Engineering Problems Illustrating Mathematics A Project of the Mathematics Division of the Society for the Promotion of Engineering Education JOHN W. CELL Chairman of Committee Associate Professor of Mathematics in the College of Engineering, North Carolina State College FIRST EDITION FIFTH IMPRESSION McGRAW-HILL ENGINEERING PROBLEMS ILLUSTRATING MATHEMATICS COPYRIGHT, 1943, BY THE EDITORIAL COMMITTEE J. W. CELL, W. C. BRENKE, G. E. MOORE, C. C. TORRANCE PRINTED IN THE UNITED STATES OP AMERICA All rights reserved. This book, or parts thereof, may not be reproduced in any form without permission of the Editorial Committee. CONTENTS FOREWORD A PROJECT OF THE MATHEMATICS DIVISION OF THE SOCIETY FOR THE PROMOTION OF ENGINEERING EDUCATION v PREFACE ix INTRODUCTION BY ROBERT E. DOHERTY xi PART I COLLEGE ALGEBRA RADICALS AND EXPONENTS 1 QUADRATIC EQUATIONS 2 COMPLEX NUMBERS 5 LOGARITHMS 5 SIMULTANEOUS EQUATIONS 7 VARIATION 8 BINOMIAL THEOREM 9 PROGRESSIONS 10 THEORY OF EQUATIONS 12 DETERMINANTS AND SIMULTANEOUS LINEAR EQUATIONS 15 PART II TRIGONOMETRY FUNDAMENTAL IDENTITIES 22 RIGHT TRIANGLES 23 OBLIQUE TRIANGLES 25 MULTIPLE ANGLE IDENTITIES 27 GRAPHS OF THE TRIGONOMETRIC FUNCTIONS. LINE VALUES 28 TRIGONOMETRIC EQUATIONS 31 COMPLEX NUMBERS. DEMOIVRES THEOREM 33 PART III ANALYTIC GEOMETRY ELEMENTARY FORMULAS 37 STRAIGHT LINES 38 CURVE SKETCHING 43 GRAPHS OF THE CURVES y ax n 46 CIRCLES 49 PARABOLAS 51 ELLIPSES 54 vii viii CONTENTS PAGE HYPERBOLAS 55 ROTATION OF AXES 57 GRAPHS OF SINE WAVES 58 EXPONENTIAL AND LOGARITHMIC GRAPHS 61 HYPERBOLIC FUNCTIONS 63 POLAR COORDINATES 64 PARAMETRIC EQUATIONS 66 EMPIRICAL CURVES 71 ELEMENTARY FORMULAS IN SOLID ANALYTIC GEOMETRY 75 SURFACES IN SOLID ANALYTIC GEOMETRY 79 PART IV DIFFERENTIAL CALCULUS USES AND INTERPRETATIONS OP DERIVATIVES 83 TANGENT AND NORMAL 86 CURVE SKETCHING. MAXIMUM, MINIMUM, AND FLEX POINTS .... 90 MAXIMUM AND MINIMUM PROBLEMS 93 VELOCITY AND ACCELERATION. RECTILINEAR MOTION 100 VELOCITY AND ACCELERATION. CURVILINEAR MOTION 102 DIFFERENTIALS 105 CURVATURE 106 INDETERMINATE FORMS 107 RELATED RATES 110 PARTIAL DIFFERENTIATION 110 PART V INTEGRAL CALCULUS CONSTANT OF INTEGRATION 114 EVALUATION OF DEFINITE INTEGRALS 122 DEFINITE INTEGRALS. SIMPSONS RULE AND TRAPEZOIDAL RULE. . . 125 AREAS. AVERAGE ORDINATE 127 LENGTH OF ARC. AREAS OF SURFACES OF REVOLUTION 131 VOLUMES 133 FIRST AND SECOND MOMENTS 135 TOTAL FORCE 140 WORK 141 OTHER APPLICATIONS OF INTEGRAL CALCULUS 143 INFINITE SERIES 149 HYPERBOLIC FUNCTIONS 153 INDEX 157 ANSWERS 161 PREFACE The problems in this collection are designed to give to the freshman and sophomore engineering student some understanding of the uses of mathematics in junior and senior engineering courses, and hence of the necessity of a thorough foundation in mathematics. These problems should help to answer the eternal question, Why study this topic The early training of an engineering student should contain an appreciation of how engineering problems are translated into mathematics problems. The Report on the Aims and Scope of Engineering Curricula in The Journal of Engineering Education , vol. 30, March, 1940, pp. 563-564 states that The scientific-technological studies should be directed toward 1. Mastery of the fundamental scientific principles and a command of basic knowledge underlying the branch of engineering which the student is pursuing. This implies a. grasp of the meaning of physical and mathematical laws, and knowledge of how they are evolved and of the limitations in their use b. knowledge of materials, machines, and structures. 2. Thorough understanding of the engineering method and elementary competence in its application. This requires a. comprehension of the interacting elements in situations which are to be analyzed 6. ability to think straight in the application of fundamental principles to new problems ... Another quotation from an address by Dr...