How to Solve Problems Donald Scarl For Success in Freshman Physics, Engineering, and Beyond @(Sat5a®QQ Dosoris Press, Glen Cove, New York How to Solve Problems Published by Dosoris Press P. O. Box 148 Glen Cove, NY 11542 516-671-0686 1st edition 1989 2nd edition 1990 3rd edition 1993 4th edition 1994 Copyright © 1994 Donald Scarl All rights reserved. No part of the contents of this book may be reproduced or transmitted in any form or by any means without the written permission of Dosoris Press. Library of Congress Catalog Card Number: 94-67968 ISBN: 0-9622008-4-0 Printed in the United States of America 987654321 Contents Preface viii 1 Why Solve Problems? Lifetime learning 1 Workmanship 2 Previously solved problems 2 Reflexes 2 Presenting work to others 3 Barriers 3 Creativity and disorganization 4 2 School Science and engineering texts 5 Learning equations 6 Homework 10 Exams 11 Efficiency 12 Teachers 12 Working in a group 13 Professional qualities 14 Exercise 15 3 Methods Example 1: World Trade Center elevator 16 Solution for Example 1 17 Divide into parts 19 Work down the page 22 Write clearly 23 Make everything explicit 23 Use symbols 24 Do one step at a time 26 Use ratios 27 Example 2: Surface area of a sphere 27 Solution for Example 2 27 Do the assigned problem 28 Example 3: Oscillating car 29 Solution for Example 3 29 Exercises 31 4 Describing the Problem Example 4: Ultralight plane takeoff 32 Solution for Example 4 33 Define the problem 36 v vi Contents______________________________________________ Write a heading 37 Title 37 Name and date 38 Draw a diagram 38 Diagram or picture 38 Axes 40 Scale 41 Three-dimensional drawings 41 Labels 42 Drawing vectors 42 Name the variables 43 Symbol definitions 43 Data equations 47 Check 48 Preliminary equations 48 Units 49 Constants 49 Trigonometry and vector components 49 Exercises 50 5 Finding the Solution Science equations 51 Choosing the right equation 51 General equations 53 Particular equations 54 Algebra 54 Small number of known operations 55 Get unknowns on left hand side alone 55 Copying symbols 57 Check die solution 57 Reread the problem 58 Check algebra 58 Check unit 58 Make a variational check 58 Check test cases 59 Check by approximation 59 Check the size of the result 60 Mistakes 60 Exercises 61 6 Presenting the Solution Numerical results 62 Put in known numbers 62 Use significant figures to indicate accuracy 63 Include units 64 Use engineering prefixes 64 Put the result in a sentence 64 Decorate your results 64 ______________________________________________Contents vii Graphs 66 Title 65 Axis labels 66 Scale divisions 66 Points and lines 66 Straight lines 66 Logarithmic axes 66 Rewriting 67 Reports and Publications 67 Exercises 68 7 Can’t Solve It Use thinking where it counts 60 Look for a similar problem 69 Simplify 69 Generalize 70 Put in numbers 70 Look for unused data 70 Try a ratio 70 Put it aside 71 Go for a little help 71 How to ask for help 71 8 Spreadsheets Calculating a single value 74 Example 5: Solar rocks 74 Solution for Example 5 75 Inputs 76 Constants 77 Unit conversion 77 Calculation 77 Results 78 Plotting data and calculating functions 78 Example 6: Optical detector power 78 Solution for Example 6 79 Graphing data 81 Graphing a function 82 Fitting a function to data 83 Numerical differentiation 83 Numerical integration 84 Other operations 84 Organizing a spreadsheet 84 Exercises 87 Problems ss Further Reading 91 Problem Solutions 97 Index ns Preface To solve science and engineering problems you need to know science and engineering. You also need a tool-kit of problem-solving methods. This book describes the methods professional problem solvers use, explains why these methods have evolved, and shows how to make them your own. You can use these problem-solving methods now for homework and examinations. You will use them in your later professional work to help you define useful problems, solve them, and convince the world that the problems are important and that your solutions are correct. Starting with the pioneering work of Polya, many authors have written self-help books and scholarly books on problem solving. These books are about the thinking part of problem solving: how to generalize, specialize, particularize, brainstorm, and so forth. They are especially useful for design problems in which part of the solution may lie in redefining the original problem. This book, in contrast, emphasizes the simple actions that professional problem solvers take to analyze and solve already defined problems. It explains how to set up and solve problems that you did not think you knew how to solve. It describes the statement, organization, and presentation of scientific and technical material. By teaching problem-solving style, it attempts to do for problem solvers what Strunk and White’s Elements of Style has done for writers. It is a pleasure to acknowledge Eric Rogers, Jay Orear, and George Wolga, who by their instruction and example helped me learn how to teach. I have enjoyed a continuing email discussion with Annalisa Crannell about her methods of teaching mathematical writing. Hilda Bass, Barbara Cohen, Lorcan Folan, Robert Folk, Judith Scarl, Romualdas Sviedrys, Robert Swart, Larry Tankersley, and Alan Van Heuvelen read and improved manuscripts of this or previous editions. viii OODOOD X O O O O D O Why Solve Problems? How can I pass freshman physics? How can I build a user- friendly computer? How can I sell a million of them? How can I get a date for tomorrow night? We all need to solve problems. This chapter explains how problem solving, in addition to helping you to pass science and engineering courses, can help you to learn technical material, to work efficiently, to think clearly, to develop self-confidence, and to convince others to work with you and to support your work. Lifetime learning Science, engineering, and life itself change from year to year, so that most of your learning will be done after you leave school and no longer have teachers, homework, tests, and grades to help you. Learn in school how to learn by yourself. A good way to learn a new subject is to make up problems to solve for yourself. By making up problems you can find the gaps in the way you—and perhaps others—understand a subject. Problem solving is a path to new knowledge and discoveries. You start out solving problems for others. Your teachers suggest that you do homework problems and require that you do exam problems. Your employer will ask you to solve problems that are part of a group engineering effort. But problem solving 1