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