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CHAPTER 1 Modeling Change 1

Introduction 1

Example 1:Testing for Proportionality 2

1.1 Modeling Change with Difference Equations 4

Example 1:A Savings Certificate 5

Example 2:Mortgaging a Home 6

1.2 Approximating Change with Difference Equations 9

Example 1:Growth of a Yeast Culture 9

Example 2:Growth of a Yeast Culture Revisited 10

Example 3:Spread of a Contagious Disease 12

Example 4:Decay of Digoxin in the Bloodstream 13

Example 5:Heating of a Cooled Object 14

1.3 Solutions to Dynamical Systems 18

Example 1:A Savings Certificate Revisited 18

Example 2:Sewage Treatment 21

Example 3:Prescription for Digoxin 25

Example 4:An Investment Annuity 26

Example 5:A Checking Account 28

Example 6:An Investment Annuity Revisited 30

Example 1:A Car Rental Company 35

1.4 Systems of Difference Equations 35

Example 2:The Battle of Trafalgar 38

Example 3:Competitive Hunter Model—Spotted Owls and Hawks 41

Example 4:Voting Tendencies of the Political Parties 44

CHAPTER 2 The Modeling Process,Proportionality,and Geometric Similarity 52

Introduction 52

2.1 Mathematical Models 54

Example 1:Vehicular Stopping Distance 59

2.2 Modeling Using Proportionality 65

Example 1:Kepler's Third Law 67

2.3 Modeling Using Geometric Similarity 75

Example 1:Raindrops from a Motionless Cloud 77

Example 2:Modeling a Bass Fishing Derby 79

2.4 Automobile Gasoline Mileage 88

2.5 Body Weight and Height,Strength and Agility 91

CHAPTER 3 Model Fitting 97

Introduction 97

3.1 Fitting Models to Data Graphically 101

3.2 Analytic Methods of Model Fitting 107

3.3 Applying the Least-SquaresCriterion 114

3.4 Choosing a Best Model 119

Example 1:Vehicular Stopping Distance 122

CHAPTER 4 Experimental Modeling 126

Introduction 126

4.1 Harvesting in the Chesapeake Bay and Other One-Term Models 127

Example 1:Harvesting Bluefish 130

Example 2:Harvesting Blue Crabs 131

4.2 High-Order Polynomial Models 138

Example 1:Elapsed Time of a Tape Recorder 140

4.3 Smoothing:Low-Order Polynomial Models 146

Example 1:Elapsed Time of a Tape Recorder Revisited 147

Example 2:Elapsed Time of a Tape Recorder Revisited Again 151

Example 3:Vehicle Stopping Distance 153

Example 4:Growth of a Yeast Culture 155

4.4 Cubic Spline Models 159

Example 1:Vehicle Stopping Distance Revisited 167

CHAPTER 5 Simulation Modeling 175

Introduction 175

5.1 Simulating Deterministic Behavior:Area Under a Curve 177

5.2 Generating Random Numbers 182

5.3 Simulating Probabilistic Behavior 186

5.4 Inventory Model:Gasoline and Consumer Demand 194

5.5 Queuing Models 205

Example 1:A Harbor System 205

Example 2:Morning Rush Hour 213

CHAPTER 6 Discrete Probabilistic Modeling 217

Introduction 217

6.1 Probabilistic Modeling with Discrete Systems 217

Example 1:Rental Car Company Revisited 217

Example 2:Voting Tendencies 219

6.2 Modeling Component and System Reliability 223

Example 1:Series Systems 224

Example 2:Parallel Systems 224

Example 3:Series and Parallel Combinations 224

6.3 Linear Regression 227

Example 1:Ponderosa Pines 229

Example 2:The Bass Fishing Derby Revisited 231

CHAPTER 7 Discrete Optimization Modeling 236

Example 1:Determining a Production Schedule 237

7.1 An Overview of Discrete Optimization Modeling 237

Example 2:Space Shuttle Cargo 240

Example 3:Approximation by a Piecewise Linear Function 241

7.2 Linear Programming 1:Geometric Solutions 250

Example 1:The Carpenter's Problem 251

Example 2:A Data-Fitting Problem 252

7.3 Linear Programming Ⅱ:Algebraic Solutions 259

Example 1:Solving the Carpenter's Problem Algebraically 261

7.4 Linear Programming Ⅲ: The Simplex Method 263

Example 1:The Carpenter's Problem Revisited 268

Example 2:Using the Tableau Format 271

7.5 Linear Programming Ⅳ:Sensitivity Analysis 273

7.6 Numerical Search Methods 279

Example 1:Using the Dichotomous Search Method 282

Example 2:Using the Golden Section Search Method 285

Example 3:Model-Fitting Criterion Revisited 287

Example 4:Optimizing Industrial Flow 288

CHAPTER 8 Dimensional Analysis and Similitude 292

Introduction 292

8.1 Dimensions as Products 295

Example 1:A Simple Pendulum 298

Example 2:Wind Force on a Van 301

8.2 The Process of Dimensional Analysis 304

Example 1:Terminal Velocity of a Raindrop 309

Example 2:Automobile Gas Mileage Revisited 311

8.3 A Damped Pendulum 313

8.4 Examples Illustrating Dimensional Analysis 319

Example 1:Explosion Analysis 319

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

8.5 Similitude 330

Example 1:Drag Force on a Submarine 331

CHAPTER 9 Graphs of Functions as Models 336

9.1 An Arms Race 336

Example 1:Civil Defense 345

Example 2:Mobile Launching Pads 346

Example 3:Multiple Warheads 347

Example 4:MIRVs Revisited:Counting Warheads 348

9.2 Modeling an Arms Race in Stages 350

9.3 Managing Nonrenewable Resources:The Energy Crisis 355

9.4 Effects of Taxation on the Energy Crisis 359

9.5 AGasoline Shortage and Taxation 364

CHAPTER 10 Modeling with a Differential Equation 368

Introduction 368

10.1 Population Growth 371

10.2 Prescribing Drug Dosage 382

10.3 Braking Distance Revisited 391

10.4 Graphical Solutions of Autonomous Differential Equations 395

Example 1:Drawing a Phase Line and Sketching Solution Curves 396

Example 2:Cooling Soup 399

Example 3:Logistic Growth Revisited 400

10.5 Numerical Approximation Methods 404

Example 1:Using Euler's Method 406

Example 2:A Savings Certificate Revisited 407

CHAPTER 11 Modeling with Systems of Differential Equations 412

Introduction 412

11.1 Graphical Solutions of Autonomous Systems of First-Order Differential Equations 413

Example 1:A Linear Autonomous System 414

Example 2:A Nonlinear Autonomous System 415

11.2 A Competitive Hunter Model 419

11.3 A Predator-Prey Model 427

11.4 Two Military Examples 437

Example 1:Lanchester Combat Models 437

Example 2:Economic Aspects of an Arms Race 444

11.5 Euler's Method for Systems of Differential Equations 450

Example 1:Using Euler's Method for Systems 451

Example 2:A Trajectory and Solution Curves 452

CHAPTER 12 Continuous Optimization Modeling 458

Introduction 458

12.1 An Inventory Problem:Minimizing the Cost of Delivery and Storage 459

12.2 A Manufacturing Problem:Maximizing Profit in Producing Competing Products 468

12.3 Constrained Continuous Optimization 474

Example 1:An Oil Transfer Company 474

Example 2:A Space Shuttle Water Container 476

12.4 Managing Renewable Resources:The Fishing lndustry 480

APPENDIX A Problems from the Mathematics Contest in Modeling,1985-2002 490

APPENDIX B An Elevator Simulation Algorithm 523

APPENDIX C The Revised Simplex Method 532

Index 535