Fundamentals of technical mathematics

Fundamentals of Technical Mathematics Sarhan M Musa Prairie View A&M University AMSTERDAM  BOSTON  HEIDELBERG  LONDON  NEW YORK  OXFORD PARIS  SAN DIEGO  SAN FRANCISCO  SINGAPORE  SYDNEY  TOKYO Academic Press is an imprint of Elsevier Academic Press is an imprint of Elsevier 125 London Wall, London EC2Y 5AS, UK 525 B Street, Suite 1800, San Diego, CA 92101-4495, USA 225 Wyman Street, Waltham, MA 02451, USA The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, UK Copyright © 2016 Elsevier Inc All rights reserved No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher Details on how to seek permission, further information about the Publisher’s permissions policies and our arrangements with organizations such as the Copyright Clearance Center and the Copyright Licensing Agency, can be found at our website: www.elsevier.com/permissions This book and the individual contributions contained in it are protected under copyright by the Publisher (other than as may be noted herein) Notices Knowledge and best practice in this field are constantly changing As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein ISBN: 978-0-12-801987-0 British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress For information on all Academic Press publications visit our website at http://store.elsevier.com/ To my late father, Mahmoud; my mother, Fatmeh; my wife, Lama; and my children, Mahmoud, Ibrahim, and Khalid Preface Fundamentals of Technical Mathematics introduces applied mathematics for engineering technologists and technicians Through a simple, engaging approach, the book reviews basic mathematics including whole numbers, fractions, mixed numbers, decimals, percentages, ratios, and proportions The text covers conversions to different units of measure (standard and/or metric) and other topics as required by specific businesses and industries Building on these foundations, it then explores concepts in arithmetic; introductory algebra; equations, inequalities, and modeling; graphs and functions; measurement; geometry; trigonometry; and matrices, determinants, and vectors It supports these concepts with practical applications in a variety of technical and career vocations, including automotive, allied health, welding, plumbing, machine tool, carpentry, auto mechanics, HVAC, and many other fields In addition, the book provides practical examples from a vast number of technologies and uses two common software programs, Maple and Matlab This book has eight chapters Chapter provides basic concepts in arithmetic Chapter introduces algebra Chapter presents equations, inequalities, and modeling Chapter presents graphs and functions Chapter presents measurement Chapter introduces geometry Chapter reviews trigonometry Finally, Chapter introduces matrices, determinants, and vectors Sarhan M Musa xi Acknowledgments It is my pleasure to acknowledge the outstanding help and support of the team at Elsevier in preparing this book, especially from Cathleen Sether, Steven Mathews, Katey Birtcher, Sarah J Watson, Amy Clark, and Anitha Sivaraj Thanks for professors John Burghduff and Mary Jane Ferguson for their support, understanding, and being great friends I would also like to thank Dr Kendall T Harris, his college dean, for his constant support Finally, this book would never have seen the light of day if not for the constant support, love, and patience of our family xiii Chapter Basic Concepts in Arithmetic You must the thing you think you cannot Eleanor Roosevelt n Alessandro Volta (1745e1827), an Italian physicist, chemist and a pioneer of electrical science He is most famous for his invention of the electric battery Alessandro has a special talent for languages Before he left school, he had learned Latin, French, English and German His language talents helped him in later life, when he traveled around Europe, discussing his work with scientists in Europe’s centers of science INTRODUCTION In this chapter, we will discuss the most useful and important basic topics in arithmetic (science of numbers), which are essential refreshment for people who have forgotten or not like math Mastering the fundamental concepts in mathematics is vital in strengthening the foundation for learning advanced material and forming a like for mathematics 1.1 BASIC ARITHMETIC In arithmetic numbers used are known The purpose of this section is to introduce the basic principles/laws of arithmetic such as properties for the Fundamentals of Technical Mathematics http://dx.doi.org/10.1016/B978-0-12-801987-0.00001-0 Copyright © 2016 Elsevier Inc All rights reserved CHAPTER Basic Concepts in Arithmetic operations of addition (ỵ), subtraction (À), multiplication (Â), and division (O) Next, we introduce fractions, decimals, and percents Numbers are categorized into six categories First is real numbers, which includes all numbers Whole numbers are numbers 0, 1, 2, 3, , while natural numbers not include zero, i.e., 1, 2, 3, Integer numbers are whole numbers including their negative counterparts, i.e., , À3, À2, À1, 0, 1, 2, À4 3, Rational numbers, i.e., ; ; 0:75; 0:111 Irrational numbers are dec2 pffiffiffi imals that are not negatives and have no end such as p (pi) and 1.1.1 Arithmetic operations 1.1.1.1 Addition (D) Addition is the process of finding the sum of two or more numbers The symbol (ỵ) represents the addition operation Example (a) How much is ỵ 5? (b) How much is ỵ ỵ 13? (c) How much is 12 ỵ 15 ỵ 28 ỵ 13? Solution (a) ỵ ẳ (b) ỵ ỵ 13 ẳ 21 (c) We can use columns of numbers to get the total or sum many numbers: 1 2 Carry numbers + sum Therefore, 12 ỵ 15 ỵ 28 ỵ 13 ẳ 68 We write the numbers in columns, and then we add the numbers in the columns from right to left If the sum of the numbers in any column is 10 or more, we write the ones number under the sum line, carry the tens number to the next column, and add it to the numbers in that column 1.1.1.2 Subtraction (L) Subtraction is taking away a number from another number The symbol (À) represents the subtraction operation 1.1 Basic arithmetic Example (a) How (b) How (c) How (d) How much is much is much is much is subtracted from 7? À 2? 21 À 72? 23 À 82? Solution (a) À ¼ When we subtract a small number from a bigger number, we get a positive number (ỵ) Note that, positive numbers not require the ỵ sign in front of the number (b) À ¼ À5 When we subtract a big number from a smaller number, we get a negative number (À) (c) We can use columns of numbers to get the subtraction: − 2 difference We write the numbers in a vertical column and then subtract the bottom numbers from the top numbers We start from right to left (d) − remainder 2 12 When the bottom number is larger than the top number, we borrow from the number in the top of the next column and add ten to the top number before subtracting, at the same time we reduce the top number in the next column by Since is smaller than 3, borrows from to become 12, and becomes 1.1.1.3 Multiplication (3) Multiplication is a quick way to add many similar numbers The symbol (Â) represents the multiplication operation For example,  is the same as adding for times, ỵ ỵ ỵ ỵ ẳ 35 This is the same as  ¼ 35 by multiplication table CHAPTER Basic Concepts in Arithmetic Example (a) Multiply 23 with 12 (b) Multiply 57 with 18 Solution (a) × multiplicand multiplier + 23 Total 276 We write the numbers in a vertical column and then we multiply each number in the multiplicand from right to left by the rightmost multiplier If the product is greater than 9, add the tens number to the product of numbers in the next column Next, we multiply each number in the multiplicand by the next number in the multiplier and we place the second partial product under the first partial product, by moving one space to the left We continue the process for each number in the multiplier, and then we add the partial products to get the final answer (b) × 5 Carry number 45 + 57 Total 1026 Signs play very important roles in arithmetic and algebra; we have summarized the sign rules for multiplication in the table below Properties of signs for multiplication Multiplication sign Result (ỵb) (ỵa) ỵab ẳ ab (b) (ỵa) ab (ỵb) (a) ab (b) (a) ỵab ẳ ab 1.1 Basic arithmetic We see from the table that, like signs give (ỵ) and unlike signs give (À) Any number written without a sign except is considered to be positive, for example, ¼ þ3.0 is a neural number, neither positive nor negative Example (a) (3)  (4) ¼ 12 (b) (À2)  (5) ¼ À10 (c) (3)  (À8) ¼ À24 (d) (À3)  (À6) ¼ 18 1.1.1.4 Division (O) Division is used for separating a number to several equal groups of numbers The symbol (O) represents the division operation Example (a) Divide 22 by (b) 634 O 28 Solution (a) 22 O ¼ 11 By dividing 22 by we get 11 without remainder When we have very large numbers to divide, long division is performed with the method as shown below: (b) divisor quotient 22 28 − − 634 56 dividend 74 56 18 remainder Long division allows us to use two numbers of the divided at a time, making the division process easier We divide 63 by 28, we get of 28 in the 63, we put the as the quotient We multiply by 28 to give us 56 Then we subtract 56 from 63 to give us The is brought down so we have numbers to divide So, 634 O 28 ẳ 22 ỵ (18/28) This means that 634 can be divided into 22 groups of 28 and there will be 18 left over, which is not enough to make a 23rd group of 28 Appendix C: Solution Manual 379 21 5x À À x2 À 2x ẳ x2 3x ỵ ẳ p p ặ 41ị5ị ặ i 11 ¼ x ¼ 2 22 À x2 À x ẳ x2 ỵ x ẳ xx ỵ 1ị ẳ x1 ẳ x2 ẳ 23 Singular because row is all zeros 24 Singular because column is all zeros 25 Singular because row is times row 26 Singular because row is times row 27 jMNj ¼ jMjjNj ¼ À4  ¼ À12 28 jMMtj ¼ jMMj ¼ jMjjMj ¼ À4  À4 ¼ 16 29 jMtNj ¼ jMjjNj ¼ À4  ¼ À12      À4 À32 ¼ ¼ À10:67 30 2MN À1  ¼ 23 jMjN À1  ¼ 3 Section 8.3 (Answers) vector scalar scalar scalar scalar vector ! u ỵ! v ẳ h3; 5i ỵ h2; 1i ẳ h1; 4i ! u À! v ¼ hÀ3; 5i À h2; À1i ¼ hÀ5; 6i 5! u À! v ¼ 5hÀ3; 5i À h2; À1i ¼ hÀ17; 6i 10 3! u þ 2! v ¼ 3hÀ3; 5i þ 2h2; À1i ¼ hÀ5; 13i qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffi 2 ! 11 k u k ẳ u1 ị ỵ u2 ị ẳ 3ị2 ỵ 5ị2 ẳ 34 q q p 12 9k! v k ẳ v1 ị2 ỵ v2 ị2 ẳ 2ị2 ỵ 1ị2 ẳ 380 Appendix C: Solution Manual p p 13 k! u k ỵ k! v k ẳ 34 ỵ ẳ 5:83 ỵ 2:24 ẳ 8:07 14 ! u ỵ! v ẳ h3; 5i ỵ h2; 1i ẳ h1; 4i q p u ỵ! v k ẳ 1ị2 ỵ 4ị2 ẳ 17 k! 15 ! u ỵ! v ẳ h3; 4i ỵ h2; 5i ẳ h1; 1i q p u ỵ! v k ẳ 1ị2 ỵ 1ị2 ẳ k! 16 ! u ỵ! v ẳ h7; 1i ỵ h3; 6i ẳ h4; 5i q p u ỵ! v k ẳ 4ị2 ỵ 5ị2 ẳ 41 k! 17 ! u ỵ! v ẳ h1; 9i þ h8; 2i ¼ h7; À7i qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffi ! v k ẳ 7ị2 ỵ 7ị2 ẳ 98 ku ỵ! u þ! v ¼ hÀ1; 3i þ hÀ2; À8i ¼ hÀ3; 5i 18 ! q p u ỵ! v k ẳ 3ị2 ỵ 5ị2 ẳ 34 k! ( ) ! w h4; 3i ! ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ ; 19 u ẳ 5 kwk 16 ỵ ) ( ! w h2; 5i ! ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ p; p 20 u ẳ kwk ỵ 25 29 29 ) ( ! w hÀ2; À6i À2 À6 ! 21 u ¼ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ pffiffiffiffiffi; pffiffiffiffiffi kwk þ 36 40 40 ) ( ! w h1; 2i 22 ! u ¼ ¼ pffiffiffiffiffiffiffiffiffiffiffi ¼ pffiffiffi; p kwk 1ỵ4 5 ! ! ! 23 w ¼ h3; À1i ¼ i À j ! ! 24 ! w ẳ h6; 4i ẳ i ỵ j ! ! 25 ! w ¼ h1; 9i ẳ i ỵ j ! ! 26 ! w ¼ hÀ17; À25i ¼ À17 i À 25 j 27 ! u $! v ẳ ỵ  À5 ¼ À26 28 ! u $! v ¼ ỵ 11 ẳ 29 ! u $! v ẳ ỵ  ¼ À9 30 ! u $! v ¼ ỵ ẳ 31 ! u $! v ẳ ỵ  ¼ À2 32 ! u $! v ¼ ỵ ẳ Appendix C: Solution Manual 381 ! u $! v 33 Since cos q ¼ ! ! , k u kk v k h1; 3i$h2; 1i 2ỵ3 cos q ẳ pq ẳ pp ẳ p 10 50 12 ỵ 32 22 ỵ 1ị2 Therefore, the angle between the vectors ! u and! v is     5 À1 + q ¼ arccos pffiffiffiffiffi ¼ cos q pffiffiffiffiffi ¼ 45 : 50 50 ! u $! v 34 Since cos q ¼ ! ! , k u kk v k h4; 1i$h1; À2i 4À2 cos q ¼ pq ẳ pp ẳ p 17 85 42 ỵ 12 12 ỵ 2ị2 Therefore, the angle between the vectors ! u and ! v is     2 À1 + q ¼ arccos pffiffiffiffiffi ¼ cos q pffiffiffiffiffi z 77:47 : 85 85 ! u $! v 35 Since cos q ¼ ! ! , k u kk v k h2; 5i$h1; 6i ỵ 30 28 cos q ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ pffiffiffiffiffipffiffiffiffiffi ¼ pffiffiffiffiffiffiffiffiffiffi 29 37 1073 2ị2 ỵ 52 12 ỵ 6ị2 Therefore, the angle between the vectors ! u and ! v is     28 28 q ¼ arccos pffiffiffiffiffiffiffiffiffiffi ¼ cosÀ1 q pffiffiffiffiffiffiffiffiffiffi z31:26+ : 1073 1073 ! u $! v 36 Since cos q ¼ ! ! , k u kk v k h2; 1i$h1; 3i ỵ cos q ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ pffiffiffipffiffiffiffiffi ¼ pffiffiffiffiffi 2 10 50 2 2ị ỵ 1 ỵ 3ị Therefore, the angle between the vectors ! u and ! v is     1 q ¼ arccos pffiffiffiffiffi ¼ cosÀ1 q pffiffiffiffiffi z 81:87 : 50 50 ! ! u$v 37 Since cos q ¼ ! ! , k u kk v k hÀ5; 2i$h0; À3i 0À6 À6 À2 cos q ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ pffiffiffiffiffipffiffiffi ¼ pffiffiffiffiffi ¼ pffiffiffiffiffi 2 29 29 29 2 5ị ỵ ỵ ðÀ3Þ Therefore, the angle between the vectors ! u and ! v is     À2 À2 q ¼ arccos pffiffiffiffiffi ¼ cosÀ1 q pffiffiffiffiffi z111:8 : 29 29 382 Appendix C: Solution Manual ! u $! v 38 Since cos q ¼ ! ! , k u kk v k h1; 2i$hÀ1; 1i À1 À À3 cos q ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ pffiffiffipffiffiffi ¼ pffiffiffiffiffi 10 12 ỵ 2ị2 1ị2 ỵ 1ị2 Therefore, the angle between the vectors ! u and! v is     À3 À3 À1 q ¼ arccos pffiffiffiffiffi ¼ cos q pffiffiffiffiffi z161:57+ : 10 10 ! u $! v 39 Since cos q ¼ ! ! , k u kk v k hÀ1; À1i$hÀ2; 2i 2À2 cos q ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ pffiffiffipffiffiffi ¼ pffiffiffiffiffi ¼ 16 1ị2 ỵ 1ị2 2ị2 ỵ 2ị2 p The angle q between the vectors ! u and ! v is q ẳ cos1 0ị ẳ Thus, the vectors ! u and ! v are orthogonal ! u $! v 40 Since cos q ¼ ! ! , k u kk v k h0; 1i$h1; À2i 0À2 À2 cos q ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ pffiffiffipffiffiffi ¼ pffiffiffi 5 0ị2 ỵ 1ị2 1ị2 ỵ 2ị2   q ¼ cosÀ1 pffiffiffi ¼ 153:43+ neither ! u $! v 41 Since cos q ¼ ! ! , k u kk v k h5; 0i$h3; 3i 15 ỵ 15 cos q ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ pffiffiffiffiffipffiffiffiffiffi ¼ pffiffiffiffiffi ¼ pffiffiffiffiffi ¼ pffiffiffi 2 2 18 18 25 18 5ị ỵ 0ị 3ị ỵ 3ị   q ẳ cos1 p ẳ 45 neither ! u $! v 42 Since cos q ¼ ! ! , k u kk v k h1; 2i$h2; 1i 2ỵ2 cos q ẳ qq ẳ pp ẳ 2 2 5 1ị ỵ 2ị 2ị ỵ 1ị   ẳ 36:87 q ¼ cosÀ1 neither Appendix C: Solution Manual 383 ! u $! v 43 Since cos q ¼ ! ! , k u kk v k h1=2; À2=3i$h4; 3i 22 cos q ẳ qq ẳ qq ẳ 1=2ị2 þ ðÀ2=3Þ2 ð4Þ2 þ ð3Þ2 ð1=2Þ2 þ ðÀ2=3Þ2 ð4Þ2 þ ð3Þ2 p The angle q between the vectors ! u and ! v is q ẳ cos1 0ị ẳ Thus, the vectors ! u and ! v are orthogonal ! u $! v 44 Since cos q ¼ ! ! , k u kk v k h2; À4i$hÀ1; 2i À2 À À10 cos q ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ pffiffiffiffiffipffiffiffi ¼ ¼ À1 10 2 2 20 2ị ỵ 4ị 1ị ỵ 2ị The angle q between the vectors ! u and ! v is q ¼ cosÀ1(À1) ¼ p ! ! Thus, the vectors u and v are parallel ! ! 45 W ¼ F $ d ẳ 70ị4ị ẳ 280 ft$lb ! ! ! ! ! ! 46 W ¼ F $ d ¼ ð22 i À j Þ$ð40 i À j ị ẳ 22ị40ị ỵ 5ị8ị ẳ 920 ! ! 47 W ¼ F $ d ¼ h20 cos 30 À 20 sin 30 i$h60; 0i W ¼ 1200 cos 30 ¼ 1039:23 ft$lb ! ! 48 W ¼ F $ d ¼ h80 cos 20 ; 80 sin 20 i$h120; 0i W ¼ 9600 cos 20 ¼ 9021:01 ft$lb 49 F2 50 lbs 60 F1 !! ! ! W ¼  F  d cos q ¼ F $ d W ¼ (50)(90)cos 60 ¼ (50)(90)(0.5) ¼ 2250 ft$lb d   ! ! ¼ 45 with i 50 The force magnitude is  F  ¼ and the q ¼ tanÀ1 pffiffi ! pffiffi ! pffiffiffi ! ! ! ! ! F ¼ 8ðcos 45 i ỵ sin 45 j ị ẳ 22 i ỵ 22 j ẳ i ỵ j ị p ! ! ẳ 32 i ỵ j ị ! ƒ! The line of motion of the object from A ¼ (0,0) to B ¼ (1,0), so AB ¼ i The work W is therefore, pffiffiffiffiffi ! ! ! pffiffiffiffiffi ! ƒ! W ¼ F $AB ¼ 32ð i ỵ j ị$ i ẳ 32 ft$lb 384 Appendix C: Solution Manual Review Exercises (Chapter 8dAnswers) ! ! ! À2 À13 18 ¼ 20 À8 ! CD ẳ ẵ ẳ ẵ7 ! CB ẳ ẵ ẳ ½ À7 22 Š AB ¼ ! ! ! À22 ¼ À5 16 ! ! ! À3 À2 AỵB ẳ ỵ ẳ 6 ! C ỵ D ẳ ẵ / not allowed ! AỵC ¼ ½ À2 Š ¼ not allowed ! ! ! À3 À2 À9 10 3A À 2B ¼ À2 ¼ À 12 " # À3 CA ỵ C ẳ ẵ ỵ ẵ Š 4 AD ¼ À3 4 CA ỵ C ẳ ẵ 18 ỵ ẵ ẳ ½ 16 ! 47 À2 ¼ 10 Ầ1 ¼ 41 35 À8 À4 À3 ! t 11 B ¼ À2 ! 12 IB ¼ 1 13 det A ẳ 3ị0ị 2ị4ị ẳ " 14 det E ẳ " det E ¼ À2 À4 10 14 # " À1 # À À1 # " # " ỵ0 ỵ0 ¼ À1 À2 " # À3 # À4 12 ! ¼ À19 10 10 À12 ! Appendix C: Solution Manual 385 15 Size of A ¼  2; Size of B ¼  2; Size of C ¼  ; Size of D ¼  ; Size of E ¼  16 Metric Metric Metric Metric Metric A is square matrix B is square matrix C is row matrix D is column matrix E is square matrix 17 Size ¼  18 Size ¼  19 Size ¼  20 Size ¼  21 Size ¼  22 Size ¼  23 Size ¼  24 Size ¼  25 Size of A ¼  Size of B ¼  Size of C ¼  2 26 (a) A ỵ C ẳ (b) A À C ¼ (c) (d) (e) (f) 27 AÀ1 3 À1 À1 À1 þ 4 55 ¼ 5; À3 À2 3 À1 À1 À 4 5 ¼ À1 À6 À3 3 À6 À3A ¼ À34 À1 ¼ À9 35 À15 À3 ! À1 ! À3 4 22 11 BC ¼ 55 ¼ 12 À3 ! ! ! À3 0 Bỵ0 ẳ þ ¼ 0 0 ! At ¼ À1 ! 0:6 0:2 ¼ À0:4 0:2 386 Appendix C: Solution Manual 28 x ỵ y ẳ / x ẳ y x ỵ 2y ẳ / y ỵ 2y ẳ / y ẳ x ¼ 10 63 37 7 ẳ ẵ z1 z2 29 zT ẳ ẵ z1 z2 41 45 5 z1 ỵ z2 ¼ z2 ¼ À z1 z1 z2 þ ¼ z1 2z1 4z2 þ ¼ z2 5z1 ỵ 3z2 z2 ẳ z1 / z1 ¼ 10 15 10 z1 ¼ ; z2 ¼ 13 13 " # " # " c1 c 2 30 M ỵ N ẳ ỵ ¼ 1 À2 # À1 c1 À ¼ / c ¼ c2 þ ¼ / c ¼ 31 x ¼ 3; y ¼ À4 32 t ¼ 10, z ẳ 12 33 ỵ x ẳ 10 / x ẳ 10 ỵ x ỵ y ẳ / y ẳ ỵ t ẳ / t ẳ 34 x ẳ 12; 12 ỵ x y ẳ / y ẳ 12 ỵ x À / y ¼ À8 35 x ¼ x ỵ y ẳ / y ẳ z ẳ 5k ỵ ẳ / k ẳ Appendix C: Solution Manual 387 ! ! ! 3y 2x ỵ ẳ z k1 " # ! 6x ỵ ỵ z 9y ỵ k 1 ẳ ẳ 4x ỵ 6y 36 AA1 ¼ AAÀ1 y ¼ x ¼ À   ỵ k ¼ 0/k ¼ À ¼ À 2   ỵ ỵ z ẳ 1/z ¼ À !! ! ! 37 W ¼  F  d cos q ¼ F $ d W ¼ (40)(10) cos 60 ¼ (40)(10)(0.5) ¼ 200 ft$lbs !! ! ! 38 W ¼  F  d cos q ¼ F $ d W ¼ (30)(8) cos 45 ¼ (30)(8)(0.7) ¼ 168 ft$lbs Glossary Common Math Symbols Math symbol Definition ¼ s z > < ! Equal to Not equal to Approximately equal to Greater than Less than Greater than or equal to Less than or equal to Plus (addition) Minus (subtraction) Time or multiply by (multiplication) Divide by or over (quotient) Ratio or proportion Implies If and only if Parenthesis Bracket Braces, indicate set membership Empty set Pi, (3.14.) Exponent, (2.72.) Logarithm Absolute value Square root Such that ỵ , $, * O, / : / () [] {} B p e log jj pffi j Set of natural number is ℕ ¼ f1; 2; 3; :::g: Set of integer numbers is ℤ ¼ f::; À3; À2; À1; 0; 1; 2; 3; :::g: m Set of rational numbers is ℚ ¼ ; m; n ˛ℤ with n s n pffiffiffi pffiffiffi pffiffiffi Note: 2; 3; 7; and p ; ℚ ℝ is set of all numbers ℚ and not rational with their negative and zeros ℂ is set of the complex Numbers written in form a ỵ bi, where a; b ˛ℝ 389 390 Glossary The Greek Alphabet Name Lower case character Upper case character Name Lower case character Upper case character Alpha Beta Gamma Delta Epsilon Zeta Eta Theta Iota Kappa Lambda Mu a b g d ε z h q i k l m A B G D E Z H Q I K L M Nu Xi Omicron Pi Rho Sigma Tau Upsilon Phi Chi Psi Omega n x o p r s s y f c j u N X O P P S T Y F X J U Index Note: Page numbers followed by “f”, “t” and “b” indicates figures, tables and boxes respectively A Absolute values, 84e85 equation, 77 Acute angle, 189 Acute triangle, 210 Addition, 2, 39e40 adding of fractions, 10 of matrices, 224 Additive identity matrix, 222, 226 Additive inverse matrix, 226 Algebra, 37 basic principles addition, 39e40 multiplication, 40e45 example, 38e39 exercises, 298e299 exponent, 45e47 expression, 38 monomials, 49, 52 base, 49 rules for division of, 51 rules for multiplication of, 50 term, 50 polynomial, 52 degree, 52 example, 52e54 types, 52 proportion, 56e59 radicals, 45e47 ratio, 55e56 shortened rules, 38 variation, 59 constant of variation, 59 direct variation, 59b example, 59e61 inverse variation, 61b joint variation, 60 Angle, 151 measurement and triangles, 160e163 exercises, 163e166 types, 189 example, 190e191 exercises, 192e197 right triangle, 191e192 units, 188 example, 188 exercise, 188e189 Arc, 157 Arcfunction, 202 Area, 173 of circular ring, 176 in geometry, 173e177 exercises, 177e178 of rectangle, 173 of sector, 177 of square, 173 of trapezoid, 174 of triangle, 175 Arithmetic numbers, 1e2 decimals, 20 addition, 23 division, 25e26 multiplication, 24e25 rules to rounding off decimals, 22e23 subtraction, 24 exercises, 289e298 operations, adding or subtracting of fractions, 10 addition, dividing fractions, 11e14 division, 5e6 fractions, 6e9 multiplication, 3e5 multiplying fractions, 11 subtraction, 2e3 percents (%), 28e30 Associative property of matrix addition, 226e227 B Base, 49 Binomial, 51e52 C Capacity measurements of liquid, 135e137 exercises, 137e138 metric units, 135t U.S and metric system, 137t U.S customary units, 136t Cartesian coordinates See Rectangular coordinates Celsius (C), 140 Center, 156 Central angle, 190 Chord, 157 Circle, 156 Circumference, 157, 169e170 Close interval, 80 Coefficient, 38 Column matrix, 222 Column vector See Column matrix Common denominator, 10 Commutative property of matrix addition, 226 Complementary angles, 190 Complex numbers, 87e90 example, 88e89 exercises, 90e92 Constant of variation, 59 D Data visualization, 97 Decagon, 155 Decimals, 20 addition, 23 division, 25e26 multiplication, 24e25 rules to rounding off decimals, 22e23 subtraction, 24 Degree, 152 Denominator, Derived units, 143e145 base and supplementary units, 143t derived units with special names, 144t exercises, 145e146 quantities of units having special names, 143t Determinants, 235 example, 238 exercises, 239e241 inverse matrix finding, 236e237 singular matrix, 238 value finding, 238 391 392 Index Diagonal matrix, 222 Diameter, 157 Direct proportion, 59 Direct variation, 59b Distance between points, 99b Distance points, 99e100 Dividing fractions, 11e14 Division, 5e6 Domain, 108b Dot product, 248 example, 249e251 exercises, 254e256 Matlab commands, 248 properties, 249b work, 252e253 E Elements of matrix, 222 Entries of matrix See Elements of matrix Equality of matrices, 224 Equations, 68 absolute value equation, 77 constructing models to solve problems with one variable, 69e71 example, 68 exercises, 77e80 quadratic equations, 72e73 radical equation, 75e76 rational equation, 73e75 with two variables, 71e72 Equilateral polygon, 167 Equilateral triangle, 169 Even and odd functions, 110 Expand commands, 266 Exponent, 45e47 Expression, 38 F Factor commands, 266 Fahrenheit (F), 140 Fractions, 6e9 Functions, 108e119 See also Graphs exercises, 119e121 inverse of, 113e119 vertical line test, 108e113 G Geometry, 149e157 angle measurement and triangles, 160e163 area in, 173e177 exercises, 157e160 perimeter and circumference in, 166e170 volume in, 179e182 Gram, 133 Graphs, 98e105 See also Functions distance points and midpoint, 99e100 exercises, 105 rectangular coordinates, 98e99 straight line equation, 100e105, 100be101b H Half-closed interval, 81 Half-open interval, 81 Hexagon, 154 Horizontal line test, 112b I Improper fraction, 7, 7b Inequality equations, 80e85 linear inequality, 83e85 Inner product See Dot product Inscribed angle, 190 Integer numbers, International System of Units (SI) Prefixes, 129t Intervals, 80e85 Inverse of function, 113e119 proportion, 59 trigonometric functions, 202e203 variation, 61b Irrational numbers, Isosceles triangle, 169 J Joint variation, 60 K Kite, 156 L Law of cosines, 211e212 Law of sines, 210e211 LCD See Lowest common denominator (LCD) Length measurements, 126e129 exercises, 130e132 metric system, 126, 128t U.S customary system, 127e129, 127te128t Like terms, 50 Line, 150 equation passing point, 101b segment, 150 Linear inequality, 83e85 absolute values, 84e85 Lowest common denominator (LCD), 10 Lowest term, M Maple, 225, 261 arithmetic, 265, 265t assignments, 266 default environment, 262f help system, 264f plots with, 268e269 solving equations, 267 started and windows of, 261 symbolic computation, 266 working with output, 267 worksheet mode, 263f Mass, 133 Mass and weight measurements, 133e134 exercises, 134e135 metric system and U.S system of weight, 133t metric units of weight, 133t U.S customary units of weight, 133t Mathematical models, 69 MATLAB, 225, 271 arithmetic operations on arrays, 278t arithmetic operators, 273t default environment, 272f elementary math functions, 274t getting started and windows, 271 in calculations, 271e279 Line Color Types, 280t Line Styles, 279t matrix operations, 275t named constants, 277t plotting, 279e286 Point Styles, 279t software, 271 symbolic computation, 287 windows, 273t Matrices, 222 example, 223e224 Index 393 exercises, 230e234 location of element in matrix, 222 operations, 224 addition of matrices, 224 additive identity matrix, 226 additive inverse matrix, 226 associative property of matrix addition, 226e227 commutative property of matrix addition, 226 equality of matrices, 224 multiplicative identity matrix, 228 multiplicative inverse matrix, 228e229 product of constant and matrix, 227 product of matrices, 227e228 subtraction of matrices, 224e225 symmetric matrix, 230 transpose of matrix, 229e230 size determination, 223 types, 222 Measure, 125 Measurement, 125 capacity measurements of liquid, 135e137 derived units, 143e145 length, 126e129 mass and weight, 133e134 temperature, 140e142 time, 108e119, 138t Meter, 126 Metric system, 126, 128t Metric units of length, 126t Midpoint, 99e100, 100b Minutes, 161 Mixed number, Monomials, 49, 52 See also Polynomial base, 49 rules for division of, 51 rules for multiplication of, 50 term, 50 Multiplication, 3e5, 24e25, 40e45 Multiplicative identity matrix, 228 Multiplicative inverse matrix, 228e229 Multiplying fractions, 11 N Negative angle, 205e206 Negative exponents, 45be46b Negative matrix See Additive inverse matrix Null matrix See Additive identity matrix Number line, 81 Numerator, O Oblique triangle, 210 Obtuse angle, 189 Obtuse triangle, 210 Octagon, 155 One-to-one function, 111, 111b Open interval, 80 Order of operations, 12 Origin, 98 Orthogonal vectors, 250 P Parallel and perpendicular lines, 104 Parallel lines, 153 Parallel vectors, 250e251 Parallelogram, 155, 168 Pentagon, 154 Percents (%), 28e30 Perimeter, 166 Perimeter and circumference in geometry, 166e173 Perpendicular lines, 153 Plane, 151 Plot functions, 276 Plotting, 279 two-dimensional, 279e286 Point, 150 of tangent, 157 Polygon, 154e155 Polynomial, 52 See also Monomials degree, 52 example, 52e54 types, 52 Positive angle, 205e206 Product of constant and matrix, 227 Product of matrices, 227e228 Proper fraction, 7, 7b Proportion, 56e59 Protractor, 160e161 Pythagorean theorem, 191b, 207e208 Q Quadratic equations, 72e73 Quadratic formula, 72 Quadrilateral, 154 R Radicals, 45e47 equation, 75e76 Radius, 157 Range, 108b Ratio, 55e56 Rational equation, 73e75 Rational numbers, Ray, 151 Real numbers, Reciprocal identities, 200 Rectangle, 168, 156 Rectangular coordinates, 98e99 Reference angle, 208e209 Rhombus, 156 Right angle, 189 Right triangle, 191e192 Row matrix, 222 Row vector See Row matrix S Scalar, 242b See also Vector(s) multiplication, 244 Scalar product See Dot product Secant, 157 Second, 161 Sector, 157 Segment, 157 Set, 81 SI See Standard International Metric system (SI) Sides, 151 Simplify command, 266 Singular matrix, 238 Slope of line, 100b, 101 Square, 155, 167 matrix, 222 Standard International Metric system (SI), 125 Standard position, 206 Standard unit vectors, 246e248 Straight angle, 189 Straight line equation, 100e105, 100be101b Subtracting of fractions, 10 Subtraction, 2e3 of matrices, 224e225 Supplementary angles, 190 Symbolic computation, 287 simplifying symbolic expressions, 287 Symbolic math, 287 394 Index Symmetric matrix, 230 Systems analysis, 221 T Tangent, point of, 157 Temperature measurement, 140e142 Terms, 38, 50 Time measurements, 138e140 Transpose of matrix, 229e230 Trapezoid, 155 Triangle, 154 Trigonometric functions, 197 complementary functions, 201 example, 198e199 inverse trigonometric functions, 202e203 reciprocal identities, 200 supplementary functions, 201e202 Trigonometry, 187e188 angles types, 189 example, 190e191 exercises, 192e197 right triangle, 191e192 complementary functions, 201 exercises, 203e205, 212e214 supplementary functions, 201e202 trigonometric functions, 197e203 unit circle, 205e212 units of angles, 188 example, 188 exercise, 188e189 Trigonometry cofunction identities functions See Trigonometryd complementary functions Trinomial, 52 Two-dimensional plotting, 279 axis commands, 283t example, 282 fplot function, 280 Greek characters, 284t Line Color Types, 280t MATLAB plots with graphics, 286t plots in MATLAB, 282e283 U examples, 242e243 operations, 244e246 standard unit vectors, 246e248 zero vector, 243 Vertex, 151 Vertical line test, 108e113 Volume, 179 of cone, 180 of cube, 179 of cylinder, 180 in geometry, 179e183 of pyramid, 181 of rectangular solid, 179 of sphere, 181 U.S customary system, 127e129, 127te128t Unit circle, 205 law of cosines, 211e212 law of sines, 210e211 oblique triangle, 210 Pythagorean theorem, 207 reference angle, 208e209 standard position, 206 unit circle diagram, 206 Unlike terms, 50 W V X Variables, 38 Variation, 59 constant of variation, 59 direct variation, 59b example, 59e61 inverse variation, 61b joint variation, 60 Vector(s), 242, 242b, 274 See also Scalar addition, 244 component form of vector, 243 dot product, 248e251 x-axis, 98 Weight, 133 Weight measurements See Mass and weight measurements Whole numbers, Work (W), 252e253 Y y-axis, 98 y-intercept, 101 Z Zero angle, 189 Zero exponents, 45be46b Zero matrix See Additive identity matrix Zero vector, 243 ... Preface Fundamentals of Technical Mathematics introduces applied mathematics for engineering technologists and technicians Through a simple, engaging approach, the book reviews basic mathematics. .. purpose of this section is to introduce the basic principles/laws of arithmetic such as properties for the Fundamentals of Technical Mathematics http://dx.doi.org/10.1016/B978-0-12-801987-0.00001-0... wire Determine the number of feet of wire that the electrician needs How many joules (J) of work (W) are done when a force of 200 newtons (N) is applied for a distance of 100 meters? If a rider