Inventory Model:Gasoline and Consumer Demand194

5.5 Queuing Models205

Example 1:A Harbor System205

Example 2:Morning Rush Hour213

CHAPTER 6 Discrete Probabilistic Modeling217

Introduction217

6.1 Probabilistic Modeling with Discrete Systems217

Example 1:Rental Car Company Revisited217

Example 2:Voting Tendencies219

6.2 Modeling Component and System Reliability223

Example 1:Series Systems224

Example 2:Parallel Systems224

Example 3:Series and Parallel Combinations224

6.3 Linear Regression227

Example 1:Ponderosa Pines229

Example 2:The Bass Fishing Derby Revisited231

CHAPTER 7 Discrete Optimization Modeling236

Example 1:Determining a Production Schedule237

7.1 An Overview of Discrete Optimization Modeling237

Example 2:Space Shuttle Cargo240

Example 3:Approximation by a Piecewise Linear Function241

7.2 Linear Programming 1:Geometric Solutions250

Example 1:The Carpenter's Problem251

Example 2:A Data-Fitting Problem252

7.3 Linear Programming Ⅱ:Algebraic Solutions259

Example 1:Solving the Carpenter's Problem Algebraically261

7.4 Linear Programming Ⅲ: The Simplex Method263

Example 1:The Carpenter's Problem Revisited268

Example 2:Using the Tableau Format271

7.5 Linear Programming Ⅳ:Sensitivity Analysis273

7.6 Numerical Search Methods279

Example 1:Using the Dichotomous Search Method282

Example 2:Using the Golden Section Search Method285

Example 3:Model-Fitting Criterion Revisited287

Example 4:Optimizing Industrial Flow288

CHAPTER 8 Dimensional Analysis and Similitude292

Introduction292

8.1 Dimensions as Products295

Example 1:A Simple Pendulum298

Example 2:Wind Force on a Van301

8.2 The Process of Dimensional Analysis304

Example 1:Terminal Velocity of a Raindrop309

Example 2:Automobile Gas Mileage Revisited311

8.3 A Damped Pendulum313

8.4 Examples Illustrating Dimensional Analysis319

Example 1:Explosion Analysis319

Example 2:How Long Should You Roast a Turkey?324

8.5 Similitude330

Example 1:Drag Force on a Submarine331

CHAPTER 9 Graphs of Functions as Models336

9.1 An Arms Race336

Example 1:Civil Defense345

Example 2:Mobile Launching Pads346

Example 3:Multiple Warheads347

Example 4:MIRVs Revisited:Counting Warheads348

9.2 Modeling an Arms Race in Stages350

9.3 Managing Nonrenewable Resources:The Energy Crisis355

9.4 Effects of Taxation on the Energy Crisis359

9.5 AGasoline Shortage and Taxation364

CHAPTER 10 Modeling with a Differential Equation368

Introduction368

10.1 Population Growth371

10.2 Prescribing Drug Dosage382

10.3 Braking Distance Revisited391

10.4 Graphical Solutions of Autonomous Differential Equations395

Example 1:Drawing a Phase Line and Sketching Solution Curves396

Example 2:Cooling Soup399

Example 3:Logistic Growth Revisited400

10.5 Numerical Approximation Methods404

Example 1:Using Euler's Method406

Example 2:A Savings Certificate Revisited407

CHAPTER 11 Modeling with Systems of Differential Equations412

Introduction412

11.1 Graphical Solutions of Autonomous Systems of First-Order Differential Equations413

Example 1:A Linear Autonomous System414

Example 2:A Nonlinear Autonomous System415

11.2 A Competitive Hunter Model419

11.3 A Predator-Prey Model427

11.4 Two Military Examples437

Example 1:Lanchester Combat Models437

Example 2:Economic Aspects of an Arms Race444

11.5 Euler's Method for Systems of Differential Equations450

Example 1:Using Euler's Method for Systems451

Example 2:A Trajectory and Solution Curves452

CHAPTER 12 Continuous Optimization Modeling458

Introduction458

12.1 An Inventory Problem:Minimizing the Cost of Delivery and Storage459

12.2 A Manufacturing Problem:Maximizing Profit in Producing Competing Products468

12.3 Constrained Continuous Optimization474

Example 1:An Oil Transfer Company474

Example 2:A Space Shuttle Water Container476

12.4 Managing Renewable Resources:The Fishing lndustry480

APPENDIX A Problems from the Mathematics Contest in Modeling,1985-2002490

APPENDIX B An Elevator Simulation Algorithm523

APPENDIX C The Revised Simplex Method532

Index535