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Applied Mathematics FOURTH EDITION For MATH 104B at the College of Southern Nevada Written by Jim Matovina and Ronald Yates, with input and assistance from several members of the CSN Math Department. APPLIED MATHEMATICS FOURTH EDITION Jim Matovina Math Professor College of Southern Nevada [email protected] Ronald Yates, PhD Math Professor College of Southern Nevada [email protected] Copyright 2011-2021 by Jim Matovina and Ronald Yates, All Rights Reserved Version 1: April 13, 2020 Version 2: September 1, 2020 Version 3: July 18, 2021 Contents I TABLE OF CONTENTS PREFACE VII MATH 104B COURSE DESCRIPTION & OUTCOMES VII COURSE DESCRIPTION VII COURSE OUTCOMES VII TO THE INSTRUCTOR VII TO THE STUDENT VIII NEW TO THE FOURTH EDITION VIII ACKNOWLEDGEMENTS IX CHAPTER 1: ARITHMETIC & PREALGEBRA 1 CHAPTER OUTLINE AND OBJECTIVES 1 1.1: ROUNDING, DECIMALS & ORDER OF OPERATIONS 2 PLACE VALUES 2 ESTIMATION VS. ROUNDING 3 ESTIMATION 3 ROUNDING 3 DON’T GET TRAPPED MEMORIZING INSTRUCTIONS 5 WHEN TO ROUND AND WHEN NOT TO ROUND 6 ADDING AND SUBTRACTING DECIMALS 7 MULTIPLYING DECIMALS 7 DIVIDING DECIMALS 7 ORDER OF OPERATIONS 8 ROUND ONLY AS THE LAST STEP 11 USING YOUR CALCULATOR 11 SECTION 1.1 EXERCISES 13 SECTION 1.1 EXERCISE SOLUTIONS 17 1.2: BINARY & HEXADECIMAL NUMBERS 19 BINARY NUMBERS 19 CONVERTING BINARY NUMBERS TO DECIMAL NUMBERS 19 CONVERTING DECIMAL NUMBERS TO BINARY NUMBERS 21 HEXADECIMAL NUMBERS 23 CONVERTING HEXADECIMAL NUMBERS TO DECIMAL NUMBERS 23 CONVERTING DECIMALS NUMBERS TO HEXADECIMAL NUMBERS 24 USING YOUR CALCULATOR 25 SECTION 1.2 EXERCISES 26 SECTION 1.2 EXERCISE SOLUTIONS 30 1.3: EXPRESSIONS & EQUATIONS 32 VARIABLES, TERMS & EXPRESSIONS 32 Applied Mathematics, Fourth Edition II Contents SIMPLIFYING EXPRESSIONS 32 EQUATIONS AND SOLUTIONS 33 THE ADDITION PROPERTY OF EQUALITY 33 THE MULTIPLICATION PROPERTY OF EQUALITY 35 USING THE ADDITION AND MULTIPLICATION PROPERTIES TOGETHER 37 ELIMINATING MISTAKES 38 USING YOUR CALCULATOR 39 SECTION 1.3 EXERCISES 41 SECTION 1.3 EXERCISE SOLUTIONS 47 1.4: FAIR DIVISION 49 THE DIVIDER-CHOOSER METHOD 49 EQUAL DIVISION OF AN ITEM THAT CAN BE LIQUIDATED 49 THE METHOD OF SEALED BIDS 50 MULTIPLYING BY A FRACTION 50 SECTION 1.4 EXERCISES 55 SECTION 1.4 EXERCISE SOLUTIONS 59 1.5: PROBLEM SOLVING 61 FROM THE WIZARD OF OZ 61 THE PROBLEM-SOLVING PROCESS 61 PROBLEM SOLVING STRATEGY: GUESS AND TEST 63 PROBLEM SOLVING STRATEGY: LOOK FOR A PATTERN 64 PROBLEM SOLVING STRATEGY: USE A VARIABLE 64 PROBLEM SOLVING STRATEGY: MAKE A LIST 65 PROBLEM SOLVING STRATEGY: SOLVE A SIMPLER PROBLEM 65 PROBLEM SOLVING STRATEGY: DRAW A PICTURE 66 PROBLEM SOLVING STRATEGY: WORK BACKWARDS 67 USING YOUR CALCULATOR 67 SECTION 1.5 EXERCISES 69 SECTION 1.5 EXERCISE SOLUTIONS 76 CHAPTER 1 CREDITS 78 CHAPTER 2: GEOMETRY & MEASUREMENT 81 CHAPTER OUTLINE AND OBJECTIVES 81 2.1: THE METRIC SYSTEM & DIMENSIONAL ANALYSIS 82 HISTORICAL MEASUREMENTS 82 THEN ALONG CAME THE FRENCH 83 SO, JUST HOW BIG ARE THESE METRIC UNITS? 83 CONVERTING BETWEEN METRIC UNITS 84 UNIT FRACTIONS 85 DIMENSIONAL ANALYSIS 85 CONVERSION FACTORS 86 USING YOUR CALCULATOR 88 USING AN ONLINE CONVERTER 88 SECTION 2.1 EXERCISES 90 Contents III SECTION 2.1 EXERCISE SOLUTIONS 95 2.2: POLYGONS, CIRCLES & SOLIDS 98 COMMON (AND SOME NOT-SO-COMMON) POLYGON NAMES 98 TYPES OF TRIANGLES AND THEIR PROPERTIES 99 TYPES OF QUADRILATERALS AND THEIR PROPERTIES 100 CIRCLES 101 INDIANA HOUSE BILL #246 102 SOLIDS 102 BOXES 103 PYRAMIDS 103 CYLINDERS 103 POLYGONS ON THE INTERNET 104 SECTION 2.2 EXERCISES 105 SECTION 2.2 EXERCISE SOLUTIONS 109 2.3: PERIMETER, AREA & VOLUME 111 PERIMETER AND AREA OF RECTANGLES 111 PERIMETER AND AREA OF PARALLELOGRAMS 112 PERIMETER AND AREA OF TRIANGLES 113 CIRCUMFERENCE AND AREA OF CIRCLES 114 PERIMETER AND AREA OF COMPOSITE FIGURES 115 VOLUME AND SURFACE AREA OF A BOX 118 VOLUME AND SURFACE AREA OF A CYLINDER 119 USING YOUR CALCULATOR 121 ON THE INTERNET 121 SECTION 2.3 EXERCISES 122 SECTION 2.3 EXERCISE SOLUTIONS 128 2.4: RATIOS, RATES & PROPORTIONS 130 RATIOS VS. RATES 130 SPEED AND MILEAGE 130 UNIT PRICING 131 PROPORTIONS 132 SIMILAR TRIANGLES 133 SECTION 2.4 EXERCISES 136 SECTION 2.4 EXERCISE SOLUTIONS 140 2.5: RIGHT TRIANGLES, DISTANCE & SLOPE 141 RIGHT TRIANGLES 141 SQUARE ROOTS 141 THE PYTHAGOREAN THEOREM 141 WHAT GOOD IS THE PYTHAGOREAN THEOREM, ANYWAY? 143 DISTANCE 143 SLOPE 144 POSITIVE SLOPE, NEGATIVE SLOPE, ZERO SLOPE AND UNDEFINED SLOPE 144 APPLICATIONS OF SLOPE 145 SECTION 2.5 EXERCISES 146 Applied Mathematics, Fourth Edition IV Contents SECTION 2.5 EXERCISE SOLUTIONS 149 CHAPTER 2 CREDITS 150 CHAPTER 3: CONSUMER MATH 153 CHAPTER OUTLINE AND OBJECTIVES 153 3.1: PERCENT 154 IDENTIFYING THE PARTS IN A PERCENT PROBLEM 154 SOLVING PERCENT PROBLEMS USING AN EQUATION 154 FROM A DIFFERENT POINT OF VIEW: WHAT, IS, AND OF 155 GENERAL APPLIED PROBLEMS INVOLVING PERCENTS 156 COMMISSIONS AND TIPS 156 SALES TAX AND DISCOUNTS 157 BE SURE TO FOLLOW THE INSTRUCTIONS 158 SECTION 3.1 EXERCISES 159 SECTION 3.1 EXERCISE SOLUTIONS 162 3.2: SIMPLE & COMPOUND INTEREST 164 SIMPLE INTEREST 164 COMPOUND INTEREST 166 USING YOUR CALCULATOR 167 SECTION 3.2 EXERCISES 169 SECTION 3.2 EXERCISE SOLUTIONS 173 3.3: INSTALLMENT BUYING & BUDGETS 175 INSTALLMENT BUYING 175 WORTH REPEATING 175 SIMPLE FINANCE CHARGES 175 CALCULATING MONTHLY PAYMENTS 176 MAKING A BUDGET 177 WORTH REPEATING… AGAIN 177 BASIC BUDGET CATEGORIES 177 USING A SPREADSHEET/CALCULATOR 179 PAYDAY LOANS 179 SECTION 3.3 EXERCISES 181 SECTION 3.3 EXERCISE SOLUTIONS 184 3.4: LEASES 185 GENERAL LEASE INFORMATION 185 RENTAL AGREEMENTS 185 WRITING A LEASE AGREEMENT 186 WORTH REPEATING… YET AGAIN 187 PROPERTY MANAGEMENT COMPANIES 187 LEASING A CAR OR EQUIPMENT 188 SECTION 3.4 EXERCISES 191 ANSWERS TO SECTION 3.4 EXERCISES 194 Contents V 3.5: MORTGAGES 195 ‘TIL DEATH DO US PART 195 AMORTIZATION 195 USING ONLINE MORTGAGE CALCULATORS 195 WORTH REPEATING… ONE MORE TIME 195 THE COMPLETE MONTHLY PAYMENT 197 HOUSING RECOMMENDATIONS 198 SECTION 3.5 EXERCISES 201 SECTION 3.5 EXERCISE SOLUTIONS 205 CHAPTER 3 CREDITS 206 CHAPTER 4: PROBABILITY & STATISTICS 207 CHAPTER OUTLINE AND OBJECTIVES 207 4.1: COUNTING, FACTORIALS, PERMUTATIONS & COMBINATIONS 208 COUNTING 208 FACTORIALS 209 PERMUTATIONS 211 COMBINATIONS 212 THE DIFFERENCE 213 USING YOUR CALCULATOR 214 SECTION 4.1 EXERCISES 215 SECTION 4.1 EXERCISE SOLUTIONS 219 4.2: SIMPLE PROBABILITY & ODDS 221 TERMINOLOGY 221 THEORETICAL PROBABILITY 221 RELATIVE FREQUENCY 221 DISCLAIMERS 221 ROLLING A PAIR OF DICE 223 DECK OF CARDS 224 OR VS. AND 225 A FEW IMPORTANT OBSERVATIONS 226 ODDS 227 SECTION 4.2 EXERCISES 229 SECTION 4.2 EXERCISE SOLUTIONS 233 4.3: EXPECTED VALUE & FAIR PRICE 235 IN THE LONG RUN... 235 EXPECTED VALUE 235 RAFFLES 237 FAIR GAMES 238 FAIR PRICE TO PLAY 238 SECTION 4.3 EXERCISES 240 SECTION 4.3 EXERCISE SOLUTIONS 243 4.4: AVERAGES 244 Applied Mathematics, Fourth Edition VI Contents MEAN 244 MEDIAN 244 MODE 246 GRADE POINT AVERAGE 246 USING YOUR CALCULATOR 248 SECTION 4.4 EXERCISES 249 SECTION 4.4 EXERCISE SOLUTIONS 251 4.5: CHARTS & GRAPHS 252 BAR GRAPHS 252 LINE GRAPHS 254 PIE CHARTS 255 GENERATING DATA FROM A PIE CHART 257 WHICH TYPE OF GRAPH IS BEST? 257 INSERTING A CHART IN MS WORD 258 SECTION 4.5 EXERCISES 259 SECTION 4.5 EXERCISE SOLUTIONS 263 CHAPTER 4 CREDITS 266 INDEX 269 Preface VII PREFACE This book represents the culmination of efforts to provide a math course to students seeking to earn Associate of Applied Science (AAS) and/or Associate of General Studies (AGS) degrees at the College of Southern Nevada. The content in the course was assembled specifically on the desires and requests of the various Program Directors and Department Chairs for the departments offering those degrees. The course, MATH 104B is intended to teach fundamental mathematical concepts and every-day uses of mathematics AAS and AGS degree-seeking students may encounter in their respective professions and lives. MATH 104B COURSE DESCRIPTION & OUTCOMES Course Description Emphasizing applications, topics include arithmetic, prealgebra, graphing, geometry, finance, probability & statistics. Course is only applicable for AAS and AGS degrees, and is not transferable for credit. Course Outcomes Upon completion of this class, students should: Be able to participate in a mathematically relevant conversation. Be able to accurately perform elementary arithmetic computations. Be able to correctly simplify simple algebraic expressions in one variable. Recognize and understand various problem-solving techniques. Recognize and understand various topics of basic geometry. Solve problems involved with the topics of consumer math. Recognize and solve elementary probability problems. Recognize and compute elementary statistical calculations. Have an ability to apply and extend these concepts. TO THE INSTRUCTOR MATH 104B does not have any prerequisites and it does not satisfy any prerequisites to other courses. It cannot be applied to a general Associate of Arts (AA) or an Associate of Science (AS), and it will not be transferable to another school in hopes of counting it as a math component for a bachelor’s degree. Thus, we are given a fair amount of flexibility in what and how we teach, and that is reflected in the survey nature of the course. In all the material, you should stress practical applications over symbolic manipulations, but that does not mean you should totally discount the algebra. Help the students learn where, when, why, and how the mathematics will help them in their lives. In general, the primary goal of this class should be to prepare the students for the math they may see on the job and in their lives moving forward, and we should make sure they know how to use the tools at hand to perform any presented calculations. Keep in mind, if an individual is faced with a calculation, the most common course of action would be to pull the phone out of their pocket to use the calculator or check the Internet.
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  • Wythoffian Skeletal Polyhedra

    Wythoffian Skeletal Polyhedra

    Wythoffian Skeletal Polyhedra by Abigail Williams B.S. in Mathematics, Bates College M.S. in Mathematics, Northeastern University A dissertation submitted to The Faculty of the College of Science of Northeastern University in partial fulfillment of the requirements for the degree of Doctor of Philosophy April 14, 2015 Dissertation directed by Egon Schulte Professor of Mathematics Dedication I would like to dedicate this dissertation to my Meme. She has always been my loudest cheerleader and has supported me in all that I have done. Thank you, Meme. ii Abstract of Dissertation Wythoff's construction can be used to generate new polyhedra from the symmetry groups of the regular polyhedra. In this dissertation we examine all polyhedra that can be generated through this construction from the 48 regular polyhedra. We also examine when the construction produces uniform polyhedra and then discuss other methods for finding uniform polyhedra. iii Acknowledgements I would like to start by thanking Professor Schulte for all of the guidance he has provided me over the last few years. He has given me interesting articles to read, provided invaluable commentary on this thesis, had many helpful and insightful discussions with me about my work, and invited me to wonderful conferences. I truly cannot thank him enough for all of his help. I am also very thankful to my committee members for their time and attention. Additionally, I want to thank my family and friends who, for years, have supported me and pretended to care everytime I start talking about math. Finally, I want to thank my husband, Keith.