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13 Ratio and Proportion Thus far, the concepts of ratio and proportion have been covered with respect to the simple idea of similarity. In this chapter, you’ll learn how ratios and proportions can be used to solve practical problems. You’ll also get a first taste of trigonometry, the branch of mathematics that relates lengths to angles. FRACTIONS, PROJECTIONS, EXTREMES, AND MEANS A fraction expresses a ratio or proportion in terms of the relationship between the numerator and denominator. Different numbers can be used to write the same value as a fraction—the same relationship between the numerator and denominator. For example, 1/2 can also be written as 2/4, 3/6, 4/8, or, in general, x/(2x), where x is any positive real number. Proportions can also be denoted by using colons. The foregoing would then be written as 1:2, 2:4, 3:6, 4:8, or, in general, x:2x. One application that sets the scene for later concepts is the use of ratio or proportion relative to projected images, as shown in Fig. 13-1. This concept is sometimes called the aspect ratio. The old-fashioned standard television (TV) screen aspect ratio is 4:3, meaning 4 units horizontally to 3 units vertically. This aspect ratio is still used in many computers today, although some of the newer computer displays have larger aspect ratios. Movie screens have various aspect ratios, particularly now that wide screens have become popular. Figure 13-2 shows several ways that ratios and proportions can be expressed, along with some of the properties of the variables in these expressions. Note especially how cross products work. In an equation containing two ratios, the left-hand numerator and the right-hand denominator are called the extremes, while the left-hand denominator and the right-hand numerator are called the means. In any equation of this form, the product of the extremes is equal to the product of the means. 173 Copyright © 2008, 1999 by The McGraw-Hill Companies, Inc. Click here for terms of use.
174 Mastering Technical Mathematics Figure 13-1 An example of ratio or proportion in a projected image. In this case the aspect ratio is 4:3, regardless of the distance from the projector. Figure 13-2 Several ways in which ratios and proportions can be written. The product of the extremes always equals the product of the means.
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Mastering Technical Mathematics
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Mastering Technical Mathematics Thi
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To Tim, Samuel, and Tony
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For more information about this tit
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Contents ix Real Vs. Imaginary Numb
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Contents xi Navigator’s Polar Vs.
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Contents xiii Law of Large Numbers
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Introduction This book is intended
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Copyright © 2008, 1999 by The McGr
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1From Counting to Addition We’ve
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From Counting to Addition 5 Figure
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From Counting to Addition 13 WEIGHT
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13. In the United States, common li
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2Subtraction Just as addition is co
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Subtraction 21 the hundreds column,
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Subtraction 23 MAKING CHANGE This i
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Subtraction 25 Figure 2-6 Two metho
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3Multiplication Suppose you go into
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Multiplication 29 In Fig. 3-2A, exa
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Multiplication 31 Figure 3-3 An exa
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Multiplication 33 Figure 3-6 It doe
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Multiplication 35 Figure 3-9 Subtra
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Multiplication 37 Figure 3-12 Multi
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Multiplication 39 3. Multiply the f
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4Division Before the days of electr
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Division 43 Figure 4-2 A pictorial
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Division 45 MULTIPLICATION CHECKS D
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Division 47 Figure 4-6 At A, we can
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Division 49 In the example of Fig.
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Division 51 7/10 0.7 17/20 0.85 3
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Division 53 QUESTIONS AND PROBLEMS
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5Fractions Chapter 4 presented some
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Fractions 57 This line of reasoning
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Fractions 59 493/23 21.xxx . . . (
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Fractions 61 The numbers in the ori
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Fractions 63 5/12 25/60 3/10 18/6
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Fractions 65 “limiting values”
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Fractions 67 QUESTIONS AND PROBLEMS
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6Plane Polygons So far, this book h
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Plane Polygons 71 WHAT IS A SQUARE?
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Plane Polygons 73 inches is not the
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Plane Polygons 75 Figure 6-7 Two wa
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Plane Polygons 77 AREA OF AN OBTUSE
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Plane Polygons 79 Figure 6-11 Some
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Plane Polygons 81 QUESTIONS AND PRO
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7Time, Percentages, and Graphs You
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Figure 7-2 A racecar driver can cal
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When a percentage expresses a chang
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Time, Percentages, and Graphs 89 Fi
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Time, Percentages, and Graphs 91 Fi
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5-mile track. Each graph gives you
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Part 2 ALGEBRA, GEOMETRY, AND TRIGO
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8First Notions in Algebra As proble
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First Notions in Algebra 99 Figure
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First Notions in Algebra 103 (a b)
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To recap, here are the rules if you
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4. Suppose x 5 and y 7. Then what
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9Some “School” Algebra In this
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Some “School” Algebra 115 DIMEN
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Some “School” Algebra 117 4 u
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Some “School” Algebra 119 20x
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Some “School” Algebra 121 Figur
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Page 156 and 157: What about the “negative solution
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Page 162 and 163: used to derive square roots to seve
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Page 168 and 169: Some Useful Shortcuts 151 which sim
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Page 214 and 215: Suppose you name the equal pairs of
Page 218 and 219: Copyright © 2008, 1999 by The McGr
Page 220 and 221: 15 Systems of Counting The decimal
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Page 226 and 227: Systems of Counting 209 DECIMAL BIN
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Progressions, Permutations, and Com
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17 Introduction to Derivatives The
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Figure 17-2 The derivative of the f
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is negative when x is negative, bec
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This rule holds true for all expres
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Introduction to Derivatives 243 CIR
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Introduction to Derivatives 245 Fig
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18 More about Differentiation The p
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More about Differentiation 253 Figu
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More about Differentiation 255 by t
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More about Differentiation 257 MULT
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In the powers table, 0 n is always
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DERIVATIVE OF FUNCTION MULTIPLIED B
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More about Differentiation 263 QUES
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19 Introduction to Integrals Now th
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Introduction to Integrals 267 Figur
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(it subtracts from itself), so you
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Introduction to Integrals 275 VOLUM
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1. Find the derivatives of the prod
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20Combining Calculus with Other Too
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plotting nine points, but to get a
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In a situation like this, you have
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Combining Calculus with Other Tools
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These principal axes can be regarde
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e reaches 1, the ellipse “breaks
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etween the plane and the double con
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21Alternative Coordinate Systems Th
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Alternative Coordinate Systems 303
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perpendicular to the other two. The
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22Complex Numbers Long ago, humanki
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Multiplying repeatedly by i works t
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Complex Numbers 327 Figure 22-4 The
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Complex Numbers 329 When you cube i
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Complex Numbers 331 Figure 22-9 The
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Complex Numbers 333 Figure 22-11 At
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Complex Numbers 335 QUESTIONS AND P
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Part 4 TOOLS OF APPLIED MATHEMATICS
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23Trigonometry in the Real World Tr
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The absolute accuracy (in fixed uni
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Trigonometry in the Real World 343
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objects, even when they are observe
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Now find the distance PV by applyin
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7. What is the range of the aircraf
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24Logarithms and Exponentials A log
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Logarithms and Exponentials 361 Fig
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Now try an example in which neither
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Let’s put a number into this form
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Logarithms and Exponentials 369 10
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Suppose x is some real number. The
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To demonstrate this, let b e, x 2
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“Inv” key followed by a “ln
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and so on. A decrease of 1 order of
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25Scientific Notation Tutorial In s
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Another alternative multiplication
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line that covers 3 orders of magnit
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Scientific Notation Tutorial 387 (3
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Scientific Notation Tutorial 389 (5
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equality symbols. It is universally
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Sometimes, the outcome of determini
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26Vectors You’ve already seen a f
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Vectors 397 MULTIPLICATION OF A VEC
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Vectors 399 The values of the angle
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Vectors 401 If a is multiplied by a
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Vectors 403 Figure 26-5 Direction a
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Vectors 405 This formula works in a
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Vectors 407 You are told that these
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Vectors 409 4. Consider the two vec
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27Logic and Truth Tables To gain a
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are true. If any one of the compone
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antecedent is (P & C). The “impli
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Logic and Truth Tables 417 X Y X &
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Logic and Truth Tables 419 (X 1 Y)
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Logic and Truth Tables 421 The same
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classes—a certain way that multip
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Table 27-9B Derivation of truth val
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Logic and Truth Tables 427 A X Y X
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Logic and Truth Tables 429 X Y Z X
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Logic and Truth Tables 431 X Y Z X
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28Beyond Three Dimensions As you ha
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Beyond Three Dimensions 435 Figure
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a little later! It “lives” for
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Beyond Three Dimensions 441 IMPOSSI
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Beyond Three Dimensions 443 Displac
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Beyond Three Dimensions 449 NO PARA
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such as the pyramid, cube, and sphe
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2. If there are no pairs of paralle
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29A Statistics Primer Statistics is
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A Statistics Primer 457 Figure 29-2
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A Statistics Primer 459 PARAMETER A
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A Statistics Primer 463 of dice, th
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To determine µ sc , check the tabl
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A Statistics Primer 471 making it m
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30Probability Basics Probability is
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Probability Basics 475 Event 1 Even
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Probability Basics 477 As with math
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Probability Basics 479 that both co
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Probability Basics 481 The notion o
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Probability Basics 483 To calculate
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Probability Basics 485 • 30 peopl
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Probability Basics 487 The resultin
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Probability Basics 489 Because the
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Probability Basics 491 Figure 30-11
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Probability Basics 493 views are re
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Final Exam Do not refer to the text
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Final Exam 497 (a) An even number t
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Final Exam 499 X Y Z ? F F F F F F
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Final Exam 501 (c) A harmonic progr
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Final Exam 503 (c) 1.8 10 6 km (d)
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Final Exam 505 47. Suppose an item
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Final Exam 507 Figure E-6 Illustrat
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Final Exam 509 (a) p q (b) p q 1
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Final Exam 511 (d) Roots of w (e) P
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Final Exam 513 82. Which of the fol
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Final Exam 515 (b) It decreases in
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Final Exam 517 95. In the situation
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Appendix 1 Solutions to End-of-Chap
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Solutions to End-of-Chapter Questio
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Figure A-30 Illustration for soluti
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Appendix 2 Answers to Final Exam Qu
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Appendix 3 Table of Derivatives Let
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Appendix 4 Table of Indefinite Inte
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Table of Indefinite Integrals 609 f
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Suggested Additional Reading Bluman
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Index absolute accuracy, 341 error,
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cosines, law of, 347-348 counter, m
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geometry elliptic, 449 Euclidean, 4
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exponential, 369-371 frequency, 255
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esonance, in system, 168-170, 255 r
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direction angles, 402-403 direction
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