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Table of Contents CHAPTER 1 CRITICAL THINKING SKILLS 1.1 Inductive Reasoning 1 1.2 Estimation 3 1.3 Problem Solving 5 Review Exercises 12 Chapter Test 14 Group Projects 16 CHAPTER 2 SETS 2.1 Set Concepts 17 2.2 Subsets 19 2.3 Venn Diagrams and Set Operations 21 2.4 Venn Diagrams With Three Sets and Verification of Equality of Sets 2.5 Applications of Sets 37 2.6 Infinite Sets 40 Review Exercises 41 Chapter Test 44 Group Projects 46 CHAPTER 3 LOGIC 3.1 Statements and Logical Connectives 47 3.2 Truth Tables for Negation, Conjunction, and Disjunction 3.3 Truth Tables for the Conditional and Biconditional 56 3.4 Equivalent Statements 64 3.5 Symbolic Arguments 73 3.6 Euler Diagrams and Syllogistic Arguments 80 3.7 Switching Circuits 82 Review Exercises 85 Chapter Test 92 Group Projects 94

CHAPTER 4 SYSTEMS OF NUMERATION 4.1 Additive, Multiplicative, and Ciphered Systems of Numeration 4.2 Place-Value or Positional-Value Numeration Systems 98 4.3 Other Bases 101 4.4 Computation in Other Bases 108 4.5 Early Computational Methods 111 Review Exercises 115 Chapter Test 119 Group Projects 121

CHAPTER 5 NUMBER THEORY AND THE REAL NUMBER SYSTEM 5.1 Number Theory 123 5.2 The Integers 127 5.3 The Rational Numbers 129 5.4 The Irrational Numbers and the Real Number System 135 5.5 Real Numbers and Their Properties 138 5.6 Rules of Exponents and Scientific Notation 140 5.7 Arithmetic and Geometric Sequences 144 5.8 Fibonacci Sequence 148 Review Exercises 150 Chapter Test 153 Group Projects 153

CHAPTER 6 ALGEBRA, GRAPHS, AND FUNCTIONS 6.1 Order of Operations 155 6.2 Linear Equations in One Variable 157 6.3 Formulas 166 6.4 Applications of Linear Equations in One Variable 176 6.5 Variation 182 6.6 Linear Inequalities 187 6.7 Graphing Linear Equations 193 6.8 Linear Inequalities in Two Variables 203 6.9 Solving Quadratic Equations by Using Factoring and by Using the Quadratic Formula 209 6.10 Functions and Their Graphs 215 Review Exercises 226 Chapter Test 238 Group Projects 241 CHAPTER 7 SYSTEMS OF LINEAR EQUATIONS AND INEQUALITIES 7.1 Systems of Linear Equations 243 7.2 Solving Systems of Equations by the Substitution and Addition Methods 250 7.3 Matrices 257 7.4 Solving Systems of Equations by Using Matrices 265 7.5 Systems of Linear Inequalities 268 7.6 Linear Programming 271 Review Exercises 275 Chapter Test 280 Group Projects 283 CHAPTER 8 THE METRIC SYSTEM 8.1 Basic Terms and Conversions Within the Metric System 285 8.2 Length, Area, and Volume 286 8.3 Mass and Temperature 288 8.4 Dimensional Analysis and Conversions to and from the Metric System Review Exercises 295 Chapter Test 297 Group Projects 298 CHAPTER 9 GEOMETRY 9.1 Points, Lines, Planes, and Angles 301 9.2 Polygons 305 9.3 Perimeter and Area 310 9.4 Volume and Surface Area 314 9.5 Transformational Geometry, Symmetry, and Tessellations 9.6 Topology 322 9.7 Non-Euclidean Geometry and Fractal Geometry 323 Review Exercises 325 Chapter Test 329 Group Projects 330 CHAPTER 10 MATHEMATICAL SYSTEMS 10.1 Groups 331 10.2 Finite Mathematical Systems 332 10.3 Modular Arithmetic 336 Review Exercises 340 Chapter Test 343 Group Projects 345

318

290

CHAPTER 11 CONSUMER MATHEMATICS 11.1 Percent 347 11.2 Personal Loans and Simple Interest 350 11.3 Compound Interest 353 11.4 Installment Buying 356 11.5 Buying a House with a Mortgage 363 11.6 Ordinary Annuities, Sinking Funds, and Retirement Investments 367 Review Exercises 368 Chapter Test 372 Group Projects 373 CHAPTER 12 PROBABILITY 12.1 The Nature of Probability 375 12.2 Theoretical Probability 376 12.3 Odds 379 12.4 Expected Value (Expectation) 382 12.5 Tree Diagrams 386 12.6 Or and And Problems 390 12.7 Conditional Probability 394 12.8 The Counting Principle and Permutations 397 12.9 Combinations 399 12.10 Solving Probability Problems by Using Combinations 12.11 Binomial Probability Formula 406 Review Exercises 407 Chapter Test 410 Group Projects 412 CHAPTER 13 STATISTICS 13.1 Sampling Techniques 413 13.2 The Misuses of Statistics 413 13.3 Frequency Distributions and Statistical Graphs 13.4 Measures of Central Tendency 423 13.5 Measures of Dispersion 427 13.6 The Normal Curve 433 13.7 Linear Correlation and Regression 439 Review Exercises 450 Chapter Test 455 Group Projects 457 CHAPTER 14 GRAPH THEORY 14.1 Graphs, Paths, and Circuits 459 14.2 Euler Paths and Euler Circuits 462 14.3 Hamilton Paths and Hamilton Circuits 14.4 Trees 472 Review Exercises 483 Chapter Test 488 Group Projects 489

415

466

CHAPTER 15 VOTING AND APPORTIONMENT 15.1 Voting Systems 491 15.2 Flaws of Voting 496 15.3 Apportionment Methods 500 15.4 Flaws of Apportionment Methods 506 Review Exercises 510 Chapter Test 514

402

CHAPTER ONE CRITICAL THINKING SKILLS Exercise Set 1.1 1. Counting 2. Divisible 3. Hypothesis 4. Counterexample 5. Inductive 6. Deductive 7. Deductive 8. Inductive 9. Inductive reasoning, because a general conclusion was made from observation of specific cases. 10. Inductive reasoning, because a general conclusion was made from observation of specific cases. 11.

5×5 = 25

12.

12×14 = 168

13.

1 5 (= 1 + 4) 10 (= 4 + 6) 10 (= 6 + 4) 5(= 4 + 1) 1

14.

100,000 = 105

15.

16.

17.

18.

19. 21.

23. 25.

10, 12, 14 (Add 2 to previous number.) 3, −3, 3 (Alternate 3 and −3.)

20.

19, 23, 27 (Add 4 to the previous number.)

22.

−3, −5, −7 (Subtract 2 from previous number.) 2500, −12,500, 62,500 (Multiply previous number by –5.)

24.

1 1 1 , , (Increase the denominator value by 1.) 5 6 7 36, 49, 64 (The numbers in the sequence are the squares of the counting numbers.)

26.

21, 28, 36 (15 + 6 = 21, 21 + 7

= 28, 28 + 8 = 36)

CHAPTER 1 Critical Thinking Skills

27.

34, 55, 89 (Each number in the sequence is the sum of the previous two numbers.)

29. There are three letters in the pattern. 39× 3 = 117 , so the 117th entry is the second R in the pattern. Therefore, the 118th entry is Y.

31.

a) 36, 49, 64 b) Square the numbers 6, 7, 8, 9 and 10. c) 8×8 = 64 9 × 9 = 81 72 is not a square number since it falls between the two square numbers 64 and 81.

33. Blue: 1, 5, 7, 10, 12

Purple: 2, 4, 6, 9, 11

28.

243 729 2187 ,− , 256 1024 4096

(Multiply previous number 3 by − .) 4

30.

a) Answers will vary. b) The sum of the digits is 9. c) When a one- or two-digit number is multiplied by 9, repeated summing of the digits in the product yields the number 9. 32. a) 28 and 36 b) To find the 7th triangular number, add 7 to the 6th triangular number. To find the 8th triangular number, add 8 to the 7th triangular number. To find the 9th triangular number, add 9 to the 8th triangular number. To find the 10th triangular number, add 10 to the 9th triangular number. To find the 11th triangular number, add 11 to the 10th triangular number. c) 36 + 9 = 45; 45 + 10 = 55; 55 + 11 = 66; 66 + 12 = 78 72 is not a triangular number since it falls between the consecutive triangular numbers 66 and 78. Yellow: 3, 8

34. a) 19 (Each new row has two additional triangles.) b) 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 = 100 35. a) ≈ $200, 000 b) We are using observation of specific cases to make a prediction. 36. a) ≈ $3.7 trillion

b) We are using observation of specific cases to make a prediction. 38.

37.

39. a) You should obtain the original number. b) You should obtain the original number. c) Conjecture: The result is always the original number. 4n + 12 4n 12 = + = n + 3, n + 3 − 3 = n 4 4 4 40. a) You should obtain twice the original number. b) You should obtain twice the original number. c) Conjecture: The result is always twice the original number. d) n, 4n, 4n + 12,

d) n, 4n, 4n + 6,

4n + 6 4n 6 = + = 2n + 3, 2n + 3 − 3 = 2n 2 2 2

SECTION 1.2

41. a) You should obtain the number 5. b) You should obtain the number 5. c) Conjecture: No matter what number is chosen, the result is always the number 5. 2n + 10 2n 10 = + = n + 5, n + 5 − n = 5 d) n, n + 1, n + ( n + 1) = 2n + 1, 2n + 1 + 9 = 2n + 10, 2 2 2 42. a) You should obtain the number 0. b) You should obtain the number 0. c) Conjecture: No matter what number is chosen, the result is always the number 0. n + 10 ⎛⎜ n + 10 ⎞⎟ , 5⎜ = n + 10, n + 10 − 10 = n, n − n = 0 d) n, n + 10, ⎜⎝ 5 ⎠⎟⎟ 5

43. 7 − 5 = 2 is one counterexample. 44. 5 ÷ 2 = 2 12 , which is not a counting number.

5 , which is not an even number. 2 46. 900 is a three-digit number. The product of 900 and 900 is 810,000, which is not a five-digit number. 47. One and two are counting numbers. The difference of 1 and 2 is 1− 2 = −1 , which is not a counting number. 48. The sum of the odd numbers 1 and 5 is 6, which is not divisible by 4. 49. a) The sum of the measures of the interior angles should be 180° . b) Yes, the sum of the measures of the interior angles should be 180° . c) Conjecture: The sum of the measures of the interior angles of a triangle is 180° . 50. a) The sum of the measures of the interior angles should be 360° . b) Yes, the sum of the measures of the interior angles should be 360° . c) Conjecture: The sum of the measures of the interior angles of a quadrilateral is 360° . a b 51. 129, the numbers in positions are found as follows: c a +b+ c

45. Two is a counting number. The sum of 2 and 3 is 5. Five divided by two is

52. 1881, 8008, 8118 (They look the same when looked at in a mirror.) 53. c

Exercise Set 1.2 (Note: Answers in this section will vary depending on how you round your numbers. The answers may differ from the answers in the back of the textbook. However, your answers should be something near the answers given. All answers are approximate.)

1. 2.

Estimation Equal

261 + 127.4 + 273.9 + 16.2 + 81.5 ≈ 260 + 127 + 274 + 16 + 82 = 759

4. 2.57 + 212.6 +176.2 + 83

198, 600×3.072 ≈ 200, 000×3.000 = 600, 000

1854 ×0.0096 ≈ 1900×0.01 = 19

405 400 ≈ = 8000 0.049 0.05

0.63×1523 ≈ 0.6×1500 = 900

≈ 0 + 210 +180 + 80 = 470

CHAPTER 1 Critical Thinking Skills

10. 51, 608× 6981 ≈ 50, 000× 7000 = 350, 000,000

11% of 8221 ≈ 10% of 8000 = 0.10×8000 = 800

11.

18% ×1576 ≈ 20%×1600 = 0.20×1600 = 320

12.

296.3 ÷ 0.0096 ≈ 300 ÷ 0.01 = 30, 000

13.

$10.49 $10 ≈ = $2 5 5

14.

$37.80 $40 ≈ = $2 20 20

15.

12 months ×$120.80 ≈12×$120 = $1440

16.

8% of $11, 250 ≈ 0.08 × $11, 000 = $880

17.

One third of an annual profit of $8795 1 ≈ × $9, 000 = $3000 3

18.

$1.29 + $6.86 + $12.43 + $25.62 + $8.99 ≈ $1+ $7 + $12 + $26 + $9 = $55

19.

95lb +127 lb + 210 lb ≈100 +100 + 200 = 400 lb

21.

20. 22.

15% of $26.32 ≈ 15% of $26 = 0.15×$26 = $3.9

3.25 lb 3.00 lb ≈ = 0.5 lb 6 6 $400 $400 ≈ = 16 $23 $25

24.

23. ($65.99 + $49.99 + $49.95) − $114.99 ≈ ($66 + $50 + $50) − $115 = $166 − $115 = $5

Team A: 189 + 172 + 191 ≈ 190 + 170 + 190 = 550 Team B: 183 + 229 + 167 ≈ 180 + 230 + 170 = 580 580 − 550 = 30 lb

25.

11 × 8 × $1.50 ≈ 10 × 8 × $1.50 = 10 × $12 = $120

26.

6 min, 25 sec×26.2 mi ≈ 6.5 min × 26 mi =169 min 169 min ≈ 3hours 60 min

27. 100 Mexican pesos = 100× 0.083 U.S. dollars

28. $973 + 6 ($61) + 6 ($97) + 6 ($200)

≈ 100× 0.08 U.S. dollars = 8 U.S. dollars $50 − $8 = $42

29. ≈ 60 miles

≈ $970 + 6 ($60) + 6 ($100) + 6 ($200) = $970 + $360 + $600 + $1200 = $3130

30.

≈ 55 miles

SECTION 1.3

32.

a) 100 b) 50 c) 125

33. a) 5 million b) 98 million c) 98 million − 33 million = 65 million d) 19 million + 79 million + 84 million + 65 million + 33 million = 280 million

34.

a) 19% b) 25% c) 20% of 179 lb ≈ 20% of 180 = 0.2 ×180 = 36 lb

35. a) 85% b) 68% − 53% = 15% c) 85% of 70 million acres = 59,500, 000 acres

36.

a) 2 ( 410) + 4 (545)

31. a) 33% of 700 ≈ 30% of 700 = 0.30×700 = 210 b) 9% of 700 ≈10% of 700 = 0.10×700 = 70 c) 24% of 700 ≈ 25% of 700 = 0.25×700 =175

≈ 2 (400) + 4 (550) = 800 + 2200 = 3000 calories b) Running: 4 (920) ≈ 4 (925) = 3700 calories

d) No, since we are not given the area of each state.

Casual bike riding: 4 (300) = 1200 calories , 3700 −1200 = 2500 calories c) 3(545) + 3(545) ≈ 3(550) + 3(550) = 1650 + 1650 = 3300 calories per week , 3300 calories per week (52 weeks) ≈ 3000×50 = 150,000 calories 38.

37.

39. 41. 43. 45. 47.

≈ 120 bananas 150° 10% 9 square units 150 feet

40. 42. 44. 46.

≈ 160 berries 315° 25% 12 square units

48.

4 (60) = 240 in. or

49.-57. Answers will vary. 59. a) Answers will vary. b) Answers will vary.

58.

There are 118 ridges around the edge.

1 in. 20.1 in. = 2.5 yd x yd

Exercise Set 1.3 1 in. 5.75 in. 1. = 12 mi x mi 1x = 12 (5.75)

1x = 2.5 (20.1)

x = 69 mi

x = 50.25 yd

240 ≈ 20 ft 12

CHAPTER 1 Critical Thinking Skills

3 ft x ft = 1.2 ft 19.36 ft 3(19.36) = 1.2 x

x bags 1 bag = 2 4000 ft 35, 000 ft 2 4000 x = 1(35, 000)

58.08 1.2 x = 1.2 1.2 58.08 = 48.4 ft x= 1.2

4000 x 35, 000 = 4000 4000 35, 000 = 8.75 bags x= 4000

5. 6.4% of $7605 = 0.064 × $7605 = $486.72 $7605 + $486.72 = $8091.72 ≈ $8092

30% of $117 = 0.30 × $117 = $35.10 $117 − $35.10 = $81.90

7. a) Ent./Misc.: 19.4% of $1750 = 0.194 × $1750 8. a) 11 − 20 years: 25% of 6.2 million = 0.25 × 6.2 = 1.55 million = $339.50 0 − 3 years: 16% of 48 million = 0.16 × 6.2 Food: 12.7% of $1750 = 0.127 × $1750 = $222.25 = 0.992 million $339.50 − $222.25 = $117.25 1.55 − 0.992 = 0.558 million = 558,000 b) Housing: 33.9% of $1750 = 0.339 × $1750 b) 4 −10 years: 33% of 6.2 million = 0.33 × 6.2 = $593.25 = 2.046 million Transportation: 17% of $1750 = 0.17 × $1750 11− 20 years: 25% of 6.2 million = 0.25 × 6.2 = $297.50 = 1.55 million $593.25 − $297.50 = $295.75 2.046 − 1.55 = 0.496 million = 496,000

54.46% of $200, 000 = 0.5446 × 200, 000 = $108,920 $200, 000 + $108,920 = $308,920

b) 2.9% of $180, 000 = 0.029 ×180,000 = $5220 $180, 000 + $5220 = $185, 220

10. 40 rides × $2 per ride = $80. In order for the cost of rides with the $81 MetroCard to be less than the cost of the rides without the MetroCard, Chandler would have to take 41 rides per month. $81 ≈ $1.98per ride 41rides

c) Flagstaff, AZ: − 1.85% of $200, 000 = −0.0185 × $200, 000 = − $3700 $200, 000 − $3700 = $196,300 Bellingham, WA: − 1.04% of $200,000 = −0.0104×$200, 000 =− $2080 $200, 000 − $2080 = $197,920 $197,920 − $196,300 = $1620 11.

$120 + $80 (15) = $120 + $1200 = $1320 Savings: $1320 − $1250 = $70

12.

2005: $20 × 2 million = $40 million 2006: $20 + 0.25 × $20 = $25 $25 × 2 million = $50 million 50 − 40 = $10 million or $10,000,000

SECTION 1.3

13.

14. 15 year mortgage: $777.83(12)(15) = $140, 009.4

Points needed for 80 average: 80 (5) = 400 points Wallace’s points so far: 79 + 93 + 91 + 68 = 331 points Grade needed on fifth exam: 400 − 331 = 69

30 year mortgage: $521.65(12)(30) = $187, 794.0 Savings: $187, 794.00 − $140, 009.40 = $47, 784.6

15.

a) 10×10×10×10 = 10, 000

16.

b) 1 in 10,000

17.

38,687.0 mi − 38, 451.4 mi = 235.6 mi

18. a) 40 ×$8.50×52 = $17, 680 b) Each week he makes 40×$8.50 = $340. $1275 = 3.75 weeks $340

235.6 mi ≈ 18.698 ≈ 18.7 mpg 12.6 gal

19.

460 = 9.2 min 50 1550 b) = 62 min 25 1400 c) = 40 min 35 1550 2200 3750 d) + = ≈ 47 min 80 80 80

By mail: ($52.80 + $5.60 + $8.56)× 4 = $66.96× 4 = $267.84 Tire store: $324 + 0.08×$324

$885 − $25 (15) = $885 − $375 = $510 20. $510 =17 hours $30

= $324 + $25.92 = $349.92 Savings: $349.92 − $267.84 = $82.08

21.

15,000 ft − 3000 ft = 12,000 ft decrease in elevation. Temperature increases 2.4° F for every 1000 ft decrease in elevation. 2.4° F×12 = 28.8° F −6° F + 28.8° F = 22.8° F The precipitation at the airport will be snow.

22. a) $620 (0.12) = $74.40 b) $1200 (0.22) = $264 c) The store lost $1200 − $1000 = $200 on the purchase. Store's profit: $264 − $200 = $64

CHAPTER 1 Critical Thinking Skills

24. a) 0.1 cm 3 × 60 sec × 60 min × 24 hr × 365 days

23. Steve and Maureen paid more than $9362.50 but less than $26,687.50, so they paid $9362.50 plus 25% of the amount over $68,000.

= 3,153,600 cm 3 b) 30 cm × 20 cm × 20 cm=12,000 cm 3

$13,365 − $9362.50 = $4002.50

0.1 cm3 × 60 sec × 60 min × 24 hr = 8640 cm3

$4002.5 = $16, 01 The amount over $68,000 was 0.25 $68, 000 + $16, 010 = $84, 010

25. a) 1 oz × 60 min × 24 hr × 365 days = 525,600 oz 525,600 = 4106.25 gal 128 b) 4106.25 ×$11.20 = 4.10625× $11.20 = $45.99 1000

27. a)

20,000 20,000 − ≈ 961.538 − 925.926 20.8 21.6 = 35.612 ≈ 35.61 gal

b) 35.61×$3.00 = $106.83 c) 140,000,000 ×35.61 = 4,985, 400, 000 gal

29. Cost after 1 year: $799 + 0.06($799) = $799 + $47.94 = $846.94 Cost after 2 years: $846.94 + 0.06 ($846.94) = $846.94 + $50.82 = $897.76

12, 000 = 1.38 ≈ 1.4 days 8640

26.

a) Short: $33 × 5 = $165 Long: $18 × 5 = $90 $165 − $90 = $75; Jeff saves $75. b) $6 for first hour, plus 6 × $3 for remaining 3 hours, for a total of $24. c) Short: $6 + 8 × $3 = $30 Long: $18 Long term is cheaper by $12.

28. a) Yes, divide the total amount spent by the amount spent per capita.

$45,592.59 ≈ 301.14 million $151.40

$11, 237.53 ≈ 60.78 million $184.90

30. Value after first year: $1000 + 0.10($1000) = $1000 + $100 = $1100 Value after second year: $1100 − 0.10($1100) = $1100 − $110 = $990

31. After paying the $100 deductible, Yungchen must pay 20% of the cost of x-rays. First x-ray: $100 + 0.20 ($540) = $100 + $108 = $208

$990 is less than the intial investment of $1000. 32. $3000 is the difference between one-fourth of the cost and one-fifth of the cost. 1 1 1 − = ; 20 × $3000 = $60,000. 4 5 20

Second x-ray: 0.20 ($920) = $184 Total: $208 + $184 = $392 ⎛1⎞ 3 salt: 3⎜⎜ ⎟⎟⎟ = tsp ⎜⎝ 8 ⎠ 8

33. a) water/milk: 3(1) = 3 cups Cream of wheat: 3(3) = 9 tbsp =

9 cup (because 16 tbsp = 1 cup) 16

SECTION 1.3

2 + 3.75 5.75 7 = = 2.875 cups = 2 cups 2 2 8 0.25 + 0.5 0.75 3 = = 0.375 tsp = tsp salt: 2 2 8 0.5 + 0.75 1.25 5 5 = = 0.625 cups = cup = (16 tbsp) = 10 tbsp cream of wheat: 2 2 8 8 3 15 4 11 3 c) water/milk: 3 −1 = − = = 2 cups 4 4 4 4 4 1 1 4 1 3 3 3 12 3 9 cream of wheat: − = − = cup = 9 tbsp salt: − = − = tsp 2 8 8 8 8 4 16 16 16 16 d) Differences exist in water/milk because the amount for 4 servings is not twice that for 2 servings. 1 Differences also exist in Cream of Wheat because cup is not twice 3 tbsp. 2 1 b) rice: 1(2) = 2 cups 34. a) rice: ( 4) = 2 cups 2 1 4 16 1 1 9 18 2 1 water: 2 (2) = ( 2) = = 4 = 4 cups water: 1 (4) = ( 4) = = 5 cups 3 3 3 3 4 4 4 4 2 1 1 salt: ( 2) = 1 tsp salt: ( 4) = 1 tsp 4 2 b) water/milk:

butter/margarine: 1(4) = 4 tsp

butter/margarine: 2 ( 2) = 4 tsp

1 1 1 3 4 + 1 = + = = 2 cups 2 2 2 2 2 1 1 4 10 14 2 water: 1 + 3 = + = = 4 cups 3 3 3 3 3 3 1 3 4 salt: + = = 1 tsp 4 4 4 butter/margarine: 1 tsp + 1 tbsp = 1 tsp + 3 tsp = 4 tsp d) rice: 3 −1 = 2 cups 1 24 9 15 3 water: 6 − 2 = − = = 3 cups 4 4 4 4 4 1 1 salt: 1 − = 1 tsp 2 2 butter/margarine: 2 tbsp = 2 (3tsp) = 6 tsp

c) rice:

6 tsp − 2 tsp = 4 tsp e)

Differences exist in water because the amount for 4 servings is not twice that for 2 servings.

$425 − $240 (one box of 20 DVDs) = $185 $185 − $180 (one box of 12 DVDs) = $5 One box of 20 DVDs and one box of 12 DVDs are the maximum number of DVDs that can be purchased. b) $240 + $180 = $420

35. a)

36. Mark will win. 38. 1 ft 3 = 12 in.×12 in.×12 in. = 1728 in.3

37. 1 ft 2 would be 12 in. by 12 in. Thus, 1 ft 2 = 12 in.×12 in. = 144 in.2

CHAPTER 1 Critical Thinking Skills

39. Area of original rectangle = lw

20 ft ×20 ft = 400 ft 2

Area of new rectangle = ( 2l )( 2 w) = 4lw Thus, if the length and width of a rectangle are doubled, the area is 4 times as large. 41. Volume of original cube = lwh Volume of new cube = (2l )(2 w)( 2h ) = 8lwh Thus, if the length, width, and height of a cube are doubled, the volume is 8 times as large or increases eight- fold.

43.

5 ft ×5 ft = 25 ft 2

40.

400 ft 2 =16 squares 25ft 2 42. 11 ft is one-sixth of the pole, so the length is 6 × 11 ft = 66 ft.

10 pieces 1000 pieces = $x $10 1000 x = 10 (10) 1000 x 100 = 1000 1000 100 x= = $0.10 = 10¢ 1000

44. Left side: 1(−6) = −6

Right side: 1(2) = 2

2 (−2) = −4

1(3) = 3

− 6 +−4 = −10

1(6) = 6

45. 3

2 + 3 + 6 = 11 Place it at −1 so the left side would total −10 +−1 = −11 46. 10;2002 , 2112 , 2222 , 2332 , 2442 , 2552 ,

47. a) ( 4× 4) + (3×3) + (2 × 2) + (1×1)

2662 , 2772 , 2882 , 2992

= 16 + 9 + 4 + 1 = 30 b) (7×7) + (6×6) + (5×5) + 30 = 49 + 36 + 25 + 30 = 140 49.

48. a) Place the object, 1 g, and 3 g on one side and 9 g on the other side. b) Place the object, 9 g, and 3 g on one side and 27 g and 1 g on the other side. 50. Eight pieces

51.

52.

53. 8 + 6 + 2 + 4 = 20;3 + 7 + 5 + 1 = 16; 10 +14 +12 + 8 = 44

The sum of the four corner entries is 4 times the number in the center of the middle row.

SECTION 1.3

54. 15,12,33 Multiply the number in the center of the middle row by 3. 56. 35 −15 = 20 cubes

45,36,99 Multiply the number in the center of the middle row by 9. 57. 3× 2×1 = 6 ways

58. Each shakes with four people.

59.

55.

Other answers are possible, but 1 and 8 must appear in the center. 61.

60.

(The diagram shows the number of times each part is used.) 62.

With umbrella policy: Mustang reduced premium: $1648 − $90 = $1558 Focus reduced premium: $1530 − 0.12 ($1530) = $1530 − $183.60 = $1346.40 Total for umbrella policy: $1558 + $1346.40 + $450 = $3354.40 Without umbrella policy: $1648 + $1530 = $3178 Net amount for umbrella policy: $3354.40 − $3178 = $176.40 64. 16 + 16 + 4 + 4 + 4 = 44

Other answers are possible. 63.

Mary is the skier.

65.

Areas of the colored regions are: 1×1, 1×1, 2× 2, 3× 3, 5×5, 8×8, 13×13, 21× 21 ; 1 + 1 + 4 + 9 + 25 + 64 + 169 + 441 = 714 square units

66.

Let x be the amount Samantha had to start. 1 1 After first store: x − x − 20 = x − 20 2 2 ⎞⎟ 1 ⎛⎜ 1 1 After second store: ⎜ x − 20⎟⎟ − 20 = x − 30 ⎠ 2 ⎜⎝ 2 4 This is equal to $0, so the original amount was $120.

67.

Thomas would have opened the box labeled grapes and cherries. Because all the boxes are labeled incorrectly, whichever fruit he pulls from the box of grapes and cherries, will be the only fruit in that box. If he pulled a grape, he labeled the box grape. If he pulled a cherry, he labeled the box cherries. That left two boxes whose original labels were incorrect. Because all labels must be changed, there was only one way for Thomas to assign the two remaining labels.

CHAPTER 1 Critical Thinking Skills

Review Exercises 1. 27, 32, 37 (Add 5 to previous number.)

−48, 96, −192 (Multiply previous number by –2.)

10, 4, − 3 (subtract 1, then 2, then 3, ...)

26, 37, 50 (17 + 9 = 26, 26 + 11 = 37, 37 + 13 = 50) 25, 32, 40 (19 + 6 = 25, 25 + 7 = 32, 32 + 8 = 40) 3 3 3 1 , , (Multiply previous number by .) 8 16 32 2

6. 8.

9. c 10. a) The final number is twice the original number. b) The final number is twice the original number. c) Conjecture: The final number is twice the original number. 10n + 5 10n 5 d) n, 10n, 10n + 5, = + = 2n + 1, 2n + 1 − 1 = 2n 5 5 5 11. This process will always result in an answer of 3. n, n + 5,6 (n + 5) = 6n + 30, 6n + 30 −12 = 6n + 18,

6n + 18 6n 18 3n + 9 3n 9 = + = 3n + 9, = + = n + 3, n + 3− n = 3 2 2 2 3 3 3

12. 12 + 22 = 5,5 is an odd number. Other answers are possible. (Note: Answers for Ex. 13 - 25 will vary depending on how you round your numbers. The answers may differ from the answers in the back of the textbook. However, your answers should be something near the answers given. All answers are approximate.) 13. 14. 215.9 + 128.752 + 3.6 + 861 + 792 ≈ 200 + 100 + 0 + 900 + 800 = 2000 210,302 ×1992 ≈ 210,000× 2000 = 420,000,000

15.

19% of 1025 ≈ 20% of 1000 = 0.20×1000 = 200

16.

17.

52 shovels ×$99.97 ≈ 50×100 = $5000

18.

Answers will vary.

7% of $1999 ≈ 7% of 2000 = 0.07 × 2000 = $140 19.

1.1 mi 1 mi 3 mi ≈ = = 3 mph 22 min 20 min 60 min

21. 5 in. =

20.

$2.49 + $0.79 + $1.89 + $0.10 + $2.19 + $6.75

22.

≈ $2 + $1 + $2 + $0 + $2 + $7 = $14.00 2.35 million − 1.95 million = 0.4 million

24.

13 square units

26.

$50 + $40 (12)= $530 Savings: $530 − $500 = $30

⎛1⎞ 20 in. = 20 ⎜⎜ ⎟⎟⎟ in. = 20 (0.1) mi = 2 mi ⎜⎝ 4 ⎠ 4

23.

2.8 million − 1.8 million = 1.0 million

25.

Length = 1.75 in., 1.75(12.5) = 21.875 ≈ 22 ft Height = 0.625 in., 0.625(12.5) = 7.8125 ≈ 8 ft

REVIEW EXERCISES

27.

4 ($2.69) = $10.76 for four six-packs Savings: $10.76 − $9.60 = $1.16

$445 = $89 5 $510 = $85 Cost per person with 6 people: 6 $89 − $85 = $4 savings

29. Cost per person with 5 people:

31.

10% of $1030 = 0.10 × $1030 = $103

30 x 30 × 24,000 = ; x= = 288 lb 2500 24,000 2500 150 b) = 5 bags, and 5 × 2500 = 12,500 ft 2 30

30. a)

32.

Savings: $721 − $60 = $661

1.5 mg x mg = 10 lb 47 lb 10 x = 47 (1.5) 10 x 70.5 = 10 10 x = 7.05 mg

$5000 − 0.30 ($5000) = $5000 − $1500

34.

9 A.M. Eastern is 6 A.M. Pacific, from 6 A.M. Pacific to 1:35 P.M. Pacific is 7 hr 35 min , 7 hr 35 min − 50 min stop = 6 hr 45 min

36.

a) 5280 ft 1hr 1min 5280 ft × × = ≈1.47 ft/sec 1 60 min 60 sec 3600sec

= $3500 take-home 28% of $3500 = 0.28×$3500 = $980 35.

Taylor: $45 for 8 hours Admar: $8 × 6 hours = $48 $48 − $45 = $3 Taylor Rental is cheaper by $3.

28.

$103× 7 = $721

33.

3 P.M. − 4 hr = 11 A.M. July 26, 11:00 A.M.

b) 55 mi 5280 ft 1hr 1min 290, 400 ft × × × = 1hr 1mi 60 min 60sec 3600sec ≈ 80.67 ft/sec 37. Each figure has an additional two dots. To get the hundredth figure, 97 more figures must be drawn, 97 ( 2) = 194 dots added to the third

38.

figure. Thus, 194 + 7 = 201.

39.

40.

59 min 59 sec Since it doubles every second, the jar was half full 1 second earlier than 1 hour.

41. 6 42. Nothing. Each friend paid $9 for a total of $27; $25 to the hotel, $2 to the clerk. $25 for the room + $3 for each friend + $2 for the clerk = $30 43. Let x = the total weight of the four women

CHAPTER 1 Critical Thinking Skills

x 520 + 180 700 = 130, x = 520, = = 140 lb 4 5 5 44. Yes; 3 quarters and 4 dimes, or 1 half dollar, 1 quarter and 4 dimes, or 1 quarter and 9 dimes. 45. 6 cm × 6 cm × 6 cm = 216 cm 3 46. Place six coins in each pan with one coin off to the side. If it balances, the heavier coin is the one on the side. If the pan does not balance, take the six coins on the heavier side and split them into two groups of three. Select the three heavier coins and weigh two coins. If the pan balances, it is the third coin. If the pan does not balance, you can identify the heavier coin.

n (n + 1)

500 (501)

250,500 = 125, 250 2 2 2 48. 16 blue: 4 green → 8 blue, 2 yellow → 5 blue, 2 white → 3 blue 49. 90: 101, 111, 121, 131, 141, 151, 161, 171, 181, 191, … 50. The fifth figure will be an octagon with sides of equal length. Inside the octagon will be a seven sided figure with each side of equal length. The figure will have one antenna. 51. 61: The sixth figure will have 6 rows of 6 tiles and 5 rows of 5 tiles (6× 6 + 5× 5 = 36 + 25 = 61). 52. Some possible answers are given below. There are other possibilities.

47.

53. a) 2 b) There are 3 choices for the first spot. Once that person is standing, there are 2 choices for the second spot and 1 for the third. Thus, 3× 2×1 = 6 . c) 4× 3× 2 ×1 = 24 d) 5× 4×3× 2 ×1 = 120 e) n (n −1)( n − 2)"1, (or n !), where n = the number of people in line

Chapter Test 1. 27, 33, 39 (Add 6 to previous number.)

1 1 1 1 , , (Multiply previous number by .) 16 32 64 2

3. a) The result is the original number plus 1. b) The result is the original number plus 1. c) Conjecture: The result will always be the original number plus 1. 5n + 10 5n 10 = + = n + 2, n + 2 −1 = n + 1 d) n,5n,5n + 10, 5 5 5 (Note: Answers for #4 - #6 will vary depending on how you round your numbers. The answers may differ from the answers in the back of the textbook. However, your answers should be something near the answers given. All answers are approximate.)

CHAPTER TEST

0.21×82, 000 ≈ 0.2×80, 000 = 16, 000

9 square units

a) 3.5 million b) 0.3 million

10.

$15 ≈ 5.79 $2.59 The maximum number of 6 packs is 5.

175, 000 170, 000 ≈ ≈ 1, 700, 000 0.09 0.1 130 lb ≈ 2.0635 a) 63 in. 2.0635 = 0.032754 63 in. 0.032754×703 ≈ 23.03 b) He is in the at risk range.

$85.99 − $59.99 = $26 9. $26 = 65 additional minutes $0.40 11. 1 cut yields 2 equal pieces. Cut each of these 2 equal pieces to get 4 equal pieces. 3 cuts → 3(2.5 min) = 7.5 min

$15.00 − (5×$2.59) = $15.00 − $12.95 = $2.05 $2.05 = 2.5625 $0.80 Thus, two individual cans can be purchased. Number of cans 6 packs Indiv. cans 5 2 32 4 5 29 3 9 27 2 12 24 1 15 21 0 18 18 The maximum number of cans is 32. 12. 2.5 in. by 1.875 in. ≈ 2.5×15.8 by 1.875×15.8 = 39.5 in. by 29.625 in. ≈ 39.5 in. by 29.6 in. (The actual dimensions are 100.5 cm by 76.5 cm.)

13. $12.75 × 40 = $510 $12.75 × 1.5 × 10 = $191.25 $510 + $191.25 = $701.25 $701.25 − $652.25 = $49.00

14.

15.

Mary drove the first 15 miles at 60 mph which took

15 1 = hr, and the second 15 miles at 30 mph which 60 4

15 1 3 = hr for a total time of hr. If she drove the entire 30 miles at 45 mph, the trip would take 30 2 4 30 2 3 = hr (40 min) which is less than hr (45 min). 45 3 4

took

CHAPTER 1 Critical Thinking Skills

16.

6 lb 1 3 1 = 3; 3× tsp = tsp or 1 tsp 2 lb 2 2 2 1

17.

1 1 tsp = tbsp 2 2

Area of lawn including walkway: (10 + 2)×(12 + 2)=12×14 =168 m 2 Area of lawn only: 10×12 = 120 m 2 Area of walkway: 168 −120 = 48 m 2

18. 243 jelly beans; 260 −17 = 243, 234 + 9 = 243, 274 − 31 = 243 19. a) 3 × $3.99 = $11.97 b) 9 ($1.75×0.75) = 11.8125 ≈ $11.81 c) $11.97 − $11.81 = $0.16 Using the coupon is least expensive by $0.16. 20. 24 (The first position can hold any of four letters, the second any of the three remaining letters, and so on. 4 × 3 × 2 × 1 = 24

Group Projects

$325 ≈ $108.33 3 b) Let x = the amount before tax x + 0.07 x = 325

1. a)

1.07 x 325 = 1.07 1.07 x = 303.7383178 ≈ $303.74 $303.74 = 101.246 ≈ $101.25 3 c) Inductive reasoning - arriving at a general conclusion from specific cases d) Combination set: $62.00 − ($62.00×0.10) = $62.00 − $6.20 = $55.80 Individual sets: 2×$36.00 = $72.00,$72.00 −($72.00× 0.20) = $72.00 − $14.40 = $57.60 Therefore, the combination set is cheaper. e) Combination with tax: $55.80 × 1.07 ≈ $59.71 Individual set with tax: $57.60 × 1.07 ≈ $61.63 $61.63 − $59.71 = $1.92 2. a) – d) Answers will vary. 400 mi ÷ 50 mi hr = 8 hrs, 9 A.M. + 8 hrs = 5 P.M. e) f) – h) Answers will vary. 3. Order 1 2 3 4

Name Ernie Zeke Jed Tex

Apparel holster vest chaps Stetson

CHAPTER TWO SETS Exercise Set 2.1 1. Set 2. Ellipsis 3. Description, Roster form, Set-builder notation 4. Finite 5. Infinite 6. Equal 7. Equivalent 8. Cardinal 9. Empty or null 10. { } ,∅ 11. Universal 12. One-to-one 13. Not well defined, “best” is interpreted differently by different people. 14. Not well defined, “most interesting” is interpreted differently by different people. 15. Well defined, the contents can be clearly determined. 16. Well defined, the contents can be clearly determined. 17. Well defined, the contents can be clearly determined. 18. Not well defined, “most interesting” is interpreted differently by different people. 19. Infinite, the number of elements in the set is not a natural number. 20. Finite, the number of elements in the set is a natural number. 21. Infinite, the number of elements in the set is not a natural number. 22. Infinite, the number of elements in the set is not a natural number. 23. Infinite, the number of elements in the set is not a natural number. 24. Finite, the number of elements in the set is a natural number. ⎧San Marino, Scotland, Serbia, Slovakia,⎪ ⎫ 25. { Maine, Maryland, Massachusetts, Michigan, ⎪ ⎪ 26. ⎪⎨ ⎬ ⎪ ⎪ Minnesota, Misssissippi, Missouri, Montana } Slovenia, Spain, Sweden, Switzerland ⎪ ⎪ ⎩ ⎭

27. 29.

{ 11,12,13,14, … ,177 } B = { 2, 4, 6, 8, …}

28.

C = {4}

30.

{ } or ∅

CHAPTER 2 Sets

31. 33.

{ } or ∅ E = { 14, 15, 16, 17, …, 84}

32. 34.

{Idaho, Oregon } { Alaska, Hawaii }

35. {Metropolitan Museum of Art, Tate Modern, National Gallery of Art, British Museum, Louvre Museum} 36. { Musee d'Art Moderne Prado, Museum of Modern Art } 37. { Museum of Modern Art, Musee d'Art Moderne Prado, Musee d'Orsay } 38.

{ National Gallery, Vatican Museums, Metropolitan Museum of Art, Tate Modern, National Gallery of Art } 39.

{ 2007, 2008 }

40.

{ 2003 }

41.

{ 2004, 2005, 2006, 2007 }

42.

{ } or ∅

43.

B = { x x ∈ N and 6 < x < 15} or

44.

A = { x x ∈ N and x < 10} or

B = { x x ∈ N and 7 ≤ x ≤ 14}

A = { x x ∈ N and x ≤ 9}

45.

C = { x x ∈ N and x is a multiple of 3}

46.

D = { x x ∈ N and x is a multiple of 5}

47.

E = { x x ∈ N and x is odd}

48.

A = { x x is Independence Day}

49.

C = { x x is February}

50.

F = { x x ∈ N and 14 < x < 101} or F = { x x ∈ N and 15 ≤ x ≤ 100}

51. 52. 53. 54. 55. 56. 57. 58.

Set A is the set of natural numbers less than or equal to 7. Set D is the set of natural numbers that are multiples of 3. Set V is the set of vowels in the English alphabet. Set S is the set of the seven dwarfs in Snow White and the Seven Dwarfs. Set T is the set of species of trees. Set E is the set of natural numbers greater than or equal to 4 and less than 11. Set S is the set of seasons. Set B is the set of members of the Beatles.

59.

{ China, India, United States }

60.

{ Pakistan, United Kingdom }

61.

{ Russia, Brazil, Indonesia, Japan, Germany }

62.

{ India, United States }

63.

{ 2008, 2009, 2010, 2011 }

64.

{ 1996, 1997, 1998, 1999 }

65. {2000, 2001, 2002, 2003, 2004, 2005, 2006, 2007}

66.

{2008, 2010}

67.

False; {e} is a set, and not an element of the set.

68.

True; b is an element of the set.

69. 71. 73.

False; h is not an element of the set. False; 3 is an element of the set. True; Titanic is an element of the set.

70. 72. 74.

True; Mickey Mouse is an element of the set. False; the Amazon is a river in South America. False; 2 is an even natural number.

75.

n ( A) = 4

76.

n ( B) = 6

77.

n (C ) = 0

78.

n (D) = 5

79. Both; A and B contain exactly the same elements. 80. Equivalent; both sets contain the same number of elements, 3. 81. Neither; the sets have a different number of elements.

SECTION 2.2

82. Neither; not all cats are Siamese 83. Equivalent; both sets contain the same number of elements, 4. 84. Equivalent; both sets contain the same number of elements, 50. 85. a) Set A is the set of natural numbers greater than 2. Set B is the set of all numbers greater than 2. b) Set A contains only natural numbers. Set B contains other types of numbers, including fractions and decimal numbers. c) A = { 3, 4, 5, 6, …} d) No 86. a) Set A is the set of natural numbers greater than 2 and less than or equal to 5. Set B is the set of numbers greater than 2 and less than or equal to 5. b) Set A contains only natural numbers. Set B contains other types of numbers, including fractions and decimal numbers. c) A = { 3, 4, 5 } d) No; because there are an infinite number of elements between any two elements in set B, we cannot write set B in roster form. 87.

Cardinal; 7 tells how many.

88.

89.

Ordinal; sixteenth tells Lincoln’s relative position.

90.

Ordinal; 25 tells the relative position of the chart. Cardinal; 35 tells how many dollars she spent.

91. Answers will vary. 92. Answers will vary. Examples: the set of people in the class who were born on the moon, the set of automobiles that get 400 miles on a gallon of gas, the set of fish that can talk 93. Answers will vary. 94. Answers will vary. Here are some examples. a) The set of men. The set of actors. The set of people over 12 years old. The set of people with two legs. The set of people who have been in a movie. b) The set of all the people in the world. Exercise Set 2.2 1. Subset 2. Proper

3. 2n , where n is the number of elements in the set. 4. 2n − 1 , where n is the number of elements in the set. 5. True; {book } is a subset of { magazine, newspaper, book } . 6. True; {Italy} is a subset of {Italy, Spain, France, Switzerland, Austria } . 7.

False; McIntosh is not in the second set.

False; pepper is not in the second set.

True; { motorboat, kayak } is a proper subset of

10.

True; {polar bear, tiger, lion } is a proper

{ kayak, fishing boat, sailboat, motorboat } .

subset of {tiger, lion, polar bear, penguin } .

11. 13.

False; no subset is a proper subset of itself. True; Xbox 360 is an element of { PSIII, Wii, Xbox 360 } .

12. 14.

False; no set is a proper subset of itself. True; LaGuardia is an element of { JFK, LaGuardia, Newark } .

15.

False; {swimming} is a set, not an element.

16.

False; { } is a set, not an element.

CHAPTER 2 Sets

17.

True; 5 is not an element of {2, 4,6} .

18.

True; the empty set is a subset of every set, including itself.

19.

True; {red} is a proper subset of

20.

True; {3,5,9} = {3,9,5} .

{red, blue, green } . 21.

False; the set {∅} contains the element ∅ .

22.

True; { } and ∅ each represent the empty set.

23.

False; the set {0} contains the element 0 .

24.

25.

False; 0 is a number and { } is a set.

26.

True; the empty set is a subset of every set, including itself. True; the elements of the set are themselves sets.

27.

B ⊆ A, B ⊂ A

28.

A = B, A ⊆ B , B ⊆ A

29.

A ⊆ B, A ⊂ B

30.

None

31.

B ⊆ A, B ⊂ A

32.

B ⊆ A, B ⊂ A

33.

A = B, A ⊆ B , B ⊆ A

34.

B ⊆ A, B ⊂ A

35.

{ } is the only subset.

36.

{ } , {○}

37.

{ } , {cow } , {horse} , {cow, horse}

38.

{ } , { steak } , { pork } , { chicken} , { steak, pork } , { steak, chicken } , { pork, chicken } , { steak, pork, chicken }

39.a) { } , {a} , {b} , {c} , {d } , {a, b} , {a, c} , {a, d } ,

40.

{b, c} , {b, d } , {c, d } , {a, b, c} , {a, b, d } ,

a) 29 = 2× 2× 2× 2× 2× 2× 2× 2× 2 = 512 subsets b) 29 −1 = 512 −1 = 511 proper subsets

{a, c, d } , {b, c, d } , {a, b, c, d } b) All the sets in part (a) are proper subsets of A except {a, b, c, d } . 41. 43. 45. 47. 49.

False; A could be equal to B . True; every set is a subset of itself. True; ∅ is a proper subset of every set except itself. True; every set is a subset of the universal set. True; ∅ is a proper subset of every set except itself and U ≠ ∅ .

42. 44. 46.

True; every proper subset is a subset. False; no set is a proper subset of itself. True; ∅ is a subset of every set.

48.

False; a set cannot be a proper subset of itself. False; the only subset of ∅ is itself and U ≠∅.

50.

51. True; ∅ is a subset of every set. 52. False; U is not a subset of ∅ . 53. The number of different variations is equal to the number of subsets of {cheese, pepperoni, sausage, onions, green peppers, mushrooms, anchovies, ham} , which is 28 = 2× 2× 2× 2× 2× 2×2× 2 = 256 . 54. The number of different variations of the house is equal to the number of subsets of {deck, jacuzzi, security system, hardwood flooring} , which is 24 = 2× 2× 2× 2 = 16 ..

SECTION 2.3

55. The number of options is equal to the number of subsets of { cucumber, onion, tomato, carrot, green pepper, olive, mushroom } , which is 27 = 2× 2 × 2× 2 × 2× 2 × 2 = 128 56. The number of different variations is equal to the number of subsets of {call waiting, call forwarding, caller identification, three way calling, voice mail, fax line} , which is 26 = 2× 2 × 2× 2× 2× 2 = 64 . 57. E = F since they are both subsets of each other. 58. If there is a one-to-one correspondence between boys and girls, then the sets are equivalent. 59. a) Yes. b) No, c is an element of set D . c) Yes, each element of {a, b} is an element of set D . 60. a) Each person has 2 choices, namely yes or no. 2× 2× 2 × 2 = 16 b) YYYY, YYYN, YYNY, YNYY, NYYY, YYNN, YNYN, YNNY, NYNY, NNYY, NYYN, YNNN, NYNN, NNYN, NNNY, NNNN c) 5 out of 16 61. A one element set has one proper subset, namely the empty set. A one element set has two subsets, namely itself and the empty set. One is one-half of two. Thus, the set must have one element. 62. Yes 63. Yes 64. No

Section 2.3 1. Complement 3. Intersection 5. Cartesian 6. m×n

2. 4.

Union Difference

7. Disjoint 8. Four 9.

10.

CHAPTER 2 Sets

11.

12.

13.

14. Or is generally interpreted to mean union. 15. And is generally interpreted to mean intersection.

16.

17.

18.

n ( A ∪ B ) = n ( A) + n ( B )− n ( A ∩ B )

19.

SECTION 2.3

20.

21. The set of animals in U.S. zoos that are not in the San Diego Zoo 22. The set of U.S. colleges and universities that are not in the state of Mississippi 23. The set of farms in the U.S. that do not produce corn 24. The set of farms in the U.S. that do not produce tomatoes 25. The set of farms in the U.S. that produce corn or tomatoes 26. The set of farms in the U.S. that produce corn and tomatoes 27. The set of farms in the U.S. that produce corn and do not produce tomatoes 28. The set of farms in the U.S. that produce corn or do not produce tomatoes 29. The set of furniture stores in the U.S. that sell mattresses or leather furniture 30. The set of furniture stores in the U.S. that sell mattresses and outdoor furniture 31. The set of furniture stores in the U.S. that do not sell outdoor furniture and sell leather furniture 32. The set of furniture stores in the U.S. that sell mattresses, outdoor furniture, and leather furniture 33. The set of furniture stores in the U.S. that sell mattresses or outdoor furniture or leather furniture 34. The set of furniture stores in the U.S. that do not sell mattresses or do not sell leather furniture 35. A = { b, c, t , w, a, h } 36. B = { a, d , f , g , h, r } 37. A ∩ B = { w, b, c, t , a, h } ∩ { a, h, f , r , d , g } = { a, h } 38. U = { c, w, b, t , a, h, d , f , g , r , p, m, z } 39. A ∪ B = { w, b, c, t , a, h } ∪ { a, h, f , r, d , g } = { w, b, c, t , a, h, f , r , d , g } 40. ( A ∪ B )′ : From #39, A ∪ B = { w, b, c, t , a, h, f , r , d , g }. ( A ∪ B )′ = { w, b, c, t , a, h, f , r , d , g }′ = { p, m, z }

41. A′ ∩ B ′ = { w, c, b, t , a, h }{ a, h, f , r, d , g } ∩ { w, c, b, t , p, m, z } = { p, m, z } 42. ( A ∩ B )′ : From #37, A ∩ B = { a, h }. ( A ∩ B )′ = { a, h }′ = { w, c, b, t , f , r , d , g , p, m, z } 43. A = { L, ∆, @, *, $ } 44. B = { *, $, R, , α } 45. U = { L, ∆, @, *, $, R, , α, ∞, Z, Σ } 46. A ∩ B = {L, ∆,@,*,$} ∩ {*,$, R, , α} = {*,$} 47. A′ ∪ B = {R, , α, ∞, Z, Σ} ∪ {*, $, R, , α} = {R, , α, ∞, Z, Σ, *, $ } 48. A ∪ B ′ = {L, ∆,@,*,$} ∪ {*,$, R, , α}′ = {L, ∆,@,*,$} ∪ {L, ∆,@, ∞, Σ, Z} = {L,∆,@,*,$,∞,Σ,Z} 49. A′ ∩ B = {L, ∆, @, *, $}′ ∩ { *, $, R, , α } = { R, , α, ∞, Z, Σ}∩ { *, $, R, , α } = { R, , α } 50. ( A ∪ B )′ : From the diagram, ( A ∪ B )′ = { ∞, Z, Σ } 51. A ∪ B = { 1, 2, 4, 5, 7 } ∪ { 2, 3, 5, 6 } = { 1, 2, 3, 4, 5, 6, 7 }

CHAPTER 2 Sets

52. A ∩ B = { 1, 2, 4, 5, 7 } ∩ { 2, 3, 5, 6 } = { 2, 5 } 53. B ′ = { 2, 3, 5, 6 }′ = { 1, 4, 7, 8 } 54. A ∪ B ′ = { 1, 2, 4, 5, 7 } ∪ { 2, 3, 5, 6 }′ = { 1, 2, 4, 5, 7 } ∪ { 1, 4, 7, 8 } = { 1, 2, 4, 5, 7, 8 } 55. ( A ∪ B )′ From #51, A ∪ B = { 1, 2, 3, 4, 5, 6, 7 }. ( A ∪ B )′ = { 1, 2, 3, 4, 5, 6, 7 }′ = { 8 } 56. A′ ∩ B ′ = { 1, 2, 4, 5, 7 }′ ∩ { 2, 3, 5, 6 }′ = { 3, 6, 8 } ∩ { 1, 4, 7, 8 } = { 8 } 57. ( A ∪ B )′ ∩ B : From #55, ( A ∪ B )′ = { 8 }. ( A ∪ B )′ ∩ B = { 8 } ∩ { 2, 3, 5, 6 } = { } 58. ( A ∪ B ) ∩ ( A ∪ B )′ = { } (The intersection of a set and its complement is always empty.) 59. ( B ∪ A)′ ∩ ( B ′ ∪ A′): From #55, ( A ∪ B )′ = ( B ∪ A)′ = { 8 }.

⎛ ⎝

⎞ ⎠

( B ∪ A)′ ∩ ( B ′ ∪ A′) = { 8 } ∩ ⎜⎜{ 2, 3, 5, 6 }′ ∪ { 1, 2, 4, 5, 7 }′ ⎟⎟⎟ = { 8 } ∩ ({ 1, 4, 7, 8 } ∪ { 3, 6, 8 }) = { 8 } ∩ { 1, 3, 4, 6, 7, 8 } = { 8 } 60.

A′ ∪ ( A ∩ B ): From #52, A ∩ B = { 2, 5 }. A′ ∪ ( A ∩ B ) = { 1, 2, 4, 5, 7 }′ ∪ { 2, 5 } = { 3, 6, 8 } ∪ { 2, 5 } = { 2, 3, 5, 6, 8 }

61. B ′ = { b, c, d , f , g }′ = { a, e, h, i, j, k } 62. B ∪ C = { b, c, d , f , g } ∪ { a, b, f , i, j } = { a, b, c, d , f , g , i , j } 63.

A ∩ C = { a, c, d , f , g , i } ∩ { a, b, f , i, j } = { a, f , i }

64.

A′ ∪ B ′ : A′ = { b, e, h, j , k } , B ′ = { a, e, h, i, j , k }. A′ ∪ B ′ = { b, e, h, j , k } ∪ { a, e, h, i, j , k } = { a, b, e, h, i, j , k }

65. ( A ∩ C )′ : From #63, A ∩ C = { a, f , i }. ( A ∩ C )′ = { a, f , i }′ = { b, c, d , e, g , h, j , k } 66. ( A ∩ B ) ∪ C = ({ a, c, d , f , g , i } ∩ { b, c, d , f , g }) ∪ { a, b, f , i, j } = { c, d , f , g } ∪ { a, b, f , i, j } = { a, b, c, d , f , g , i, j } 67.

A ∪ (C ∩ B )′ = { a, c, d , f , g , i } ∪ ({ a, b, f , i, j } ∩ { b, c, d , f , g })′ = { a, c, d , f , g , i } ∪ { b, f }′ = { a, c, d , f , g , i } ∪ { a, c, d , e, g , h, i, j, k } = { a, c, d , e, f , g , h, i , j , k }

68.

⎛ ⎞ A ∪ (C ′ ∪ B ′) = { a, c, d , f , g , i } ∪ ⎜⎜{ a, b, f , i, j }′ ∪ { b, c, d , f , g }′ ⎟⎟⎟ ⎝ ⎠ = { a, c, d , f , g , i } ∪ ({ c, d , e, g , h, k } ∪ { a, e, h, i, j, k }) = { a, c, d , f , g , i } ∪ { a, c, d , e, g , h, i, j, k } = { a, c, d , e, f , g , h, i , j , k }

69.

⎛

⎞

( A′ ∪ C ) ∪ ( A ∩ B) = ⎜⎜⎝{ a, c, d , f , g , i }′ ∪ { a, b, f , i, j }⎠⎟⎟⎟ ∪ ({ a, c, d , f , g , i } ∩ { b, c, d , f , g }) = ({ b, e, h, j, k } ∪ { a, b, f , i, j }) ∪ { c, d , f , g } = { a, b, e, f , h, i, j, k } ∪ { c, d , f , g } = { a, b, c, d , e, f , g , h, i, j, k } , or U

SECTION 2.3

70.

(C ∩ B) ∩ ( A′ ∩ B): From #67, C ∩ B = { b, f }. ⎛ ⎝

⎞ ⎠

(C ∩ B) ∩ ( A′ ∩ B) = {b, f } ∩ ⎜⎜{ a, c, d , f , g , i }′ ∩ { b, c, d , f , g }⎟⎟⎟ = { b, f } ∩ ({ b, e, h, j, k } ∩ { b, c, d , f , g }) = { b, f } ∩ { b } = { b } For exercises 71-78: U = { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 } , A = { 1, 2, 4, 6, 9 } , B = { 1, 3, 4, 5, 8} , C = { 4, 5, 9 } 71. A − B = { 1, 2, 4, 6, 9 } − { 1, 3, 4, 5, 8} = { 2, 6, 9 } 72. A − C = { 1, 2, 4, 6, 9 } − { 4, 5, 9 } = { 1, 2, 6 } 73. A − B ′ : This leaves only A ∩ B, which is { 1, 4 } 74. A′ − C = { 3, 5, 7, 8, 10 } − { 4, 5, 9 } = { 3, 7, 8, 10 } 75. ( A − B )′ = { 2, 6, 9 }′ = { 1, 3, 4, 5, 7, 8, 10 } 76. ( A − B )′ − C = { 1, 3, 4, 5, 7, 8, 10 } − { 4, 5, 9 } = { 1, 3, 7, 8,10 } 77. C − A′ = { 4, 5, 9 } − { 3, 5, 7, 8, 10 } = { 4, 9 } 78. (C − A)′ − B = { 5 }′ − { 1, 3, 4, 5, 8 } = { 2, 6, 7, 9, 10 } For exercises 79-84: A = { a, b, c } and B = { 1, 2 } 79. { (a, 1), (a, 2), (b, 1), (b, 2), ( c, 1), (c, 2) } 80.

{ (1, a ), (1, b), (1, c), (2, a ), (2, b), (2, c ) }

81. No; the ordered pairs are not the same. 82. 6 83. 6 84. Yes 85. A ∩ B = { 1, 3, 5, 7, 9 } ∩ { 2, 4, 6, 8 } = { } 86. A ∪ B = { 1, 3, 5, 7, 9 } ∪ { 2, 4, 6, 8 } = { 1, 2, 3, 4, 5, 6, 7, 8, 9} , or U 87. A′ ∪ B = { 1, 3, 5, 7, 9 }′ ∪ { 2, 4, 6, 8 } = { 2, 4, 6, 8 } ∪ { 2, 4, 6, 8 } = { 2, 4, 6, 8 } , or B 88. ( B ∪ C )′ = ({ 2, 4, 6, 8 } ∪ { 1, 2, 3, 4, 5 })′ = { 1, 2, 3, 4, 5, 6, 8 }′ = { 7, 9 } 89.

A ∩ C ′ = { 1, 3, 5, 7, 9 } ∩ { 1, 2, 3, 4, 5 }′ = { 1, 3, 5, 7, 9 } ∩ { 6, 7, 8, 9} = { 7, 9 }

90. A ∩ B ′ = { 1, 3, 5, 7, 9 } ∩ { 2, 4, 6, 8 }′ = { 1, 3, 5, 7, 9} ∩ { 1, 3, 5, 7, 9 } = { 1, 3, 5, 7, 9 } , or A 91. ( B ∩ C )′ = ({ 2, 4, 6, 8 } ∩ { 1, 2, 3, 4, 5 })′ = { 2, 4 }′ = { 1, 3, 5, 6, 7, 8, 9 } 92. ( A ∪ C ) ∩ B = ({ 1, 3, 5, 7, 9 } ∪ { 1, 2, 3, 4, 5 }) ∩ { 2, 4, 6, 8 } = { 1, 2, 3, 4, 5, 7, 9} ∩ { 2, 4, 6, 8 } = { 2, 4 } 93.

⎛

⎞

(C ′ ∪ A) ∩ B = ⎜⎜⎝{ 1, 2, 3, 4, 5 }′ ∪ { 1, 3, 5, 7, 9 }⎠⎟⎟⎟ ∩ { 2, 4, 6, 8 } = ({ 6, 7, 8, 9 } ∪ { 1, 3, 5, 7, 9 }) ∩ { 2, 4, 6, 8 } = { 1, 3, 5, 6, 7, 8, 9 } ∩ { 2, 4, 6, 8 } = { 6, 8 }

94. (C ∩ B ) ∪ A : From #91, C ∩ B = { 2, 4 }. (C ∩ B ) ∪ A = { 2, 4 } ∪ { 1, 3, 5, 7, 9 } = { 1, 2, 3, 4, 5, 7, 9 }

CHAPTER 2 Sets

95. ( A ∩ B )′ ∪ C : From #85, A ∩ B = { }.

( A ∩ B )′ ∪ C = { }′ ∪ { 1, 2, 3, 4, 5 } = { 1, 2, 3, 4, 5, 6, 7, 8, 9 } ∪ { 1, 2, 3, 4, 5 } = { 1, 2, 3, 4, 5, 6, 7, 8, 9 } , or U 96.

⎛

⎞

( A′ ∪ C ) ∩ B = ⎜⎜⎝{ 1, 3, 5, 7, 9 }′ ∪ { 1, 2, 3, 4, 5 }⎠⎟⎟⎟ ∩ { 2, 4, 6, 8 } = ({ 2, 4, 6, 8 } ∪ { 1, 2, 3, 4, 5 }) ∩ { 2, 4, 6, 8 } = { 1, 2, 3, 4, 5, 6, 8 } ∩ { 2, 4, 6, 8 } = { 2, 4, 6, 8 } , or B

97.

⎛

⎞

( A′ ∪ B′) ∩ C = ⎜⎜⎝{ 1, 3, 5, 7, 9 }′ ∪ { 2, 4, 6, 8 }′ ⎠⎟⎟⎟ ∩ { 1, 2, 3, 4, 5 } = ({ 2, 4, 6, 8 } ∪ { 1, 3, 5, 7, 9 }) ∩ { 1, 2, 3, 4, 5 } = { 1, 2, 3, 4, 5, 6, 7, 8, 9 } ∩ { 1, 2, 3, 4, 5 } = { 1, 2, 3, 4, 5 } , or C

98.

( A′ ∩ C ) ∪ ( A ∩ B): From #83, A ∩ B = { }. ⎛

⎞

( A′ ∩ C ) ∪ ( A ∩ B) = ⎜⎜⎝{ 1, 3, 5, 7, 9 }′ ∩ { 1, 2, 3, 4, 5 }⎠⎟⎟⎟ ∪ { } = ({ 2, 4, 6, 8 } ∩ { 1, 2, 3, 4, 5 }) ∪ { } = { 2, 4 } ∪ { } = { 2, 4 } 99. A set and its complement will always be disjoint since the complement of a set is all of the elements in the universal set that are not in the set. Therefore, a set and its complement will have no elements in common. For example, if A ∩ B = { }. n ( A ∩ B ) = 0 100. n ( A ∩ B ) = 0 when A and B are disjoint sets. For example, if U = {1, 2,3, 4,5,6} , A = {1,3} , B = {2, 4} , then A ∩ B = { }. n ( A ∩ B ) = 0 101. Let A = { customers who owned dogs } and B = { customers who owned cats} . n ( A ∪ B ) = n ( A) + n ( B ) − n ( A ∩ B ) = 27 + 38 −16 = 49 102. Let A = {students who sang in the chorus} and B = {students who played in the stage band}. n ( A ∪ B ) = n ( A) + n ( B ) − n ( A ∩ B ) 46 = n ( A) + 30 − 4 46 = n ( A) + 26 20 = n ( A) 103. a) A ∪ B = {a, b, c, d } ∪ {b, d , e, f , g , h} = {a, b, c, d , e, f , g , h} , n ( A ∪ B ) = 8, A ∩ B = {a, b, c, d } ∩ {b, d , e, f , g , h} = {b, d } , n ( A ∩ B ) = 2. n ( A) + n ( B ) − n ( A ∩ B ) = 4 + 6 − 2 = 8 Therefore, n ( A ∪ B ) = n ( A) + n ( B ) − n ( A ∩ B ). b) Answers will vary. c) Elements in the intersection of A and B are counted twice in n ( A) + n ( B ) . 104. A ∩ B ′ defines Region I. A ∩ B defines Region II. A′ ∩ B defines Region III. A′ ∩ B ′ or ( A ∪ B )′ defines Region IV. 105. A ∪ B = { 1, 2, 3, 4, …} ∪ { 4, 8, 12, 16, …} = { 1, 2, 3, 4, …} , or A 106. A ∩ B = { 1, 2, 3, 4, …} ∩ { 4, 8, 12, 16, …} = { 4, 8, 12, 16, …} , or B 107. B ∪ C = { 4, 8, 12, 16, …} ∪ { 2, 4, 6, 8, …} = { 2, 4, 6, 8, …} , or C 108. B ∩ C = { 4, 8, 12, 16, …} ∩ { 2, 4, 6, 8, …} = { 4, 8, 12, 16, …} , or B

SECTION 2.4

109.

A ∩ C = { 1, 2, 3, 4, …} ∩ { 2, 4, 6, 8, …} = { 2, 4, 6, 8, …} , or C

110.

A′ ∩ C = { 1, 2, 3, 4, …}′ ∩ { 2, 4, 6, 8, …} = { 0 } ∩ { 2, 4, 6, 8, …} = { }

111. B ′ ∩ C = { 4, 8, 12, 16, …}′ ∩ { 2, 4, 6, 8, …} = { 0, 1, 2, 3, 5, 6, 7, 9, 10, 11, 13, 14, 15, …} ∩ { 2, 4, 6, 8, …} = { 2, 6, 10, 14, 18, …} 112. ( B ∪ C )′ ∪ C : From #107, B ∪ C = C. ( B ∪ C )′ ∪ C = C ′ ∪ C = { 2, 4, 6, 8, …}′ ∪ { 2, 4, 6, 8, …} = { 0, 1, 2, 3, 4, …} , or U 113.

( A ∩ C ) ∩ B ′ : From #109, A ∩ C = C. ( A ∩ C ) ∩ B ′ = C ∩ B ′. From #111, B ′ ∩ C = C ∩ B ′ = { 2, 6, 10, 14, 18, …}

114.

U ′ ∩ ( A ∪ B ): From #103, A ∪ B = A. U ′ ∩ ( A ∪ B ) = U ′ ∩ A = { } ∩ { 1, 2, 3, 4, …} = { }

115. A ∩ A′ = { }

116. A ∪ A′ = U

117. A ∪ ∅ = A

118. A∩ ∅ = ∅

119. A′ ∪ U = U

120. A ∩ U = A

121. A ∪ U = U

122.

A∩ A = A

123.

If A ∩ B = B , then B ⊆ A .

124. If A ∪ B = B, then A ⊆ B.

125.

If A ∩ B = ∅, then A and B are disjoint sets.

126. If A ∪ B = A, then B ⊆ A .

127.

If A ∩ B = A, then A ⊆ B.

128. If A ∪ B = ∅, then A = ∅ and B = ∅. Therefore, they are equal sets.

Exercise Set 2.4 1. 8 2.a) V b) VI 3.a) A′ ∩ B ′

b) A′ ∪ B ′ 4.Deductive 5. A ' ∩ B' is represented by regions V and VI. If B ∩ C contains 12 elements and region V contains 4 elements, then region VI contains 12 − 4 = 8 elements. 6. A ∩ B is represented by regions II and V. If A ∩ B contains 9 elements and region V contains 4 elements, then region II contains 9 − 4 = 5 elements.

CHAPTER 2 Sets

a) Yes A ∪ B = {1, 4,5} ∪ {1, 4,5} = {1, 4,5} A ∩ B = {1, 4,5} ∩ {1, 4,5} = {1, 4,5} b) No, one specific case cannot be used as proof. c) No, not equal

Set A B A∪ B

A∪ B Regions I, II II, III I, II, III

A∩ B Set A B A∩ B

Regions I, II II, III II

Since the two statements are not represented by the same regions, A ∪ B ≠ A ∩ B for all sets A and B . 8.

10.

11.

12.

SECTION 2.4

13.

14.

15. 17. 19. 21. 23. 25. 27.

Italy, II Canada, VIII Spain, III VI III III V

16. United States, V 18. Portugal, VII 20. Mexico, VI 22. VIII 24. IV 26. I 28. III

29. 31. 33. 35. 37.

II VII I VIII VI

30. 32. 34. 36. 38.

39. 41. 43. 45.

47. 49. 51.

A = { 1, 2, 3, 4, 5, 7 }

40.

B = { 3, 4, 5, 6, 8, 9, 12, 14 }

42.

A ∩ B = { 3, 4, 5 }

44.

( B ∩ C )′ = { 1, 2, 3, 7, 9, 10, 11, 12, 13, 14 } A ∪ B = { 1, 2, 3, 4, 5, 6, 7, 8, 9, 12, 14 }

( A ∪ C )′ = { 9, 10, 12, 13, 14 }

46.

48. 50.

A′ = { 6, 8, 9, 10, 11, 12,13, 14 } 52.

VIII VI VII V III U = { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 } C = { 4, 5, 6, 7, 8, 11 } A ∩ C = { 4, 5, 7 } A ∩ B ∩ C = { 4, 5 }

B ∪ C = { 3, 4, 5, 6, 7, 8, 9, 11,12, 14 } A ∩ ( B ∪ C ) = { 3, 4, 5, 7 }

( A ∪ B ∪ C )′ = { 10, 13 }

CHAPTER 2 Sets

53.

( A ∩ B )′

A′ ∪ B ′

Set A

Regions I, II

B A∩ B

( A ∩ B )′

54.

( A ∩ B )′

Regions I, II

Set A

Regions I, II

II, III II

Set A A′ B

III, IV II, III

B A∩ B

II, III II

I, III, IV

B′

I, IV

( A ∩ B )′

I, III, IV

A′ ∪ B Set A A′ B

Regions I, II III, IV II, III

A′ ∪ B

II, III, IV

A′ ∪ B ′ I, III, IV Both statements are represented by the same regions, I, III, IV, of the Venn diagram. Therefore,

Since the two statements are not represented by the

( A ∩ B )′ = A′ ∪ B ′ for all sets A and B.

all sets A and B.

55. Set A A′ B B′

A′ ∪ B ′ Regions I, II III, IV II, III

A∩ B Set A

Regions I, II

B A∩ B

II, III II

I, IV ′ ′ A ∪B I, III, IV Since the two statements are not represented by the same regions, it is not true that A′ ∪ B ′ = A ∩ B for all sets A and B. 57.

same regions, it is not true that ( A ∩ B )′ = A′ ∪ B for

( A ∪ B )′

A′ ∪ B ′

Set A A′

Regions I, II

Set A

Regions I, II

III, IV

II, III

( A ∪ B )′

I, II, IV

56.

( A ∪ B )′

( A ∩ B )′

Set A B A∪ B

Regions I, II II, III I, II, III

Set A B A∩ B

Regions I, II II, III II

( A ∪ B )′

( A ∩ B )′

I, III, IV

Since the two statements are not represented by the same regions, it is not true that ( A ∪ B )′ = ( A ∩ B )′ for all sets A and B. 58. A′ ∩ B ′

A ∪ B′

Set A A′

Regions I, II

Set A

Regions I, II

III, IV

II, III

B′

I, IV

B′ A′ ∩ B ′

I, IV

A ∪ B′

I, II, IV

B′

I, IV ′ ′ A ∪B I, III, IV Since the two statements are not represented by the same

IV Since the two statements are not represented by the Same regions, it is not true that A′ ∩ B ′ = A ∪ B ′

regions, it is not true that A′ ∪ B′ = ( A ∪ B )′ for all sets

for all sets A and B.

A and B.

SECTION 2.4

59. ( A′ ∩ B )′

A ∪ B′ Regions I, II

Set A

Regions I, II

III, IV

II, III

B′

I, IV

A′ ∩ B

III

A ∪ B′

I, II, IV

( A′ ∩ B)′

( A′ ∩ B′)′

60. A′ ∩ B ′

Set A A′

I, II, IV

Set A A′ B B′

Regions I, II

A′ ∩ B ′

Regions I, II

I, IV

Set A A′ B B′

A′ ∩ B ′

( A′ ∩ B′)′

I, II, III

III, IV II, III

III, IV II, III I, IV

Both statements are represented by the same regions, I, II, IV, of the Venn diagram. Therefore,

Since the two statements are not represented by the same

( A′ ∩ B )′ = A ∪ B′ for all sets A and B.

regions, it is not true that A′ ∩ B ′ = ( A′ ∩ B ′)′ for all sets A and B.

A ∩(B ∪ C)

61.

( A ∩ B) ∪ C Regions II, III, V, VI IV , V, VI, VII II, III, IV, V, VI, VII I, II, IV, V II, IV, V

Set B C B∪C A A ∩(B ∪ C)

Set A B A∩ B C

( A ∩ B) ∪ C

Regions I, II, IV, V II, III, V, VI II, V IV, V, VI, VII II, IV, V, VI, VII

Since the two statements are not represented by the same regions, it is not true that A ∩ ( B ∪ C ) = ( A ∩ B ) ∪ C for all sets A, B , and C. 62.

A ∪(B ∩ C)

(B ∩ C)∪ A Regions II, III, V, VI IV , V, VI, VII V, VI I, II, IV, V I, II, IV, V, VI

Set B C B∩C A A ∪(B ∩ C)

Set B C B∩C A

(B ∩ C)∪ A

Regions II, III, V, VI IV, V, VI, VII V, VI I, II, IV, V I, II, IV, V, VI

Both statements are represented by the same regions, I, II, IV, V, VI, of the Venn diagram. Therefore, A ∪ ( B ∩ C ) = ( B ∩ C ) ∪ A for all sets A, B , and C. 63. Set B C B∪C A A ∩(B ∪ C)

A ∩(B ∪ C)

(B ∪ C)∩ A Regions II, III, V, VI IV , V, VI, VII II, III, IV, V, VI, VII I, II, IV, V II, IV, V

Set B C B∪C A

(B ∪ C)∩ A

Regions II, III, V, VI IV, V, VI, VII II, III, IV, V, VI, VII I, II, IV, V II, IV, V

Both statements are represented by the same regions, II, IV, V, of the Venn diagram. Therefore, A ∩ ( B ∪ C ) = ( B ∪ C ) ∩ A for all sets A, B , and C.

CHAPTER 2 Sets

64.

A ∪ ( B ∩ C )′

A′ ∩ ( B ′ ∪ C )

Set B C B∩C

Regions II, III, V, VI

( B ∩ C )′

IV , V, VI, VII V, VI

Set B′ C

Regions I, IV, VII, VIII IV, V, VI, VII I, IV, V, VI, VII, VIII

I, II, III, IV, VII, VIII

B′ ∪ C A

I, II, IV, V

A′

III, VI, VII, VIII

A ∪ ( B ∩ C )′

I, II, III, IV, V, VII, VIII

A′ ∩ ( B ′ ∪ C )

VI, VII, VIII

Since the two statements are not represented by the same regions, it is not true that A ∪ ( B ∩ C )′ = A′ ∩ ( B ′ ∪ C ) for all sets A, B, and C. 65.

A ∩(B ∪ C)

Set B C B∪C A A ∩(B ∪ C)

( A ∩ B) ∪ ( A ∩ C ) Regions II, III, V, VI IV, V, VI, VII II, III, IV, V, VI, VII I, II, IV, V II, IV, V

Set A B A∩ B C A∩ C

Regions I, II, IV, V II, III, V, VI II, V IV, V, VI, VII IV, V

( A ∩ B) ∪ ( A ∩ C )

II, IV, V

Both statements are represented by the same regions, II, IV, V, of the Venn diagram. Therefore, A ∩ ( B ∪ C ) = ( A ∩ B ) ∪ ( A ∩ C ) for all sets A, B , and C. 66. Set B C B∩C A A ∪(B ∩ C)

A ∪(B ∩ C)

( A ∪ B) ∩ ( A ∪ C ) Regions II, III, V, VI IV, V, VI, VII V, VI I, II, IV, V I, II, IV, V, VI

Set A B A∪ B C A∪ C

Regions I, II, IV, V II, III, V, VI I, II, III, IV, V, VI IV, V, VI, VII I, II, IV, V, VI, VII

( A ∪ B) ∩ ( A ∪ C )

I, II, IV, V, VI

Both statements are represented by the same regions, I, II, IV, V, VI, of the Venn diagram. Therefore, A ∪ ( B ∩ C ) = ( A ∪ B ) ∩ ( A ∪ C ) for all sets A, B , and C.

SECTION 2.4

A ∪ ( B ∪ C )′

67. Set B C B∪C

( B ∪ C )′ A A ∪ ( B ∪ C )′

A ∪ ( B ′ ∩ C ′) Regions II, III, V, VI IV, V, VI, VII II, III, IV, V, VI, VII I, VIII

Set B B′ C C′

Regions II, III, V, VI I, IV, VII, VIII IV, V, VI, VII I, II, III, VIII

I, II, IV, V I, II, IV, V, VIII

B′ ∩ C ′ A

I, VIII I, II, IV, V

A ∪ ( B ′ ∩ C ′)

I, II, IV, V, VIII

Both statements are represented by the same region, I, II, IV, V, VIII of the Venn diagram. Therefore, A ∪ ( B ∪ C )′ = A ∪ ( B ′ ∩ C ′) for all sets A, B, and C.

( A ∪ B) ∩ ( B ∪ C )

68.

B ∪( A∩ C)

Set A B A∪ B C B∪C

Regions I, II, IV, V II, III, V, VI I, II, III, IV, V, VI IV, V, VI, VII II, III, IV, V, VI, VII

( A ∪ B) ∩ ( B ∪ C )

II, III, IV, V, VI

Set A C A∩ C B B ∪( A∩ C)

Regions I, II, IV, V IV, V, VI, VII IV, V II, III, V, VI II, III, IV, V, VI

Both statements are represented by the same regions, II, III, IV, V, VI, of the Venn diagram. Therefore, ( A ∪ B ) ∩ ( B ∪ C ) = B ∪ ( A ∩ C ) for all sets A, B , and C.

69.

( A ∪ B )′ ∩ C

( A′ ∪ C ) ∩ ( B ′ ∪ C )

Set A B

Regions I, II, IV, V II, III, V, VI

A∪ B

I, II, III, IV, V, VI VII, VIII

( A ∪ B )′ C

( A ∪ B )′ ∩ C

IV, V, VI, VII VII

Set A A′ C A′ ∪ C

Regions I, II, IV, V III, VI, VII, VIII IV, V, VI, VII III, IV, V, VI, VII, VIII

B′

II, III, V, VI I, IV, VII, VIII

B′ ∪ C

I, IV, V, VI, VII, VIII

( A′ ∪ C ) ∩ ( B ′ ∪ C )

IV, V, VI, VII, VIII

Since the two statements are not represented by the same regions, it is not true that ( A ∪ B )′ ∩ C = ( A′ ∪ C ) ∩ ( B ′ ∪ C ) for all sets A, B, and C.

CHAPTER 2 Sets

(C ∩ B )′ ∪ ( A ∩ B )′

70.

A ∩(B ∩ C)

Regions IV, V, VI, VII II, III, V, VI V, VI I, II, III, IV, VII, VIII

Set B C B∩C A

Regions II, III, V, VI IV, V, VI, VII V, VI I, II, IV, V

I, II, IV, V

A ∩(B ∩ C)

A∩ B

II, V I, III, IV, VI, VII, VIII

Set C B C∩B

(C ∩ B )′

( A ∩ B )′ (C ∩ B )′ ∪ ( A ∩ B )′

I, II, III, IV, VI, VII, VIII

Since the two statements are not represented by the same regions, it is not true that (C ∩ B )′ ∪ ( A ∩ B )′ = A ∩ ( B ∩ C ) for all sets A, B, and C. 71.

( A ∪ B )′

72.

( A ∩ B)′

73.

( A ∪ B) ∩ C ′

74.

( A ∩ B) ∪ (B ∩ C )

75.

a) ( A ∪ B ) ∩ C = ({1, 2,3, 4} ∪ {3,6,7}) ∩ {6,7,9} = {1, 2,3, 4,6,7} ∩ {6,7,9} = {6,7}

( A ∩ C ) ∪ ( B ∩ C ) = ({1, 2,3, 4} ∩ {6,7,9}) ∪ ({3,6,7} ∩ {6,7,9}) = ∅∪ {6,7} = {6,7} Therefore, for the specific sets, ( A ∪ B ) ∩ C = ( A ∩ C ) ∪ ( B ∩ C ). b) Answers will vary. c) Set A B A∪ B C

( A ∪ B) ∩ C

( A ∩ C)∪(B ∩ C) Regions I, II, IV, V II, III, V, VI I, II, III, IV, V, VI IV, V, VI, VII IV, V, VI

Set A C A∩ C B B∩C

Regions I, II, IV, V IV, V, VI, VII IV, V II, III, V, VI V, VI

( A ∩ C)∪(B ∩ C)

IV, V, VI

Both statements are represented by the same regions, IV, V, VI, of the Venn diagram. Therefore, ( A ∪ B ) ∩ C = ( A ∩ C ) ∪ ( B ∩ C ) for all sets A, B, and C.

SECTION 2.4

76. . a) ( A ∪ C )′ ∩ B = ({a, c, d, e, f } ∪ {a, b, c, d, e})′ ∩ {c, d} = {a, b, c, d, e, f }′ ∩ {c, d}

= {g, h, i} ∩ {c, d} = ∅

( A ∩ C )′ ∩ B = ({a, c, d, e, f } ∩ {a, b, c, d, e})′ ∩ {c, d} = {a, c, d, e}′ ∩ {c, d} = {b, f, g, h, i} ∩ {c, d} = ∅ Therefore, for the specific sets, ( A ∪ C )′ ∩ B = ( A ∩ C )′ ∩ B. b) Answers will vary.

( A ∪ C )′ ∩ B

( A ∩ C )′ ∩ B

Set

Regions

Set

Regions

I, II, IV, V

IV, V, VI, VII

A∪ C

I, II, IV, V, VI, VII

A∩ C

IV, V

( A ∪ C )′

III, VIII

( A ∩ C )′

I, II, III, VI, VII, VIII

II, III, V, VI

( A ∪ C )′ ∩ B

III

( A ∩ C )′ ∩ B

II, III, VI

Since the two statements are not represented by the same regions, ( A ∪ C )′ ∩ B ≠ ( A ∩ C )′ ∩ B for all sets A, B , and C. 77.

78. Region

Set

Region

Set

A ∩ B′ ∩ C ′

A∩ B ∩ C

A∩ B ∩ C′

A′ ∩ B ∩ C

III

A′ ∩ B ∩ C ′

VII

A′ ∩ B ′ ∩ C

A ∩ B′ ∩ C

VIII

A′ ∩ B ′ ∩ C ′

CHAPTER 2 Sets

79.

a) A : Office Building Construction Projects, B : Plumbing Projects, C : Budget Greater Than $300,000

b) Region V; A ∩ B ∩ C c) Region VI; A′ ∩ B ∩ C d) Region I; A ∩ B ′ ∩ C ′

80. n ( A ∪ B ∪ C ) = n ( A) + n ( B ) + n (C )− 2n ( A ∩ B ∩ C )− n ( A ∩ B ∩ C ′)− n ( A ∩ B ′ ∩ C )− n ( A′ ∩ B ∩ C ) 81.

b) A ∩ B′ ∩ C ′ ∩ D′

Region IX

A ∩ B′ ∩ C ∩ D′

A ∩ B ∩ C ′ ∩ D′

A ∩ B ∩ C ∩ D′

III

A′ ∩ B ∩ C ′ ∩ D ′

A′ ∩ B ∩ C ∩ D ′

A ∩ B′ ∩ C ′ ∩ D

XII

A′ ∩ B ∩ C ∩ D

A∩ B ∩ C′ ∩ D

XIII

A′ ∩ B ′ ∩ C ∩ D ′

A′ ∩ B ∩ C ′ ∩ D

XIV

A′ ∩ B ′ ∩ C ∩ D

VII

A ∩ B′ ∩ C ∩ D

A′ ∩ B ′ ∩ C ′ ∩ D

VIII

A∩ B ∩ C ∩ D

XVI

A′ ∩ B ′ ∩ C ′ ∩ D ′

Region I

Set

SECTION 2.5

Exercise Set 2.5

a) 48 b) 37 c) 200 − (48 + 61 + 37), or 54

a) 33 b) 29 c) 27

a) 17 b) 12 c) 59, the sum of the numbers in Regions I, II, III

a) 47 b) 38 c) 140, the sum of the numbers in Regions I, II, III d) 150 −140, or 10. 6.

a) 3 b) 6 c) 3 + 2 + 6 + 5 + 2 + 4, or 22

a) 30 b) 8 + 30 + 16, or 54

d) 3 + 6 + 2, or 11

c) 85 − 3, or 82

e) 2 + 6 + 4, or 12

d) 3 + 6 + 12, or 21 8.

a) 22 b) 11 c) 85 −15 − 6, or 64

a) 20 b) 121 c) 121 + 83 + 40, or 244

d) 22 + 11 + 17, or 50

d) 16 + 38 + 11, or 65

e) 9 + 11 + 3, or 23

e) 350 − 20 − 40, or 290

e) 3

CHAPTER 2 Sets

10.

a) 3 b) 12 c) 3 d) 12 + 3 + 2, or 17

496, the sum of the numbers in all the regions b) 132 c) 29 d) 132 + 125 + 71, , or 328 e) a)

e) 8

496 − 26, or 470 11.

12. No. The sum of the numbers in the Venn diagram is 99. Dennis claims he surveyed 100 people.

a) 30 + 37, or 67 b) 350 − 25 − 88, or 237 c) 37

d) 25

13. The Venn diagram shows the number of cars driven by women is 37, the sum of the numbers in Regions II, IV, V. This exceeds the 35 women the agent claims to have surveyed.

U Women

U.S. I

III

V IV 10

VI 5

13 Two or more

VII VIII

SECTION 2.5

14. First fill in 15, 20 and 35 on the Venn diagram. Referring to the labels in the Venn diagram and the given information, we see that a + c = 140 b + c = 125 a + b + c = 185 −15 = 170 Adding the first two equations and subtracting the third from this sum gives c = 125 + 140 −170 = 95. Then a = 45 and b = 30. Then d = 210 − 45 − 95 − 20 = 50. We now have labeled all the regions except the region outside the three circles, so the number of parks with at least one of the features is 15 + 45 + 20 + 30 + 95 + 50 + 35, or 290. Thus the number with none of the features is 300 − 290, or 10. a) 290 b) 95 c) 10 d) 30 + 45 + 50, or 125.

15. First fill in 15, 20 and 35 on the Venn diagram. Referring to the labels in the Venn diagram and the given information, we see that a + c = 60 b + c = 50 a + b + c = 200 −125 = 75 Adding the first two equations and subtracting the third from this sum gives c = 60 + 50 − 75 = 35. Then a = 25 and b = 15. Then d = 180 −110 − 25 − 35 = 10. We now have labeled all the regions except the region outside the three circles, so the number of farmers growing at least one of the crops is 125 + 25 + 110 + 15 + 35 + 10 + 90, or 410. Thus the number growing none of the crops is 500 − 410, or 90. a) 410 b) 35 c) 90 d) 15 + 25 + 10, or 50

16. 16

CHAPTER 2 Sets

17. From the given information we can generate the Venn diagram. First fill in 4 for Region V. Then since the intersections in pairs all have 6 elements, we can fill in 2 for each of Regions II, IV, and VI. This already accounts for the 10 elements A ∪ B ∪ C , so the remaining 2 elements in U must be in Region VIII.

U A

B I

II 2

IV 2

III

VI 2

VII

VIII

a) 10, the sum of the numbers in Regions I, II, III, IV, V, VI b) 10, the sum of the numbers in Regions III, IV, V, VI, VIII c) 6, the sum of the numbers in Regions I, III, IV, VI, VII

Exercise Set 2.6 1. Infinite 2. Countable 3. {3, 4, 5, 6, 7, …, n + 2, …} ↓ ↓ ↓ ↓ ↓ ↓ {4, 5, 6, 7, 8, …, n + 3, …}

{30, 31,32, 33, 34, …, n + 29, …} ↓ ↓ ↓ ↓ ↓ ↓ {31, 32, 33, 34, 35, …, n + 30, …}

5. {3, 5, 7, 9, 11, …, 2n + 1, …} ↓ ↓ ↓ ↓ ↓ ↓ {5, 7, 9, 11, 13, …, 2n + 3, …}

{20, 22, 24, 26, 28, …, 2n + 18, …} ↓ ↓ ↓ ↓ ↓ ↓ {22, 24, 26, 28, 30, …, 2n + 20, …}

7. {5, 9, 13, 17, 21 …, 4n  1, …} ↓ ↓ ↓ ↓ ↓ ↓ {9, 13, 17, 21, 25, …, 4n + 5, …}

8. 

{6, 11, 16, 21, 26, …, 5n+1, …} ↓ ↓ ↓ ↓ ↓ ↓ {11, 16, 21, 26, 31, …, 5n+6, …}

⎧⎪ 1 1 1 1 1 ⎫ 1 ,…⎪⎬ ⎨ , , , , ,… , ⎪⎩⎪ 2 4 6 8 10 2n ⎪⎭⎪ ↓ ↓ ↓ ↓ ↓ ⎧ ⎫ 1 1 1 1 1 ⎪ ,…⎪ ⎨ , , , ,… , ⎬ ⎪ 2n + 2 ⎪ ⎪ 4 6 8 10 ⎪ ⎩ ⎭

11.

⎧ n + 3 ⎫⎪ 4 5 6 7 ⎪ ,…⎬ ⎨ , , , ,… , ⎪ ⎪⎭⎪ 11 ⎪11 11 11 11 ⎩ ↓ ↓ ↓ ↓ ↓ ⎧⎪ 5 6 7 8 n + 4 ⎫⎪ ,…⎬ ⎨ , , , ,… , ⎪⎩⎪11 11 11 11 ⎪⎭⎪ 11

⎧ 1 1 1 1 1 ⎫ ⎪ ⎨1, , , , ,… , ,…⎪ ⎬ ⎪ 2 3 4 5 n ⎪ ⎪ ⎪ ⎭ 10. ⎩ ↓ ↓ ↓ ↓ ↓ ↓ ⎧ ⎫ 1 1 1 1 1 1 ⎪ ,…⎪ ⎨ , , , , ,… , ⎬ ⎪ n +1 ⎪ ⎪2 3 4 5 6 ⎪ ⎩ ⎭

12.

⎧ 6 7 8 9 10 n +5 ⎫ ⎪ ,…⎪ ⎨ , , , , ,… , ⎬ ⎪ ⎪ 13 13 13 13 13 13 ⎪ ⎪ ⎩ ⎭ ↓ ↓ ↓ ↓ ↓ ↓ ⎧ 7 8 9 10 11 n+6 ⎫ ⎪ ,…⎪ ⎨ , , , , ,… , ⎬ ⎪ ⎪ 13 13 13 13 13 13 ⎪ ⎪ ⎩ ⎭

REVIEW EXERCISES

13. {1, 2, 3, 4, 5, …, n, …} ↓ ↓ ↓ ↓ ↓ ↓ {3, 6, 9, 12, 15, …, 3n, …}

14.

{1, 2, 3, 4, 5, …, n, …} ↓ ↓ ↓ ↓ ↓ ↓ {40, 41, 42, 43, 44, …, n + 39, …}

15. {1, 2, 3, 4, 5, …, n, …} ↓ ↓ ↓ ↓ ↓ ↓ {4, 6, 8, 10, 12, …, 2n + 2, …}

16.

{1, 2, 3, 4, 5, …, n, …} ↓ ↓↓ ↓↓ ↓ {0, 2, 4, 6, 8, …, 2n - 2, …}

17. {1, 2, 3, 4, 5, …, n, …} ↓↓ ↓ ↓ ↓ ↓ {2, 5, 8, 11, 14, …, 3n  1, …}

18.

{1, 2, 3, 4, 5, …, n, …} ↓ ↓ ↓ ↓ ↓ ↓ {7, 11, 15, 19, 23, …, 4n + 3, …}

19. {1, 2, 3, 4, 5, …, n , …} ↓ ↓ ↓ ↓ ↓ ↓ 1 ⎪⎧ 1 1 1 1 1 ⎪⎫ ,…⎬ ⎨ , , , , ,… , ⎪⎩⎪ 3 6 9 12 15 3n ⎪⎭⎪

20.

{ 1, 2, 3, 4, 5, …, n, …} ↓ ↓ ↓ ↓ ↓ ↓ ⎧1 1 1 1 1 ⎫ 1 ⎪ ⎪ ⎨ , , , , ,… , ,…⎬ ⎪ 2n ⎪ ⎪ 2 4 6 8 10 ⎪ ⎩ ⎭

21. { 1, 2, 3, 4, 7, …, n, …} ↓ ↓ ↓ ↓ ↓ ↓ 1 ⎪⎧ 1 1 1 1 1 ⎪⎫ ,…⎬ ⎨ , , , , ,… , ⎪⎩⎪ 3 4 5 6 7 n+2 ⎪ ⎭⎪

22. {1, 2, 3, 4, 5, …, n, …} ↓ ↓ ↓ ↓ ↓ ↓ n ⎪⎧ 1 2 3 4 5 ⎪⎫ ,…⎬ ⎨ , , , , ,… , ⎪⎩⎪ 2 3 4 5 6 n +1 ⎪ ⎭⎪

23. {1, 2, 3, 4, 5, …, n, …} ↓ ↓ ↓ ↓ ↓ ↓

24.

{1, 2, 3, 4, 5, …, n, …} ↓ ↓ ↓ ↓ ↓

↓

{1, 4, 9 , 16, 25, …, n , …} 25. {1, 2, 3, 4, 5, …, n, …} ↓ ↓ ↓ ↓ ↓ ↓

{2, 4, 8, 16, 32, …, 2 , …} { 1, 2, 3, 4, 5, …, n, …} ↓ ↓ ↓ ↓ ↓ ↓ ⎧ ⎫ 1 1 1 1 1 1 ⎪ ,…⎪ ⎨ , , , , ,…, ⎬ n− 1 ⎪ ⎪ 3× 2 ⎪ ⎪ ⎩ 3 6 12 24 48 ⎭ 28. = 30. = 32. a) Answers will vary. b) No

26.

{3, 9, 27, 81, 243, …, 3n , …} 27. = 29. = 31. =

Review Exercises 1. True

3. True 5. False; the elements 6, 12, 18, 24, … are members of both sets. 7. False; the two sets do not contain exactly the same elements. 9. True

2. False; the word best makes the statement not well defined. 4. False; no set is a proper subset of itself. 6. True 8. True 10. True

CHAPTER 2 Sets

11. True 13. True

12. True 14. True 16. B = { Colorado, Nebraska, Missouri, Oklahoma }

15. A = { 7, 9, 11, 13, 15 }

17. C = { 1, 2, 3, 4, …, 161 }

18. D = { 9, 10, 11, 12, … , 80 }

19. A = { x x ∈ N and 50 < x < 150}

20. B = { x x ∈ N and x > 42}

21. C = { x x ∈ N and x < 7}

22. D = { x x ∈ N and 27 ≤ x ≤ 51}

23. 24. 25. 26.

A is the set of capital letters in the English alphabet from E through M, inclusive. B is the set of U.S. coins with a value of less than one dollar. C is the set of the first three lowercase letters in the English alphabet. D is the set of numbers greater than or equal to 3 and less than 9.

27. A ∩ B = { 1, 3, 5, 7 } ∩ { 3, 7, 9, 10 } = { 3, 7 } 28. A ∪ B ′ = { 1, 3, 5, 7 } ∪ { 3, 7, 9, 10 }′ = { 1, 3, 5, 7 } ∪ { 1, 2, 4,5, 6, 8 } = { 1, 2, 3, 4, 5, 6, 7, 8 }

29. A′ ∩ B = { 1, 3, 5, 7} ′ ∩ { 3, 7, 9, 10 } = { 2, 4, 6, 8, 9, 10} ∩ { 5, 7, 9, 10 } = { 9, 10 }

30. ( A ∪ B )′ ∪ C = ({ 1, 3, 5, 7 } ∪ { 3, 7, 9, 10 })′ ∪ { 1, 7, 10 } = { 1, 3, 5, 7, 9, 10 }′ ∪ { 1, 7, 10 } = { 2, 4, 6, 8 } ∪ { 1, 7, 10 } = { 1, 2, 4, 6, 7, 8, 10 }

31. A − B = { 1, 3, 5, 7 } − { 3, 7, 9, 10 } = { 1, 5 } 32. A − C ′ = { 1, 3, 5, 7 } − { 1, 7, 10 }′ = { 1, 3, 5, 7 } − { 2, 3, 4, 5, 6, 8, 9 } = { 1, 7 } 33. { (1, 1), (1, 7), (1, 10), (3, 1), (3, 7), (3, 10), (5, 1), (5, 7), (5, 10), (7, 1), (7, 7), (7, 10) } 34. { (3, 1), (3, 3), (3, 5), (3, 7), (7, 1), (7, 3), (7, 5), (7, 7), (9, 1), (9, 3), (9, 5), (9, 7), (10, 1), (10, 3), (10, 5), (10, 7) } 36. 24 −1 = (2× 2 × 2× 2)−1 = 16 −1 = 15

35. 2 4 = 2× 2× 2 × 2 = 16 37.

39. 41.

38.

A ∩ B ′ = { i, k } A∩ B ∩ C = { f }

40. 42.

A ∪ B = { a, c, d, f, g, i, k, l }

A ∪ B ∪ C = { a, b, c, d, f, g, h, i, k, l }

( A ∪ B ) ∩ C = { a, f, i }

REVIEW EXERCISES

43.

44.

( A ∩ B ) ∪ C = { a, b, d, f, h, i, l }

( A′ ∪ B′)′

Set A A′ B B′

A∩ B Regions I, II

Set A

Regions I, II

III, IV II, III

B A∩ B

II, III II

I, IV

A′ ∪ B ′

I, III, IV

( A′ ∪ B′)′

Both statements are represented by the same region, II, of the Venn diagram. Therefore, ( A′ ∪ B ′)′ = A ∩ B for all sets A and B. 45.

( A ∪ B ′ ) ∪ ( A ∪ C ′)

A ∪ ( B ∩ C )′

Regions I, II, IV, V II, III, V, VI I, IV, VII, VIII I, II, IV, V, VII, VIII

Set B C B∩C

C′

IV, V, VI, VII I, II, III, VIII

A∪ C′

I, II, III, IV, V, VIII

( A ∪ B ′ ) ∪ ( A ∪ C ′)

I, II, III, IV, V, VII, VIII

Set A B B′ A ∪ B′ C

Regions II, III, V, VI IV, V, VI, VII V, VI I, II, III, IV, VII, VIII

( B ∩ C )′

I, II, IV, V I, II, III, IV, V, VII, VIII

A ∪ ( B ∩ C )′

Both statements are represented by the same regions, I, II, III, IV, V, VII, VIII, of the Venn diagram. Therefore, ( A ∪ B ′) ∪ ( A ∪ C ′) = A ∪ ( B ∩ C )′ for all sets A, B, and C. 46. II 48. I 50. IV 52. II

47. 49. 51.

53. The company paid $450 since the sum of the numbers in Regions I through IV is 450.

III IV II

U Thin

Thick I

130

III

200

CHAPTER 2 Sets

54. a) 131, the sum of the numbers in Regions I through VIII b) 32, Region I c) 10, Region II d) 65, the sum of the numbers in Regions I, IV, VII

55. a) 38, Region I b) 298, the sum of the numbers in Regions I, III, VII c) 28, Region VI d) 236, the sum of the numbers in Regions I, IV, VII e) 106, the sum of the numbers in Regions II, IV, VI

56. {2, 4, 6, 8, 10, …, 2n, …} ↓ ↓ ↓ ↓ ↓ ↓ {4, 6, 8, 10, 12, …, 2n + 2, …} 58. {1, 2, 3, 4, 5, …, n, …} ↓ ↓ ↓ ↓ ↓ ↓ {5, 8, 11, 14, 17, …, 3n + 2, …}

57.

59.

Chapter Test 1. True

{3, 5, 7, 9, 11, …, 2n + 1, …} ↓↓ ↓ ↓ ↓ ↓ {5, 7, 9, 11, 13, …, 2n + 3, …} {1, 2, 3, 4, 5, …, n, …} ↓ ↓ ↓ ↓ ↓ ↓ {4, 9, 14, 19, 24, …, 5n - 1, …}

5. False; the set has 24 = 2× 2× 2× 2 = 16 subsets.

2. False; the sets do not contain exactly the same elements. 4. False; the second set has no subset that contains the element 7. 6. True

7. False; for any set A , A ∪ A′ = U , not { }.

8. True

3. True

A = { 1, 2, 3, 4, 5, 6, 7, 8,9 }

10. Set A is the set of natural numbers less than 10.

11. A ∩ B = { 3, 5, 7, 9 } ∩ { 7, 9, 11, 13 } = { 7, 9 } 12. A ∪ C ′ = { 3, 5, 7, 9 } ∪ { 3, 11, 15 }′ = { 3, 5, 7, 9 } ∪ { 5, 7, 9, 13 } = { 3, 5, 7, 9, 13 }

(

)

13. A ∩ B ∩ C ′ = { 3, 5, 7, 9 } ∩ ({ 7, 9, 11, 13 } ∩ { 5, 7, 9, 13}) = { 3, 5, 7, 9 } ∩ { 7, 9 } = { 7, 9}

CHAPTER TEST

⎛ ⎞ 14. n ( A ∩ B ′) = n ⎜⎜{ 3, 5, 7, 9 } ∩ { 7, 9, 11, 13 }′ ⎟⎟⎟ = n ({ 3, 5, 7, 9 } ∩ { 3, 5, 15 }) = n ({ 3, 5 }) = 2 ⎝ ⎠ 15. A − B = { 3, 5, 7, 9 } − { 7, 9, 11, 13 } = { 3, 5 } 16. A×C = { (3, 3), (3, 11), (3, 15), (5, 3), (5, 11), (5, 15), (7, 3), (7, 11), (7, 15), (9, 3), (9, 11), (9, 15) } 17.

A ∩ ( B ∪ C ′)

18. Set B C

( A ∩ B ) ∪ ( A ∩ C ′)

C′

Regions II, III, V, VI IV, V, VI, VII I, II, III, VIII

Set A B A∩ B

Regions I, II, IV, V II, III, V, VI II, V

B ∪ C′

I, II, III, V, VI, VIII

IV, V, VI, VII

I, II, IV, V

C′

I, II, III, VIII

A ∩ ( B ∪ C ′)

I, II, V

A∩ C′

I, II

( A ∩ B ) ∪ ( A ∩ C ′)

I, II, V

Both statements are represented by the same regions, I, II, V, of the Venn diagram. Therefore, A ∩ ( B ∪ C ′) = ( A ∩ B ) ∪ ( A ∩ C ′) for all sets A, B, and C. 19.

a) 52, the sum of the numbers in Regions I, III, VII b) 10, Region VIII c) 93, the sum of the numbers in Regions II, IV, V, VI d) 22, Region II e) 69, the sum of the numbers in Regions I, II, III f) 5, Region VII

20. {7, 8, 9, 10, 11, …, n + 6, …} ↓ ↓ ↓ ↓ ↓ ↓ {8, 9, 10, 11, 12, …, n + 7, …}

CHAPTER EIGHT THE METRIC SYSTEM Exercise Set 8.1 1. Metric 3. a) Meter 4. a) Measurement 5. a) Deka b) Deci 6. a) Hecto

b) Kilogram b) Quantity

c) Liter c) 10

b) Centi

7. a) Kilo

2. Customary d) Celsius

b) Milli

8. a) Mega b) Micro 9. 100 10. 10,000 11. a) 0o C b) 100o C c) 37o C 15. 22o C 13. a) Yard b) Quart 14. 30o C 12. 2.2 16. 2 m 17. kilo d 18. milli b 19. hecto c 20. deka e 21. deci f 22. centi a 23. a) 0.001 gram b) 100 grams c) 1000 grams d) 0.01 gram e) 10 grams f) 0.1 gram 24. a) 1/100 liter b) 10 liters c) 1/1000 liter d) 1/10 liter e) 1000 liters f) 100 liters 25. mg 1/1000 gm 26. cg 1/100 gm 27. dg 1/10 gm 28. dag 10 gm 29. hg 100 gm 30. kg 1000 gm 31. Max. mass 3000 kg = (3000 x 1000) g = 3 000 000 g

32. Max. mass 3000 kg = (3000 x 1,000,000) mg = 3 000 000 000 mg

33. 7 m = (7 x 1000) mm = 7000 mm

34. 42.9 hg = (42.9 x 100) g = 4290 g

35. 0.085 hl = (0.085)(0.1) kl = 0.0085 kl 37. 186.2 cm = (186.2)(0.0001) hm = 0.01862 hm

36. 8 dam = (8 x 10) m = 80 m 38. 9.32 mA = (9.32)(0.001) A = 0.00932 A

39. 9.2 m = (9.2)(1000) mm = 9200 mm 41. 895 l = 895,000 ml 43. 40,302 ml = 4.0302 dal 45. 500 m = (500)(0.001) km = 0.5 km 47. 500 m = (500) (0.01) hm = 5 hm 49. 80 km/hr = (80)(1 000 000) mm/km 80 000 000 mm/hr 51. 620 cm, 4.4 dam, 0.52 km 53. 1.4 kg, 1600 g, 16,300 dg 55. 105,000 mm, 2.6 km, 52.6 hm

40. 69.3 kg = (69.3)(10) hg= 693 hg 42. 24 dm = 0.0024 km 44. 0.034 ml = 0.000034 l 46. 500 m = (500)(100) cm = 50 000 cm 48. 80km = (80)(1000) m = 80 000 m/hr 50. 80 km/hr = (80)(10) hm/km = 800 hm/hr 52. 680 m, 514 hm, 62 km 54. 420 cl, 4.3 l, 0.045 kl 56. 0.032 kl, 460 dl, 48,000 cl 285

286

CHAPTER 8

The Metric System

57. Jim, since a meter is longer than a yard.

58. 1 hectometer in 10 min. 1 hm > 1 dm

59. The pump that removes 1 daA of water per min. 1 dekaliter > 1 deciliter

60. The side with the 15 lb. weight would go down. 5 kg = 5(2.2 lbs.) = 11 lbs.

61. a) Perimeter= 2l + 2w= 2(74) + 2(99)= 346 cm b) 346 cm = (346 x 10) mm = 3,460 mm

62. a) (2)(250)(7) = 3,500 mg / week b) 3,500 mg / week = 3.5 g / week

63. a) 200 km / 12.21 l ≈ 16.4 km/l b) 12.21 km/l(1000) m/km= 12 210 m/l

64. 8 (400) m = 3,200 m;

65. a) 6(360) ml = 2,160 ml b) 2160(1000) = 2.16 l c) 2.45 / 2.16 = $1.13 per liter

66. a) (4)(27 m) = 108 m b) 108 m = 0.108 km c) 108 m = 108 000 mm

3,200 m = 3.2 km

67. a) 1944 – 1558 = 386 m b) 386 m = 0.386 km 68. a) (16950 – 5830) km = 11,120 km b) 11,120,000 m c) 386 m = 38 600 cm 69. 1 gigameter = 1000 megameters 70. 1 nanogram = .001 microgram 71. 1 teraliter = 1 x 1024 picoliters 72. 1 megagram = 1 x 1015 73. 9000 cm = 9 dam nanogms 74. 2000 mm = 2 m 75. 0.00006 hg = 6 mg 76. 3000 dm = 3 hm 77. 0.02 kl = 2 dal 78. 500 cm = 5 m 79. magr gram 80. migradec decigram 83. terem meter 86. timenceret centimeter

Exercise Set 8.2 1. Volume 5. Length 9. Volume 13. Centimeters 17. Centimeters 21. Kilometers 25. c 130 km 29. c 828 m

33. cm or m AWV 37. sq. m 41. sq. mm or sq. cm 45. a 50 cm2 49. a 3 cm2 53. AWV 57. AWV 61. Kiloliters

81. rteli liter 84. leritililm milliliter 87. greseed sulesic degrees celsius

2. Length 6. Volume 10. Length 14. Kilometers 18. Meters 22. cm or m 26. a 800 cm 30. b 1000 m 34. mm or cm AWV 38. sq. cm 42. sq. cm 46. a 800 m2 50. c 1200 cm2 54. AWV 58. AWV 62. Cubic meters or cubic centimeters

82. raktileed dekaliter 85. reketolim kilometer 88. togmeharc hectogram

3. Area 7. Volume 11. Volume 15. cm or mm 19. Kilometers 23. c 110 m 27. b 4 cm 31. mm Answers will vary (AWV). 35. cm AWV 39. sq. cm or sq. m 43. sq. km or hectares 47. b 1/8 ha 51. c 4900 km2 55. AWV 59. Milliliters 63. Cubic meters

4. Length 8. Area 12. Volume 16. Centimeters 20. cm or mm 24. a 2 cm x 3 cm 28. b 8 cm 32. cm AWV 36. mm AWV 40. sq. m or hectares 44. sq. m 48. c 930 cm2 52. b 2.2 m2 56. AWV 60. Liters 64. Liters or milliliters

SECTION 8.2

65. Cubic meters 69. c 55 kA 73. b 5000 cm3

66. Cubic centimeters 70. b 355 mA 74. b 2600 m3

67. c 7780 cm3 71. c 0.04 m3 75. a) AWV

68. a 3 mA 72. b 120 mA 75. b) 10,580 cm3

76. a) AWV b) (2)(1.5)(.25) = 0.75 m3 78. a) AWV b) v = π r2h ≈ (3.14)(0.20)2(2) = 0.25m3 80. A kiloliter

77. a) AWV b) v ≈ (3.14)(0.25)2 (1) = 0.20 m3 79. 1 cubic decimeter

82. 2.5 acres

83. Longer side = 4cm; Shorter side = 2.2 cm A = lw = (4)(2.2) = 8.8 cm2

84. r ≈ 1.2 cm, A = π r 2 ≈ 3.14 (1.22 ) ≈ 4.52 cm 2

85. A = π r 2 ≈ 3.14 (152 ) ≈ 706.86 m 2

86. (37)(28) = 1036 cm2 2540 – 1036 = 1504 cm2 88. a) (3.75)(1.4) = 5.25 km2 b) (5.25)(100 ha) = 525 ha 90. a) (18)(10)(2.5) = 450 m3 b) 450 m3 = 450 kl

87. a) (73)(53) = 3869 m2 b) 3869 – (70)(50) = 3869 – 3500 = 369 m2 89. a) (42.4)(32.5) = 1378 m2 b) (1378)(0.0001) ha = 0.1378 ha

81. A cubic centimeter

91. Total Surface Area of 4 walls = 2lh + 2wh = 2(20)(6) + 2(12)(6) = 384 m2 ⎛ 1 A ⎞⎟ ⎛ 1 A ⎞⎟ = 38.4 l Liters for second coat = (384 m2) ⎜⎜ = 25.6 l Liters for first coat = (384 m2) ⎜⎜ ⎜⎝10 m 2 ⎠⎟⎟ ⎜⎝15 m 2 ⎠⎟⎟

Total liters = 38.4 + 25.6 = 64 l

Total cost = (64)($4.75) = $304

92. V = π r2h ≈ (3.14)(9.0)2(9) = 2290 cm3 93. a) V = lwh = (70)(40)(20) = 56,000 cm3

b) 56,000 cm3 = 56,000 ml

⎛ 56000 ⎞⎟ l = 56 l c) 56 000 ml = ⎜⎜ ⎜⎝ 1000 ⎠⎟⎟

0.56 m = 0.28 m; V = πr 2 h ≈ 3.14 (0.282 ) (1.17) ≈ 0.29 m3 2 b) (0.29)(1000) l = 290 l 95. 102 = 100 times larger 96. 1002 = 10,000 times larger 94. a) radius =

97. 103 = 1000 times larger 100. 10,000 dam2 103. 1,000,000 cm3 106. 435 cm3 = 0.435 l 109. 60 m3 = 60 kl

98. 103 = 1000 times larger 101. 0.0001 m2 104. 1 hm3 = 0.001 km3 107. 76 kl = 76 m3 110. (600,000)(100) = 60,000,000 ml = 60,000,000 cm3

111. 6.7 kl = 6.7 m3 = (6.7 x 103) dm3 = 6,700 dm3

99. 100 mm2 102. 0.000001 dm3 105. 419 cm3 = 419 mA 108. 4.2 l = 4,200 cm3

112. 1.4 ha = 14,000 m2 = (14000 x 1002) cm2 = 140,000,000 cm2

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ft 113. a) 1 sq mi = (1 mi2)(5280)2 2 = 27,878,400 ft2 mi in 2 = 4,014,489,600 in2 ft 2 b) It is easier to convert in the metric system because it is a base 10 system. 27,878,400 ft2 x (12)2

114. a) AWV; the average use is 590 liters / day b) AWV; the average use is 75 liters / day

Exercise Set 8.3 1. Kilogram 5. Celsius

9. Kilograms 13. Grams 17. Kilograms or metric tonnes 21. b 1.4 kg 25. AWV 29. c 0o C 33. c 1200o C 37. c 40o C

2. Grams 6. 0o C

3. Mass 7. 100 o C

4. Kilogram 8. 37 o C

10. Grams or milligrams 14. Kilograms or metric tonnes 18. Grams

11. Grams 15. Kilograms

12. Grams 16. Milligrams

19. b

20. c

22. c 0.45 kg 26. AWV 30. b 21o C

23. b 2800 kg 27. AWV 31. b Dress warmly and walk. 35. c 40o C

34. b

5o C

38. b

−5o C

9 (20) + 32 = 36 + 32 = 68o F 5 5 5 41. C = (92 − 32) = (60) = 33.3o C 9 9 5 5 43. C = (0 − 32) = (−32) ≈ −17.8o C 9 9 9 45. F = (37) + 32 = 66.6 + 32 = 98.6o F 5 5 5 47. C = (17 − 32) = (−15) ≈ –8.3o C 9 9 9 49. F = (0) + 32 = 0 + 32 = 32o F 5 5 5 51. C = (−20 − 32) = (−52) = −28.9o C 9 9 9 53. F = (22) + 32 = 39.6 + 32 = 71.6o F 5 9 55. F = (33.6) + 32 = 60.48 + 32 = 92.48o F 5 39. F =

2.27 kg

24. c 1.6 t 28. AWV 32. b 96o C 36. c

177o C

9 (−5) + 32 = – 9 + 32 = 23o F 5 5 5 42. C = (−10 − 32) = (−42) = –23.3o C 9 9 5 5 44. C = (100 − 32) = (68) = 37.8o C 9 9 9 46. F = (−4) + 32 = – 7.2 + 32 = 24.8o F 5 5 5 48. C = (75 − 32) = (43) = 23.9o C 9 9 9 50. F = (50) + 32 = 90 + 32 = 122o F 5 5 5 52. C = (425 − 32) = (393) = 218.3o C 9 9 9 54. F = (35.1) + 32 = 63.2 + 32 = 95.2o F 5 9 56. F = (32.3) + 32 = 58.1 + 32 = 90.1o F 5 40. F =

SECTION 8.3

9 (17.8) + 32 = 32 + 32 = 64.04o F 5 9 high: F = (23.5) + 32 = 42.3 + 32 = 74.3o F 5 Range = 74.30 – 64.04 = 10.26o F

57. low: F =

59. a) cost = (9)(2.50) = 22.5 euros

9 (22) + 32 = 39.6 + 32 = 71.6o F 5 9 high: F = (34) + 32 = 61.2 + 32 = 93.2o F 5 Range = 93.2 – 71.6 = 21.6o F

58. low: F =

60. (180)(.001)t + 2.92t = 3.1t

b) cost = (9)($1.46) = $13.14

3.5t – 3.1t = 0.4t

c) $5.97/lb 61. total mass = 45 g + 29 g + 370 ml = 45 g + 29 g + 370 g = 444 g

62. fuel used = (4320)(17) = 73,440 kg ⎛ 1 t ⎞⎟ ⎟ = 73.44 t 73, 440 kg ⎜⎜ ⎜⎝1000 kg ⎠⎟⎟

63. a) V= lwh,

l = 16 m, w = 12 m, h = 12 m V = (16)(12)(12) = 2304 m3

b) 2304 m3= 2304 kl c) 2304 kl = 2304 t

64. a) V = π r2h

r = 50 cm = 0.50 m h = 150 cm = 1.5 m V = (3.14)(0.50)2(1.50) = 1.1775 m3

b) 1.1775 m3 = 1.1775 kl = 1177.5 l c) 1177.5 l = 1177.5 kg

65. a) Yes; mass is a measure of the amount of matter in an object. b) No; weight is a measure of gravitational force. ⎛ 1 t ⎞⎟ ⎟ = 0.0062 t 67. 6.2 kg = (6.2 kg) ⎜⎜ ⎜⎝1000 kg ⎠⎟⎟ ⎛1000 kg ⎞⎟ = 17,400 kg = 69. 17.4 t = (17.4 t) ⎜⎜ ⎜⎝ 1 t ⎠⎟⎟

66. Yes, 78o F =

⎛1000 kg ⎞⎟ = 9520 kg 68. 9.52 t = (9.52 t) ⎜⎜ ⎜⎝ 1 t ⎠⎟⎟ ⎛ 1 t ⎞⎟ ⎟= 70. 1,460,000 mg = 1.46 kg = (1.46 kg) ⎜⎜ ⎜⎝1000 kg ⎠⎟⎟

17,400,000 g

0.00146 t

71. 1.2 l = 1200 ml

a) 1200 g

72. a) V = lwh

b) 1200 cm3

l = 1 yd = 3 ft w = 15 in = 1.25 ft h = 1.5 ft V = (3)(1.25)(1.5) = 5.625 cubic feet ⎛ ⎝

b) (5.625 ft3) ⎜⎜⎜62.5

lbs ⎞⎟ ⎟ = 351.6 lb ft 3 ⎠⎟

⎛ 1 gal ⎞⎟ c) (351.6 lb) ⎜⎜ = 42.4 gal ⎜⎝ 8.3 lb ⎠⎟⎟ 74. a) −62.11o C b) 2.5o C c) 56.7o C

5 (78 – 32) ≈ 25.6o C, not 20o C 9

73. 3 kg

(3)(2) = 6 = 4x x = 6/4 = 3/2 = 1.5 1.5 kg = 1500 g

9 (−62.11) + 32 = −111.798 + 32 = −79.8o F 5

9 (2.5) + 32 = 4.5 + 32 = 36.5o F 5 9 F = (56.7) + 32 = 102.06 + 32 ≈ 134.1o C 5

1.5 kg

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Exercise Set 8.4 1. Dimensional 2. Unit 60 seconds 1 minute 3. or because 60 seconds = 1 minute 1 minute 60 seconds 12 in 1 ft , 4. 1 ft 12 in 5.

100 cm 1 m , 1 m 100 cm

1000 mA 1A , 1A 1000 mA

2.2 lb Since we need to eliminate kilograms, kg must appear in the denominator. Since we 1 kg need to convert to pounds, lb must appear in the numerator.

1 kg Since we need to eliminate pounds; lb must appear in the denominator. Since we 2.2 lb need to convert to kilograms, kg must appear in the numerator. 1 ft 8. a) Since we need to eliminate centimeters, cm must appear in the denominator. Since we 30 cm need to convert to feet, ft must appear in the numerator. b)

30 cm Since we need to eliminate feet, ft must appear in the denominator. Since we 1 ft need to convert to centimeters, cm must appear in the numerator. 3.8 A 9. a) Since we need to eliminate gallons, gal must appear in the denominator. Since we 1 gal b)

need to convert to liters, l must appear in the numerator. 1 gal Since we need to eliminate liters, l must appear in the denominator. Since we 3.8 A need to convert to gallons, gal must appear in the numerator.

10. a)

0.8 m 2 Since we need to eliminate square yards, yd 2 must appear in the denominator. Since we 1 yd 2

need to convert to square meters, m 2 must appear in the numerator. b)

1 yd 2 Since we need to eliminate square meters, m 2 must appear in the denominator. Since we 0.8 m 2

need to convert to square yards, yd 2 must appear in the numerator.

⎛ 2.54 cm ⎞⎟ = 208.28 cm 11. 82 in. = (82 in.)⎜⎜ ⎜⎝ 1 in. ⎠⎟⎟ ⎛ 0.45 kg ⎞⎟ 12. 9 lb = (9 lb)⎜⎜ = 4.05 kg ⎜⎝ 1 lb ⎠⎟⎟ ⎛ 30 cm ⎞⎛ ⎟⎟⎜⎜ 1 m ⎞⎟⎟ = 1.26 m 13. 4.2 ft = (4.2 ft )⎜⎜ ⎟⎜100 cm ⎠⎟ ⎝⎜ 1 ft ⎠⎝

SECTION 8.4

⎛ 1 oz ⎞⎟ ⎟ = 15.25 oz 14. 427 g = ( 427 g)⎜⎜ ⎜⎝ 28 g ⎠⎟⎟ ⎛ 1 lb ⎞⎟ ⎟ = 360 lb 15. 162 kg = (162 kg )⎜⎜ ⎜⎝ 0.45 kg ⎠⎟⎟ ⎛ 0.8 m 2 ⎞⎟ ⎟ = 16 m 2 16. 20 yd 2 = ( 20 yd 2 )⎜⎜ ⎜⎝ 1 yd 2 ⎠⎟⎟ ⎛1.6 km ⎞⎟ = 62.4 km 17. 39 mi = (39 mi)⎜⎜ ⎜⎝ 1 mi ⎠⎟⎟ ⎛ 1 cm ⎞⎛ ⎟⎟⎜⎜ 1 in. ⎞⎟⎟ = 30.11811024 ≈ 30.12 in. 18. 765 mm = (765 mm)⎜⎜ ⎟⎜ 2.54 cm ⎠⎟ ⎝⎜10 mm ⎠⎝ ⎛ 1 acre ⎞⎟ 19. 675 ha = (675 ha )⎜⎜ = 1687.5 acres ⎜⎝ 0.4 ha ⎠⎟⎟ ⎛ 28 g ⎞⎟ 20. 253 oz = (253 oz)⎜⎜ = 7084 g ⎜⎝ 1 oz ⎠⎟⎟ ⎛ 1 pt ⎞⎟ = 43.19148936 ≈ 43.19 pints 21. 20.3 A = (20.3 A)⎜⎜ ⎜⎝ 0.47 A ⎠⎟⎟ ⎛ 0.9 t ⎞⎟ = 3.6 t 22. 4 T = ( 4 T )⎜⎜ ⎜⎝ 1 T ⎠⎟⎟ ⎛ 1 mi 2 ⎞⎟ ⎟ = 1.4615... ≈ 1.46 mi 2 23. 3.8 km2 = (3.8 km 2 )⎜⎜ ⎜⎝ 2.6 km 2 ⎠⎟⎟ ⎛1 fl oz ⎞⎟ = 0.853 ≈ 0.85 fl oz 24. 25.6 mA = ( 25.6 mA)⎜⎜ ⎜⎝ 30 mA ⎠⎟⎟ ⎛ 0.45 kg ⎞⎟ = 54 kg 25. 120 lb = (120 lb)⎜⎜ ⎜⎝ 1 lb ⎠⎟⎟ ⎛ 0.4 ha ⎞⎟ 26. 6.2 acres = (6.2 acres)⎜⎜ = 2.48 ha ⎜⎝ 1 acre ⎠⎟⎟ ⎛ 1 yd ⎞⎟ 27. 505 m = (505 m)⎜⎜ = 561. 1 ≈ 561.11 yd ⎜⎝ 0.9 m ⎠⎟⎟ ⎛ 1 yd ⎞⎟ 28. 175 m = (175 m)⎜⎜ = 194.4 ≈ 194.44 yd ⎜⎝ 0.9 m ⎠⎟⎟ ⎛100 cm ⎞⎛ ⎟⎟⎜⎜ 1 ft ⎞⎟⎟ = 1146.6 ≈ 1146.67 ft 29. 344 m = (344 m)⎜⎜ ⎟⎜ 30 cm ⎠⎟ ⎝⎜ 1 m ⎠⎝ ⎛100 cm ⎞⎛ ⎟⎟⎜⎜ 1 ft ⎞⎟⎟ = 1010 ft 30. 303 m = (303 m)⎜⎜ ⎟⎜ 30 cm ⎠⎟ ⎝⎜ 1 m ⎠⎝ ⎛ 1 mi ⎞⎟ 31. 30 kph = (30 km)⎜⎜ = 18.75 mph ⎜⎝1.6 km ⎠⎟⎟ ⎛1.6 km ⎞⎟ = 168 km 32. 105 mi = (105 mi)⎜⎜ ⎜⎝ 1 mi ⎠⎟⎟

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33. (6 yd )(9 yd ) = 54 yd 2

⎛ 0.8 m 2 ⎞⎟ ⎟ = 43.2 m 2 54 yd 2 = (54 yd 2 )⎜⎜ ⎜⎝ 1 yd 2 ⎠⎟⎟ ⎛1.6 km ⎞⎟ = 1147.2 km 34. 717 mi = (717 mi)⎜⎜ ⎜⎝ 1 mi ⎠⎟⎟ ⎛ 1 mi ⎞⎟ 35. 80 km = (80 km)⎜⎜ = 50 mph ⎜⎝1.6 km ⎠⎟⎟ ⎛ 30 mA ⎞⎟ 36. 8 fl oz = (8 fl oz )⎜⎜ = 240 mA ⎜⎝1 fl oz ⎠⎟⎟ ⎛ 1 oz ⎞⎟ ⎟ ≈ 0.21 oz 37. 6 g = (6 g)⎜⎜ ⎜⎝ 28 g ⎠⎟⎟ ⎛ 3.8 A ⎞⎟⎛ 1 kA ⎞ ⎟⎟ = 47.5 kA ⎟⎜⎜ 38. 12,500 gal = (12,500 gal)⎜⎜ ⎝⎜ 1 gal ⎠⎟⎟⎝⎜1000 A ⎠⎟ ⎛ 0.4 ha ⎞⎟ 39. (91,696 acres)⎜⎜ ⎟ = 36 678.4 ha ⎝⎜ 1 acre ⎠⎟ 40. (50 ft )(30 ft )(8 ft ) = 12,000 ft 3 ⎛ 0.03 m3 ⎞⎟ ⎟⎟ = 360 m3 12,000 ft 3 = (12, 000 ft 3 )⎜⎜⎜ 3 ⎝ 1 ft ⎠⎟ ⎛ 1 lb ⎞⎟ ⎟ = 2.2 lb 41. 1 kg = (1 kg)⎜⎜ ⎜⎝ 0.45 kg ⎠⎟⎟ $1.10 = $0.495 per pound 2.2 ⎛ 1 T ⎞⎟ = 8.3 ≈ 8.33 T 42. a) 7.5 t = (7.5 t )⎜⎜ ⎜⎝ 0.9 t ⎠⎟⎟ ⎛ 2000 lb ⎞⎟ b) 8.3 T = (8.3 T)⎜⎜ = 16, 666.7 ≈ 16, 666.67 lb ⎜⎝ 1 T ⎠⎟⎟ ⎛1000 A ⎞⎛ ⎟⎟⎜⎜ 1 gal ⎞⎟⎟ = 9078.947368 ≈ 9078.95 gal 43. 34.5 kA = (34.5 kA)⎜⎜ ⎟⎜ 3.8 A ⎠⎟ ⎜⎝ 1 kA ⎠⎝ ⎛ 28 g ⎞⎟ 44. 0.25 oz = (0.25 oz )⎜⎜ =7 g ⎜⎝ 1 oz ⎠⎟⎟

$80 = 11.42857143 ≈ $11.43 per gram 7 ⎛ 1 qt ⎞⎟ = 6 qt 45. 5.7 A = (5.7 A)⎜⎜ ⎜⎝ 0.95 A ⎠⎟⎟ ⎛ 30 cm ⎞⎟ 46. a) −282 ft = (−282 ft )⎜⎜ = −8460 cm ⎜⎝ 1 ft ⎠⎟⎟ ⎛ 1 m ⎞⎟ b) −8460 cm = (−8460 cm)⎜⎜ = −84.6 m ⎜⎝100 cm ⎠⎟⎟

SECTION 8.4

⎛ 1 km ⎞⎟ 47. a) 550 m = (550 m)⎜⎜ = 0.55 km ⎜⎝1000 m ⎠⎟⎟ ⎛ 1 km ⎞⎛ ⎟⎟⎜⎜ 1 mi ⎞⎟⎟ ≈ 0.34 mi b) 550 m = (550 m)⎜⎜ ⎟⎜1.6 km ⎠⎟ ⎜⎝1000 m ⎠⎝

48.

⎛ 1 g ⎞⎟ 1 carat = (0.125 carat )⎜⎜ = 0.025 g ⎜⎝ 5 carat ⎠⎟⎟ 8

⎛(3.3)2 ft 2 ⎞⎟ ⎜ ⎟⎟ = 10.89 ft 2 49. a) 1 m 2 = (1 m 2 )⎜⎜ ⎜⎜⎝ 1 m 2 ⎠⎟⎟ ⎛ (3.3)3 ft 3 ⎞⎟ ⎜ 3 b) 1 m3 = (1 m3 )⎜⎜ ⎟⎟ = 35.937 ft 3 ⎜⎜⎝ 1 m ⎠⎟⎟ ⎛ 0.4 ha ⎞⎟ ⎟ = 6.12 ha 50. a) 15.3 acres = (15.3 acres)⎜⎜ ⎜⎝ 1 acre ⎠⎟⎟ ⎛10,000 m 2 ⎞⎟ ⎟ = 61, 200 m 2 b) 6.12 ha = (6.12 ha )⎜⎜ ⎜⎝ 1 ha ⎠⎟⎟ ⎛ 0.45 kg ⎞⎟⎛⎜1 mg ⎞⎟ ⎟ = 25.2 mg 51. 56 lb = (56 lb)⎜⎜ ⎟⎜ ⎝⎜ 1 lb ⎠⎟⎜⎝ 1 kg ⎠⎟⎟ ⎛ 0.45 kg ⎞⎟⎛⎜1.5 mg ⎞⎟ 52. 170 lb = (170 lb)⎜⎜ ⎟⎜⎝ 1 kg ⎠⎟⎟⎟ = 114.75 mg ⎜⎝ 1 lb ⎠⎟⎜ ⎛ 1 g ⎞⎟ ⎛ 0.45 kg ⎞⎟⎛⎜ 200 mg ⎞⎟ ⎜ 53. 76 lb = (76 lb)⎜⎜ ⎟⎜⎝ 1 kg ⎠⎟⎟⎟ = 6840 mg; 6840 mg = (6840 mg)⎝⎜⎜1000 mg ⎠⎟⎟⎟ = 6.84 g ⎜⎝ 1 lb ⎠⎟⎜ ⎛ 0.45 kg ⎞⎟⎛⎜ 5 mg ⎞⎟ ⎟ = 184.5 mg 54. 82 lb = (82 lb)⎜⎜ ⎟⎜ ⎝⎜ 1 lb ⎠⎟⎝⎜ 1 kg ⎠⎟⎟ ⎛ 12.5 mg ⎞⎟ ⎟ = 25 mg 55. a) 2 teaspoons = ( 2 teaspoons)⎜⎜ ⎜⎝1 teaspoon ⎠⎟⎟ ⎛ 30 mA ⎞⎛ ⎟⎟⎜⎜12.5 mg ⎞⎟⎟ = 900 mg b) 12 fl oz = (12 fl oz )⎜⎜ ⎟⎜ 5 mA ⎠⎟ ⎝⎜1 fl oz ⎠⎝ ⎛ 236 mg ⎞⎟ ⎟ = 472 mg 56. a) 2 tablespoons = (2 tablespoons)⎜⎜ ⎜⎝1 tablespoon ⎠⎟⎟ ⎛ ⎞ ⎛ 30 mA ⎞⎛ ⎟⎜1 tablespoon ⎞⎟⎜ 236 mg ⎟ b) 8 fl oz = (8 fl oz )⎜⎜ ⎟⎟⎜⎜ 15 mA ⎠⎟⎟⎜⎜⎝1 tablespoon ⎠⎟⎟⎟ = 3776 mg ⎜⎝ 1 fl oz ⎠⎝ ⎛ 1 mi ⎞⎟ 57. a) 1.2 km = (1.2 km)⎜⎜ ≈ 0.75 mi ⎜⎝1.6 km ⎠⎟⎟ ⎛ 1 m 2 ⎞⎟ ⎟ ≈ 2.63 m 2 b) 6.84 km 2 = (6.84 km 2 )⎜⎜⎜ 2⎟ ⎝ 2.6 km ⎠⎟ ⎛ 1 yd ⎞⎟⎛⎜ 3 ft ⎞⎟ c) 426 m = (426 m)⎜⎜ ⎟⎜⎝1 yd ⎠⎟⎟⎟ = 1420 ft ⎜⎝ 0.9 m ⎠⎟⎜

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CHAPTER 8

The Metric System

⎛ 30 cm ⎞⎛ ⎟⎟⎜⎜ 1 m ⎞⎟⎟ = 289.2 m 58. a) 964 ft = (964 ft )⎜⎜ ⎟⎜100 cm ⎠⎟ ⎜⎝ 1 ft ⎠⎝ ⎛ 0.9 tonne ⎞⎟ b) 85,000 tons = (85,000 tons)⎜⎜ = 76 500 t ⎜⎝ 1 ton ⎠⎟⎟ ⎛1.6 km ⎞⎟ c) 28 mi = (28 mi)⎜⎜ = 44.8 kph ⎜⎝ 1 mi ⎠⎟⎟ ⎛ 1 km ⎞⎛ ⎟⎟⎜⎜ 1 mi ⎞⎟⎟ ≈ 6.08 mi 59. a) 9730 m = (9730 m)⎜⎜ ⎟⎜1.6 km ⎠⎟ ⎝⎜1000 m ⎠⎝ ⎛ 1 mi ⎞⎟ b) 709 kph = (709 kph )⎜⎜ ≈ 443.13 kph ⎜⎝1.6 km ⎠⎟⎟ ⎛ 1 mi ⎞⎟ c) 7181 kph = (181 kph )⎜⎜ ≈ 113.13 kph ⎜⎝1.6 km ⎠⎟⎟ 9 d) −45° C = (−45) + 32 = −49°F 5 ⎛ 2.00 €€ 2.00 €€ ⎞⎛ ⎟⎟⎜⎜ 1 kg ⎞⎟⎟ ≈ 0.9 €€ 60. a) = ⎜⎜ ⎟⎟⎜ 2.2 lb ⎟⎟ 1 lb ⎜⎝ 1 kg ⎠⎝ 1 kg ⎠ 61. a)

⎛ 7.00 €€ ⎞⎛ 7.00 €€ ⎟⎟⎜⎜1000 g ⎞⎟⎟ ≈ 70 €€ = ⎜⎜ ⎟ ⎟ ⎜ 100 g ⎝ 100 g ⎠⎝⎜ 1 kg ⎠⎟⎟ 1 kg

⎛ 31.5 €€ ⎞⎛ 31.5 €€ ⎟⎟⎜⎜ $1.40 ⎞⎟⎟ = $44.10 = ⎜⎜⎜ ⎟⎟⎜ 1 €€ ⎠⎟⎟ 1 lb 1 lb ⎝ 1 lb ⎠⎝

62. a)

⎛ 0.9 €€ ⎞⎛ 0.9 €€ ⎟⎟⎜⎜ $1.40 ⎞⎟⎟ = $1.26 = ⎜⎜ ⎟⎟⎜ 1 €€ ⎟⎟ ⎜⎝ 1 lb ⎠⎝ 1 lb 1 lb ⎠

⎛ 70 €€ ⎞⎛ 70 €€ ⎟⎟⎜⎜ 0.45 kg ⎞⎟⎟ ≈ 31.5 €€ = ⎜⎜ ⎟ ⎟ ⎜ 1 kg ⎝ 1 kg ⎠⎝⎜ 1 lb ⎠⎟⎟ 1 lb

12.85 pesos 17.35 pesos ≈ 0.74 euro 1 euro

0.74 euro 0.70 euro ≈ 1.06 Canadian dollars 1 Canadian dollar

12.85 pesos 12.12 pesos ≈ 1.06 Canadian dollars 1 Canadian dollar

63. a)

745 kmph 1.85 kmph ≈ 402 knots 1 knot

402 knots 0.87 knots ≈ 463 mph 1 mph

463 mph 0.62 mph ≈ 745 kph 1 kmph

⎛ 0.24 A ⎞⎟ 64. (0.5 c)⎜⎜ = 0.12 A graham cracker crumbs ⎜⎝ 1 c ⎠⎟⎟ ⎛ 28 g ⎞⎟ ⎟ = 336 g nuts ⎝ 1 oz ⎠⎟

(12 oz)⎜⎜⎜

⎛ 28 g ⎞⎟ ⎟ = 224 g chocolate pieces ⎝ 1 oz ⎠⎟

(8 oz )⎜⎜⎜

⎛ 4 ⎞⎛ ⎞ ⎜⎜ c⎟⎟⎜⎜ 0.24 A ⎟⎟ = 0.32 A flaked coconut ⎟ ⎝⎜ 3 ⎠⎝⎜ 1 c ⎠⎟

REVIEW EXERCISES

295

⎛ 4 ⎞⎛ ⎞ ⎜⎜ c⎟⎟⎜⎜ 0.24 A ⎟⎟ = 0.32 A condensed milk ⎟⎜ 1 c ⎠⎟ ⎝⎜ 3 ⎠⎝ ⎛ 2.54 cm ⎞⎟ ⎛ 2.54 cm ⎞⎟ ⎟×(13 in.)⎜⎜⎜ ⎟ = 22.86 cm × 33.02 cm baking pan ⎝ 1 in. ⎠⎟ ⎝ 1 in. ⎠⎟

(9 in.)⎜⎜⎜

5 350° F = (350 − 32) = 176.6 ≈ 176.7° C 9 ⎛ 2.54 cm ⎞⎟ ⎛ 2.54 cm ⎞⎟ (1.5 in.)⎜⎜⎜ ⎟⎟×(3 in.)⎜⎜⎜ ⎟ = 3.81 cm × 7.62 cm bars ⎝ 1 in. ⎠ ⎝ 1 in. ⎠⎟

⎛ ⎞⎟ ⎜ ⎛1 grain ⎞⎟⎜⎜ 1 mA ⎟⎟⎟ ⎟⎜ ⎟⎟ = 1.0 cc, or b) 65. (0.2 mg)⎜⎜ ⎜⎝ 60 mg ⎠⎟⎟⎜⎜ 1 ⎟⎟ ⎜ grain ⎜⎝ 300 ⎠⎟⎟ ⎛ 0.18 kg ⎞⎟⎜⎛ 1 lb ⎞⎟ ⎟ = 7.8 lb 66. 15(130 lb) = 1950 lb; (1950 lb)⎜⎜ ⎜⎝ 100 lb ⎠⎟⎟⎜⎜⎝ 0.45 kg ⎠⎟⎟ ⎛1000 mA ⎞⎟⎛⎜1 cm3 ⎞⎟ ⎟ = 3000 cc 67. a) (3.0 A )⎜⎜ ⎟⎜ ⎝⎜ 1 l ⎠⎟⎜⎜⎝ 1 mA ⎠⎟⎟ ⎛ 1 in.3 ⎞⎟ ⎜ ⎟⎟ = 3000 = 183.071... ≈ 183.07 in.3 b) (3000 cm3 )⎜⎜ ⎜⎜⎝(2.54)3 cm3 ⎠⎟⎟ 16.387064 68. 1 microscope

69. wonton

⎛ ⎞ ⎛ 28 g ⎞⎟ = 448 g⎟⎟⎟ 70. 1 pound cake ⎜⎜1 lb = 16 oz; 16 oz ⎜⎜ ⎜⎝ 1 oz ⎠⎟⎟ ⎜⎝ ⎠⎟

71. 1 kilohurtz

72. 2 megacycles 74. 1 decacards

73. 1 megaphone 75. 2 kilomockingbirds

Review Exercises

1. 1000× base unit 4. 100× base unit 7. 82 mg = 0.082 g 10. 1 000 000 mg = 1 kg 13. 2.67 kl = 2 670 000 ml 14 630 cl = 146 300 ml 3000 ml, 14 630 cl, 2.67 kl 16. Degrees Celsius 19. Millimeters 22. Centimeters 25. c 28. a

1 of base unit 100 5. 10 times base unit

8. 3.2 l = 320 cl 11. 4.62 kl = 4620 l 14. 0.047 km = 47 m 47 000 cm = 470 m 0.047 km, 47 000 cm, 4700 m 17. Square meters 20. Kilograms or tonnes 23. a) and b) Answers will vary. 26. b 29. a

1 of base unit 1000 1 6. of base unit 10 9. 0.0035 cm = 0.035 mm 12. 192.6 dag = 19 260 dg 15. Meters 3.

18. Milliliters or cubic cm 21. Kilometers 24. a) and b) AWV 27. c 30. c

296

CHAPTER 8

The Metric System

⎛ 1 t ⎞⎟ ⎟ = 3.6 t 31. 3600 kg = (3600 kg)⎜⎜ ⎜⎝1000 kg ⎠⎟⎟

⎛1000 kg ⎞⎛ ⎟⎟⎜1000 g ⎞⎟⎟ = 4 300 000 g 32. 4.3 t = (4.3 t )⎜⎜ ⎜ ⎜⎝ 1 t ⎠⎟⎟⎜⎝ 1 kg ⎠⎟⎟

9 33. 24° C = (24) + 32 = 75.2° F 5 5 35. −6° F = (−6 − 32) = −21. 1 ≈ −21.1° C 9

5 34. 68° F = (68 − 32) = 20° C 9 9 36. 39° C = (39) + 32 = 102.2° F 5

37. l = 4 cm, w = 1.6 cm; A = lw = 4 (1.6) = 6.4 cm 2 38. r = 1.5 cm; A = π r 2 ≈ 3.14 (1.5) = 7.065 ≈ 7.07 cm 2 2

39. a) V = πr 2 h = π (1)(1) ≈ 3.14 m3

⎛1000 kg ⎞

(3.14 m3 )⎜⎜⎜⎝ 1 m3 ⎠⎟⎟⎟ ≈ 3140 kg

40. a) A = lw = (33.7)(26.7) = 899.79 cm 2 ⎛ 1 m2 ⎞⎟ ⎟ = 0.089979 m 2 b) 899.79 cm 2 = (899.79 cm 2 )⎜⎜ 2⎟ ⎜⎝10,000 cm ⎠⎟

41. a) V = lwh = (80)(40)(30) = 96 000 cm3 ⎛ 1 m3 ⎞⎟ ⎜ ⎟⎟ = 0.096 m3 b) 96 000 cm3 = (96 000 cm3 )⎜⎜ 3 ⎜⎜⎝ (100) cm3 ⎠⎟⎟ ⎛ 1 mA ⎞⎟ = 96 000 mA c) 96 000 cm3 = (96 000 cm3 )⎜⎜ ⎜⎝1 cm3 ⎠⎟⎟

⎛ 1 kA ⎞⎟ = 0.096 kA 0.096 m 3 = (0.096 m 3 )⎜⎜ ⎜⎝1 m3 ⎠⎟⎟

42. Since 1 km = 100×1 dam, 1 km 2 = 1002 ×1 dam 2 = 10 000 dam 2 . Thus, 1 square kilometer is 10,000 times larger than a square dekameter. ⎛ 1 in. ⎞⎟ ⎛ 1 lb ⎞⎟ = 933.858... ≈ 33.86 in. 43. (86 cm)⎜⎜ ⎟ = 233.3 ≈ 233.33 lb 44. (105 kg)⎜⎜ ⎜⎝ 2.54 cm ⎠⎟⎟ ⎜⎝ 0.45 kg ⎠⎟⎟ ⎛ 0.9 m ⎞⎟ ⎟ = 74.7 m 45. (83 yd)⎜⎜ ⎜⎝ 1 yd ⎠⎟⎟

⎛ 1 yd ⎞⎟ 46. (100 m)⎜⎜ = 111. 1 ≈ 111.11 yd ⎜⎝ 0.9 m ⎠⎟⎟

⎛1.6 km ⎞⎟ = 72 kph 47. ( 45 mi)⎜⎜ ⎜⎝ 1 mi ⎠⎟⎟

⎛ 1 qt ⎞⎟ = 44.210... ≈ 44.21 qt 48. (42 A )⎜⎜ ⎜⎝ 0.95 A ⎠⎟⎟

⎛ 3.8 A ⎞⎟ ⎟ = 76 A 49. ( 20 gal)⎜⎜ ⎜⎝ 1 gal ⎠⎟⎟

50.

⎛ 1 in.2 ⎞

51.

(96 cm 2 )⎜⎜⎜ 6.5 cm2 ⎟⎟⎟⎟ = 14.76923077 ≈ 14.77 in.2

53.

(15 yd3 )⎜⎜⎜ 1 yd 3

⎝

⎠

⎛ 0.729 m3 ⎞⎟ ⎟⎟ = 10.9 m 3 ⎝ ⎠⎟

⎛ 1 yd 3 ⎞

(60 m3 )⎜⎜⎜ 0.76 m3 ⎟⎟⎟⎟ = 78.947... ≈ 78.95 yd 3 ⎝

⎠

⎛ 0.95 A ⎞⎟ ⎟ = 3.8 A 52. ( 4 qt )⎜⎜ ⎜⎝ 1 qt ⎠⎟⎟ ⎛1.6 km ⎞⎟ 54. (62 mi)⎜⎜ = 99.2 km ⎜⎝ 1 mi ⎠⎟⎟

⎛1.6 km ⎞⎟ 55. (241 mi)⎜⎜ ⎟ = 385.6 km ⎝⎜ 1 mi ⎠⎟

⎛ 1 yd 3 ⎞⎟ ⎟ ≈ 429,322.37 yd 3 56. (326,285 m3 )⎜⎜ ⎜⎝ 0.76 m3 ⎠⎟⎟

57. a) 700 (1.5 kg) = 1050 kg ⎛ 1 lb ⎞⎟ ⎟ = 2333.3 ≈ 2333.33 lb b) 1050 kg = (1050 kg)⎜⎜ ⎜⎝ 0.45 kg ⎠⎟⎟

CHAPTER TEST

297

⎛ 0.09 m 2 ⎞⎟ ⎟ = 32.4 m 2 58. A = lw = ( 24)(15) = 360 ft 2 ; 360 ft 2 = (360 ft 2 )⎜⎜ ⎜⎝ 1 ft 2 ⎠⎟⎟ ⎛ 3.8 A ⎞⎟⎛ 1 kA ⎞ ⎟ = 190 kA ⎟⎜ 59. a) (50,000 gal)⎜⎜ ⎜⎝ 1 gal ⎠⎟⎟⎜⎝⎜1000 A ⎠⎟⎟

⎛1000 A ⎞⎛ 1 kg ⎞⎟ b) (190 kA)⎜⎜ ⎟⎟⎟⎜⎜⎜ ⎟ = 190 000 kg ⎜⎝ 1 kA ⎠⎝ 1 A ⎠⎟

⎛1.6 km ⎞⎟ 60. a) 65 mi/hr = (65 mi/hr )⎜⎜ = 104 km/hr ⎜⎝ 1 mi ⎠⎟⎟ ⎛1000 m ⎞⎟ ⎟ = 104,000 m/hr b) 104 km/hr = (104 km/hr )⎜⎜ ⎜⎝ 1 km ⎠⎟⎟ ⎛ 1 mA ⎞⎛ ⎟⎟⎜⎜ 1 A ⎞⎟⎟ = 252 A 61. a) V = lwh = (90)(70)(40) = 252 000 cm 3 ; 252 000 cm 3 = (252 000 cm3 )⎜⎜ ⎟⎜1000 mA ⎠⎟ ⎜⎝1 cm3 ⎠⎝ ⎛1 kg ⎞⎟ = 252 kg b) 252 A = (252 A)⎜⎜ ⎜⎝ 1 A ⎠⎟⎟ ⎛ 1 lb ⎞⎟ $3.50 ⎟ = 2.2 lb; 62. 1 kg = (1 kg)⎜⎜ = $1.575 ≈ $1.58 per pound ⎜⎝ 0.45 kg ⎠⎟⎟ 2.2

Chapter Test 1. 759 cA = 0.759 daA 2. 273 hm = 2,730,000 m

⎛100 dam ⎞⎟ = 100 dam or 100 times greater 3. 1 km = (1 km)⎜⎜ ⎜⎝ 1 km ⎠⎟⎟ ⎛ 1 km ⎞⎟ = 2.4 km 4. 400 (6) = 2400 m; (2400 m)⎜⎜ ⎜⎝1000 m ⎠⎟⎟

5. b

6. a 8. b

7. c 9. b

⎛1002 cm 2 ⎞⎟ ⎟⎟ = 10 000 cm 2 or 10,000 times greater 10. 1 m 2 = (1 m 2 )⎜⎜⎜ 2 ⎝ 1m ⎠⎟ ⎛10002 mm3 ⎞⎟ ⎟⎟ = 1 000 000 cm3 or 1,000,000 times greater 11. 1 m3 = (1 m3 )⎜⎜ ⎜⎝ 1 m3 ⎠⎟ ⎛ 28 g ⎞⎟ = 6300 g 12. 225 oz = (225 oz )⎜⎜ ⎜⎝ 1 oz ⎠⎟⎟ ⎛ 1 yd ⎞⎛ ⎟⎟⎜ 3 ft ⎞⎟⎟ = 564.3 ft 13. 169.29 m = (169.29 m)⎜⎜ ⎟⎜⎜1 yd ⎟⎟ ⎜⎝ 0.9 m ⎠⎟⎜ ⎝ ⎠ ⎛ 1 mi 2 ⎞⎟ ⎟ 14. 53,321 km 2 = (53,321 km 2 )⎜⎜ ⎜⎝ 2.6 km 2 ⎠⎟⎟

5 15. 10° F = (10 − 32) = −12.2 ≈ −12.22° C 9

≈ 20,508.1 mi 2

9 16. 20° C = ( 20) + 32 = 68° F 5

298

CHAPTER 8

The Metric System

⎛1000 g ⎞⎟ ⎟ = 300,000 g 17. a) 300 kg = (300 kg)⎜⎜ ⎜⎝ 1 kg ⎠⎟⎟ ⎛ 1 lb ⎞⎟ ⎟ = 666.6 lb ≈ 670 lb b) 300 kg = (300 kg)⎜⎜ ⎜⎝ 0.45 kg ⎠⎟⎟

18. a) V = lwh = 20 (20)(8) = 3200 m3

⎛1000 kA ⎞⎟ = 3200 k A b) 3200 m 3 = (3200 m 3 )⎜⎜ ⎜⎝ 1 m 3 ⎠⎟⎟

⎛1000 A ⎞⎛ 1 kg ⎞⎟ ⎟⎟⎟⎜⎜ ⎟ = 3 200 000 kg c) 3200 kA = (3200 kA) ⎜⎜ ⎟⎜ 1 A ⎠⎟⎟ ⎜⎝ 1 kA ⎠⎝

19. Total surface area: 2lh + 2 wh = 2 ( 20)(6) + 2 (15)(6) = 420 m 2 ⎛ 1 A ⎞⎟ = 42 A Liters needed for first coat: ( 420 m 2 )⎜⎜ ⎜⎝10 m 2 ⎠⎟⎟ ⎛ 1 A ⎞⎟ = 28 A Liters needed for second coat: ( 420 m 2 )⎜⎜ ⎜⎝15 m 2 ⎠⎟⎟

Total liters needed: 42 + 28 = 70 A ⎛ $3.50 ⎞⎟ = $245 Total cost: (70 A)⎜⎜ ⎜⎝ 1 A ⎠⎟⎟ ⎛1.65 euros ⎞⎟⎛⎜ 3.8 A ⎞⎟ ⎟ = 6.27 euros per gallon 20. a) ⎜⎜ ⎟⎜ ⎝⎜ ⎠⎟⎜⎝ 1 gal ⎠⎟⎟ 1A ⎛ ⎞⎟ $1 b) 6.27 euros per gallon= (6.27 euros per gallon )⎜⎜ ≈ $9.36 per gallon ⎜⎝ 0.67 euros ⎠⎟⎟

Group Projects

⎛ 0.45 kg ⎞⎟⎛⎜ 20 mg ⎞⎟ ⎟ = 1764 mg 1. a) (196 lb)⎜⎜ ⎟⎜ ⎝⎜ 1 lb ⎠⎟⎝⎜ 1 kg ⎠⎟⎟

⎛ 250 cc ⎞⎛ ⎟⎟⎜⎜ 1 hr ⎞⎟⎟ = 4.16 ≈ 4.17 cc/min b) ⎜⎜ ⎟⎜ 60 min ⎠⎟ ⎝⎜ 1 hr ⎠⎝

⎛ 0.45 kg ⎞⎟ 27 kg = 27 kg; child's dose: 2. a) (60 lb)⎜⎜ (70 mg) = 28 mg ⎜⎝ 1 lb ⎠⎟⎟ 67.5 kg

child's weight in kg child's weight in kg ×70 mg = 70 mg; =1 67.5 kg 67.5 kg ⎛ 1 lb ⎞⎟ ⎟ = 150 lb Child's weight: 67.5 kg = (67.5 kg)⎜⎜ ⎜⎝ 0.45 kg ⎠⎟⎟

⎛ 1 lb ⎞⎟ ⎟⎟ ≈ 555,555.6 lbs 3. Payload: (250, 000 kg)⎜⎜⎜ ⎝ 0.45 kg ⎠⎟ ⎛ 1 ft ⎞⎟ Length: 84 m = (84 m)⎜⎜ = 280 ft ⎜⎝ 0.3 m ⎠⎟⎟ ⎛ 1 yd ⎞⎟⎛⎜ 3ft ⎞⎟ ⎟ ≈ 294.67 ft Wingspan: 88.4 m = (88.4 m)⎜⎜ ⎟⎜ ⎝⎜ 0.9 m ⎠⎟⎜⎝1 yd ⎠⎟⎟ ⎛ 1 yd ⎞⎟⎜⎛ 3ft ⎞⎟ ⎟ ≈ 60.3 ft Height: 18.1 m = (18.1 m)⎜⎜ ⎟⎜ ⎝⎜ 0.9 m ⎠⎟⎜⎝1 yd ⎠⎟⎟

GROUP PROJECTS

⎛ 1 ft 3 ⎞⎟ ⎟ ≈ 43,333.33 ft 3 Cargo volume: (1300 m3 )⎜⎜⎜ 3⎟ ⎝ 0.03 m ⎠⎟ ⎛ 1 lb ⎞⎟ ⎟ = 388,888.8 ≈ 388,888.9 lbs Empty weight: (175,000 kg)⎜⎜ ⎜⎝ 0.45 kg ⎠⎟⎟ ⎛ 0.45 kg ⎞⎟ Maximum take-off -weight: (1,323, 000 lb)⎜⎜ = 595,350 kg ⎜⎝ 1 lb ⎠⎟⎟ ⎛1 yd ⎞⎟⎜⎛ 0.9 m ⎞⎟ Take-off-run with maximum weight: (11,500 ft )⎜⎜ ⎟⎜⎝ 1 yd ⎠⎟⎟⎟ = 3450 m ⎜⎝ 3 ft ⎠⎟⎜ ⎛1.6 km ⎞⎟ Maximum speed: (530 mph )⎜⎜ = 848 kmph ⎜⎝ 1 mi ⎠⎟⎟ ⎛1.6 km ⎞⎟ = 15,312 km Range with maximum fuel: (9570 mi)⎜⎜ ⎜⎝ 1 mi ⎠⎟⎟ ⎛1.6 km ⎞⎟ = 4000 km Range with maximum payload: (2500 mi)⎜⎜ ⎜⎝ 1 mi ⎠⎟⎟ ⎛1 yd ⎞⎟⎛⎜ 0.9 ⎞⎟ ⎟ = 10,830 m Service ceiling: (36,100 ft )⎜⎜ ⎟⎜ ⎝⎜ 3 ft ⎠⎟⎝⎜1 yd ⎠⎟⎟

299

CHAPTER NINE GEOMETRY Exercise Set 9.1 1. Parallel 2. Skew 3. Angle 4. Complementary 5. Supplementary 6. Adjacent 7. Straight 8. Right 9. Obtuse 10. Acute 11. Vertical 12. Transversal JJJG 13. Ray, AB 17.

JJJG Ray, BA

14.

Open line segment, HJJG 18. Line, AB

15. 19.

Half line, Half open line segment,

22. 30.

HJJG EG + BCF

31.

33.

AD (CBF or (FBC HJJG DE

34.

{ B}

35.

37.

(ABE

38.

(FBE or (EBF ∅

39. 43.

42.

Straight None of these 90°− 32 43 ° = 57 14 °

21. 29.

41.

23.

HJJG BC JJJG BC

45. 49. 53.

Right Obtuse 90°−10° = 80°

46. 50. 54.

Acute None of these 90°−15° = 75°

47. 51. 55.

56.

90°− 31 25 ° = 58 53 °

57.

90°− 64.7° = 25.3°

58. 90°− 0.01° = 89.99°

59.

180°− 80° = 100°

60.

180°−150° = 30°

61. 180°− 20.5° = 159.5°

63. 180°− 43 75 ° = 136 72 °

64. 180°− 64 167 ° = 115 169 °

62. 180°−148.7° = 31.3°

301

16.

Line segment,

20.

AB Half line,

24.

32.

40.

∅ HJJG DE JJJG DE

44. 48. 52.

FG Straight Right

36.

28.

302

CHAPTER 9 Geometry

65. f 68. d 71. Let x = measure of ( 2 5x = measure of ( 1 x + 5 x = 90

66. 69.

c a

72.

67. b 70. e Let x = measure of ( 1 90 − x = measure of ( 2

x − (90 − x ) = 22

6 x = 90 90 x= = 15°, m( 2 6 5 x = (5)(15) = 75°, m(1

73.

Let x = measure of ( 1 180 − x = measure of ( 2

x − 90 + x = 22 2 x − 90 = 22 2 x = 112 112 = 56°, m(1 2 90 − x = 90 − 56 = 34°, m( 2 x=

74.

x −(180 − x ) = 88

Let x = measure of ( 1 17x = measure of ( 2 x + 17 x = 180 18 x = 180 180 x= = 10°, m(1 18 17 x = 17 (10) = 170°, m(2

x −180 + x = 88 2 x −180 = 88 2 x = 268 268 = 134°, m(1 2 180 − x = 180 −134 = 46°, m(2 x=

75. m( 1 + 125° = 180°

76. m( 3 + 120° = 180°

m( 1 = 55° m( 2 = m( 1 (vertical angles)

m( 3 = 60°

m( 3 = 125° (vertical angles) m( 5 = m( 2 (alternate interior angles) m( 4 = m( 3 (alternate interior angles) m( 7 = m( 4 (vertical angles) m( 6 = m( 5 (vertical angles)

m( 4 = 120° (vertical angles) m( 7 = m( 3 (vertical angles) m( 6 = m( 3 (alternate interior angles) m( 1 = m( 6 (vertical angles) m( 5 = m( 4 (alternate exterior angles) m( 2 = m( 5 (vertical angles)

Measures of angles 3, 4, and 7 are each 125°. Measures of angles 1, 2, 5, and 6 are each 55°.

Measures of angles 2, 4, and 5 are each 120°. Measures of angles 1, 3, 6, and 7 are each 60°.

77. m( 3 + 30° = 180°

78. m( 1 + 25° = 180°

m( 3 = 150° m( 1 = 30° (vertical angles) m( 2 = m( 3 (vertical angles) m( 4 = m( 1 (corresponding angles) m( 7 = m( 4 (vertical angles) m( 6 = m( 3 (alternate interior angles) m( 5 = m( 6 (vertical angles) Measures of angles 1, 4, and 7 are each 30°. Measures of angles 2, 3, 5, and 6 are each 150°.

m( 1 = 155° m( 3 = m( 1 (vertical angles) m( 2 = 25° (vertical angles) m( 4 = m( 3 (alternate interior angles) m( 7 = m( 4 (vertical angles) m( 5 = m( 2 (corresponding angles) m( 6 = m( 5 (vertical angles) Measures of angles 2, 5, and 6 are each 25°. Measures of angles 1, 3, 4, and 7 are each 155°.

SECTION 9.1

303

79.

x + 2x + 12 = 90 3x + 12 = 90 3x = 78 78 x = = 26°, m( 2 3 90 − x = 90 − 26 = 64°, m( 1

80.

x + 5x − 6 = 90 6x − 6 = 90 6x = 96 96 = 16°, m( 1 x = 6 90 − x = 90 – 16 = 74°, m( 2

81.

x + 2x - 9 = 90 3x - 9 = 90 3x = 99 99 = 33°, m( 1 x = 3 90 – x = 90 – 33 = 57°, m( 2

82.

x + 8x - 9 = 90 9x - 9 = 90 9x = 99 99 = 11°, m( 2 x = 9 90 – x = 90 – 11 = 79°, m( 1

83.

x + 3x – 4 = 180 4x – 4 = 180 4x = 184 184 = 46°, m( 2 x = 4 180 – x = 180 – 46 = 134°, m( 1

84.

x + 7x – 12 = 180 8x – 12 = 192 8x = 192 192 = 24°, m( 2 x = 8 180 – x = 180 – 24 = 156°, m( 1

85.

x + 5x + 6 = 180 6x + 6 = 180 6x = 174 174 = 29°, m( 1 x = 6 180 – x = 180 – 29 = 151°, m( 2

86.

x + 6x + 5 = 180 7x + 5 = 180 7x = 175 175 = 25°, m( 1 x = 7 180 – x = 180 – 25 = 155°, m( 2

For Exercises 87 - 94, the answers given are one of many possible answers. 87.

Plane ABG and plane JCD

88.

HJJG HJJG EF and DG

89.

HJJG HJJG BG and DG

90.

Plane ABG and plane BCD

91.

Plane AGB ∩ plane ABC ∩ plane BCD = {B}

92.

HJJG Plane HGD ∩ plane FGD ∩ plane BGD = GD

93.

HJJG BC ∩ plane ABG = {B}

94.

HJJG HJJG AB ∩ plane ABG = AB

95. a) Undefined terms, definitions, postulates (axioms), and theorems b) First, Euclid introduced undefined terms. Second, he introduced certain definitions. Third, he stated primitive propositions called postulates (axioms) about the undefined terms and definitions. Fourth, he proved, using deductive reasoning, other propositions called theorems. 96. An axiom (postulate) is a statement that is accepted as being true on the basis of its "obviousness" and its relation to the physical world. A theorem is a statement that has been proven using undefined terms, definitions, and axioms. 97. a) An infinite number of lines can be drawn through a given point. b) An infinite number of planes can be drawn through a given point.

304

CHAPTER 9 Geometry

98. If the two planes are not parallel, the intersection is a straight line. 99. An infinite number of planes can be drawn through a given line. 100. a) Yes, any three noncollinear points always determine a plane. b) No, the plane determined is unique. c) An infinite number of planes can be drawn through three collinear points. 101. Always true. If any two lines are parallel to a third line, then they must be parallel to each other. 102. Sometimes true. A triangle must always contain at least two acute angles. Some triangles contain three acute angles. 103. Sometimes true. Vertical angles are only complementary when each is equal to 45°. 104. Sometimes true. Alternate exterior angles are only supplementary when each is equal to 90°. 105. Sometimes true. Alternate interior angles are only complementary when each is equal to 45°. 106. Never true. The sum of two obtuse angles is greater than 180°. 107. Answers will vary. 108. No. Line m and line n may intersect. 109. No. Line l and line n may be parallel or skew. 110.

m(1 + m(2 = 180°

111.

m(3 + m(4 = 180°

1 2

180° + 180° = 360°

SECTION 9.2

305

c) m(CBD = y

112. a)

y = 2 x = 2 (30°) = 60° d) m(ABD + m(DBE = 180° m(ABD = x + y = 30° + 60° = 90°. 90° + m(DBE = 180° m(DBE = 180°− 90° = 90°.

Other answers are possible. b) Let m(ABC = x and m(CBD = y.

x + y = 90° and y = 2 x Substitute y = 2 x into x + y = 90°. x + 2 x = 90° 3x = 90° 3x 90° = 3 3 x = 30° = m(ABC

Exercise Set 9.2 1. Polygon 2. Regular 3. Proportion 4. 180° 5. Congruent 6. 360°

7. 11. 15. 19. 23. 27.

a) Octagon b) Regular a) Parallelogram b) Not regular a) Isosceles b) Right a) Scalene b) Acute Trapezoid Rhombus

8. 12. 16. 20. 24. 28.

a) Trapezoid b) Not regular a) Dodecagon b) Regular a) Isosceles b) Obtuse a) Equilateral b) Acute Rectangle Trapezoid

9. 13. 17. 21. 25.

a) Triangle b) Regular a) Heptagon b) Not regular a) Isosceles b) Acute a) Scalene b) Right Square

10. 14. 18. 22. 26.

a) Rectangle b) Not regular a) Pentagon b) Regular a) Scalene b) Right a) Scalene b) Obtuse Parallelogram

29. The measures of the other two angles of the triangle are 37° and 180° − 133° (supplementary angles). The measure of the third angle of the triangle is 180° − (37°) – (180° − 133°) = 96. Since angle x is a vertical angle with the 96° angle, the measure of angle x is 96°. 30. The measures of two angles of the triangle are 180° − 105° and 180° − 133° (supplementary angles). The measure of the third angle of the triangle is 180° − (180° − 105°) – (180° − 133°) = 58°. Since angle x is a vertical angle with the 58° angle, the measure of angle x is 58°.

306

CHAPTER 9 Geometry

31. The measure of one angle of the triangle is 27° (by vertical angles). The measure of another angle of the triangle is 180° - 57° = 123°. The measure of the third angle of the triangle is 180° - 27° - 123° = 30°. The measure of angle x is 180° - 30° = 150° (The 30° angle and angle x form a straight angle.). 32. The given measure of one angle of the triangle is 35°. The measure of another angle of the triangle is 30° (by vertical angles). The measure of the third angle of the triangle is 180° - 35° - 30° = 115°. The measure of angle x is 180° - 115° = 65° (The 115° angle and angle x form a straight angle.). 33.

34.

35. 37.

39.

Angle 1 2 3 4 5 6 7 8 9 10 11 12

Measure 90° 50° 130° 50° 50° 40° 90° 130° 140° 40° 140° 40°

Angle 1 2 3 4 5 6 7 8 9 10 11 12

Measure 51° 59° 70° 70° 51° 110° 51° 129° 70° 110° 129° 51°

Reason ( 1 and ( 7 are vertical angles ( 2 and ( 4 are corresponding angles ( 3 and ( 4 form a straight angle Vertical angle with the given 50° angle ( 2 and ( 5 are vertical angles Vertical angle with the given 40° angle ( 2, ( 6, and ( 7 form a straight angle ( 3 and ( 8 are vertical angles ( 9 and ( 10 form a straight angle ( 10 and ( 12 are vertical angles ( 9 and ( 11 are vertical angles ( 6 and ( 12 are corresponding angles Reason ( 1 and ( 5 are vertical angles Vertical angle with the given 59° angle ( 1, ( 2, and ( 3 form a straight angle ( 3 and ( 4 are vertical angles ( 5 and ( 12 are corresponding angles ( 6 and the given 67° angle form a straight angle The sum of the measures of the interior angles of a triangle is 180° ( 8 and ( 12 form a straight angle ( 4 and ( 9 are corresponding angles ( 6 and ( 10 are vertical angles ( 8 and ( 11 are vertical angles ( 7 and ( 12 are vertical angles

n=5 (5 − 2) × 180° = 3 × 180° = 540° n=9 (9 − 2) × 180° = 7 × 180° = 1260°

36.

n = 20 (20 − 2) × 180° = 18 × 180° = 3240°

40.

38.

n=7 (7 − 2) × 180° = 5 × 180° = 900° n = 10 (10 − 2) × 180° = 8 × 180° = 1440° n = 12 (12 − 2) × 180° = 10 × 180° = 1800°

SECTION 9.2

41. a) The sum of the measures of the interior angles of a triangle is 180°. Dividing by 3, the number of angles, each interior angle measures 60°. b) Each exterior angle measures 180° − 60° = 120°.

43. a) The sum of the measures of the interior angles of an octagon is (8 − 2) × 180° = 6 × 180° = 1080°. Dividing by 8, the number of angles, each interior angle measures 135°. b) Each exterior angle measures 180° − 135° = 45°.

45. a) The sum of the measures of the interior angles of a decagon is (10 − 2) × 180° = 8 × 180° = 1440°. Dividing by 10, the number of angles, each interior angle measures 144°. b) Each exterior angle measures 180° − 144° = 36°.

47. Let x = A′ C ′ A′ C ′

42. a) The sum of the measures of the interior angles of a quadrilateral is (4 − 2) × 180° = 2 × 180° = 360°. Dividing by 4, the number of angles, each interior angle measures 90°. b) Each exterior angle measures 180° − 90° = 90°. 44. a) The sum of the measures of the interior angles of a nonagon is (9 − 2) × 180° = 7 × 180° = 1260°. Dividing by 9, the number of angles, each interior angle measures 140°. b) Each exterior angle measures 180° − 140° = 40°. 46. a) The sum of the measures of the interior angles of an icosagon is (20 − 2) × 180° = 18 × 180° = 3240°. Dividing by 20, the number of angles, each interior angle measures 162°. b) Each exterior angle measures 180° − 162° = 18°. Let y = B ′C ′

A′ B ′ AC AB x 1 = 5 2.5 2.5 x = 5 x = 2

B ′C ′ A′ B ′ = BC AB y 1 = 2.5 4 2.5y = 4

y =

48. Let x = BC BC B ′C ′ x 12 2x x

4 = 1.6 2.5

Let y = A′ C ′ AB A′ B ′ 50 5 = = 20 2 = 60 = 30

A′ C ′ A′ B ′ = AC AB y 2 = 5 40 5y = 80

y = 16

307

308

CHAPTER 9 Geometry

Let y = B ′C ′

49. Let x = DC DC AB = D ′C ′ A′ B ′ x 4 = 10 6 10 x = 24 24 12 x = = 10 5

B ′C ′ A′ B ′ = BC AB y 10 = 4 3 4 y = 30 y =

x 15 = 24 45 45 x = 360 x = 8

50.

30 15 = 4 2

12 y = 24 45 24 y = 540 y = 22

51. Let x = D ′C ′ D ′C ′

Let y = A′ B ′

A′ D ′ AD DC x 22.5 = 16 18 18x = 360 x = 20

52. Let x = B ′C ′ BC B ′C ′ 12 x 27x x

A′ B ′ A′ D ′ = AB AD y 22.5 = 17 18 18 y = 382.5

y = 21.25

Let y = D ′C ′ AB A′ B ′ 27 = 36 = 432 = 16

DC AB = A′ B ′ D ′C ′ 27 15 = 36 y

27 y = 540

y = 20

53. Let x = BC BC AB = EC DE x 3 = 1 1 x = 3

55. 57.

1 2

5 15 5 10 AD = AC − DC = 5 − = − = 3 3 3 3 AC = A′ C ′ = 14

54. Let x = DC DC DE = AC AB x 1 = 5 3 3x = 5 5 x = 3 56. BE = BC − EC = 3 −1 = 2 58.

A′ B ′ = AB = 7

SECTION 9.2

309

59.

B ′C ′ = BC = 15

60.

m(B ′A′ C ′ = m(BAC = 84°

61.

m(ACB = m(A′ C ′B ′ = 28°

62.

m(ABC = m(A′ B ′C ′ = 180°− 84°− 28° = 68°

63.

AD = A′ D ′ = 9

64.

B ′C ′ = BC = 11

65.

A′ B ′ = AB = 10

66.

m(BCD = m(B ′C ′D ′ = 50°

67.

m(A′ D ′C ′ = m(ADC = 70°

68.

m(DAB = m(D ′A′ B ′

= 360°−130°− 70°− 50° = 110° 55° 90° + 35° = 125°

69.

180°−125° = 55°

70.

71.

180°− 90°− 55° = 35°

72.

73.

Let x = height of silo x 105 = 6 9 9 x = 630

74.

m(BAC + m(BCA + 80° = 180° m(BAC + m(BCA = 100°

m(BAC = m(BCA m(BAC = 50°, m(BCA = 50° m(x = 50° since (x and (BAC are alternate interior angles. The measure of the angle adjacent to (y is 180°− 50°− 80° = 50°. m(y = 180°− 50° = 130°

x = 70 ft

44 mi SP-A = 0.875 in. 2.25 in. (44)(2.25) SP-A = mi = 113.14 mi 0.875 44 mi SP-R = b) 0.875 in. 1.5 in. (44)(1.5) SP-R = mi = 75.43 mi 0.875

76. a)

75. a)

90 mi C-B = 2 in. 3 in. (90)(3) C-B = mi = 135 mi 2 90 mi B-R = 2 in. 3 in. (90)(3.1) C-B = mi = 139.5 mi 2

77. The different types of triangles are acute, obtuse, right, isosceles, equilateral, and scalene. Descriptions will vary. 78 The different types of quadrilaterals are trapezoid, parallelogram, rhombus, rectangle, and square. Descriptions will vary. EF DF DE =3 =3 =3 79. E ′F ′ D ′F ′ D ′E ′

80.

12 =3 D ′E ′ 3D ′E ′ = 12

15 =3 E ′F ′ 3E ′F ′ = 15

9 =3 D ′F ′ 3D ′F ′ = 9

D ′E ′ = 4

E ′F ′ = 5

D ′F ′ = 3

E ′F ′ 1 = EF 3

F ′G ′ 1 = FG 3

G ′H ′ 1 = GH 3

E ′H ′ 1 = EH 3

310

CHAPTER 9 Geometry

E ′F ′ 1 = 21 3 ′ ′ 3E F = 21

F ′G ′ 1 = 9 3 ′ ′ 3F G = 9

G ′H ′ 1 = 9 3 ′ ′ 3G H = 9

E ′H ′ 1 = 12 3 ′ ′ 3E H = 12

E ′F ′ = 7

F ′G ′ = 3

G ′H ′ = 3

E ′H ′ = 4

81. a) m( HMF = m( TMB, m( HFM = m( TBM, m( MHF = m( MTB b) Let x = height of the wall x 5.5 = 20 2.5 2.5 x = 110 110 x= = 44 ft 2.5 82. a) m(CED = m(ABC ; m(ACB = m(DCE (vertical angles); m(BAC = m(CDE (alternate interior angles) b) Let x = DE x CE = AB BC x 1404 = 543 356 356 x = 762,372; x = 2141.494... ≈ 2141.49 ft

Exercise Set 9.3 Throughout this section, on exercises involving π , we used the π key on a scientific calculator to determine the answer. If you use 3.14 for π , your answers may vary slightly.

1. a) Perimeter b) Area 2. Hypotenuse 3. Circle 4. a) Radius b) Diameter c) Circumference 1 1 A = bh = (6)( 4) = 12 cm 2 2 2 1 1 7. A = bh = (7)(5) = 17.5 cm 2 2 2

9. A = lw = 10 (5) = 50 ft 2

10.

P = 2l + 2 w = 2 (10) + 2 (5) = 30 ft 11. 3 m = 3(100) = 300 cm

1 1 A = bh = (1)(9) = 4.5 ft 2 2 2 1 1 A = bh = (2) 3 = 3 m 2 2 2

( )

A = bh = (13)(6) = 78 in.2 P = 2l + 2 w = 2 (13) + 2 (8) = 42 in.

12.

2 yd = 2 (3) = 6 ft

A = bh = 300 (20) = 6000 cm 2

A = s 2 = (6) = 36 ft 2

P = 2l + 2 w = 2 (300) + 2 (27) = 654 cm

P = 4 s = 4 (6) = 24 ft

SECTION 9.3

13.

2 ft = 2 (12) = 24 in. 1 1 A = h (b1 + b2 ) = (24)(5 + 19) = 288 in.2 2 2 P = s1 + s2 + b1 + b2 = 25 + 25 + 5 + 19 = 74 in.

15. A = π r 2 = π (5) = 25π ≈ 78.54 m 2 2

311

1 1 14. A = h (b1 + b2 ) = (12)(6 + 16) 2 2 1 = (12)(22) = 132 in.2 2 P = s1 + s2 + b1 + b2 = 13 + 13 + 6 + 16 = 48 in. 16. r =

C = 2πr = 2π (5) = 10π ≈ 31.42 m

16 = 8 yd 2

A = πr 2 = π (8) = 64π ≈ 201.06 yd 2 2

C = πd = π (16) ≈ 50.27 yd 17.

13 = 6.5 ft 2

18.

A = πr 2 = π (13) = 169π ≈ 530.93 mm 2 2

C = 2πr = 2π (13) = 26π ≈ 81.68 mm

A = πr 2 = π (6.5) = 42.25π ≈ 132.73 ft 2 2

C = πd = π (13) ≈ 40.84 ft 19.

a) c 2 = 152 + 82

20.

a 2 + 144 = 225

c 2 = 225 + 64

21.

a) a 2 + 12 2 = 152

c 2 = 289

a 2 = 81

c = 289 = 17 yd

a = 81 = 9 in.

b) P = s1 + s2 + s3 = 8 + 15 + 17 = 40 yd

b) P = s1 + s2 + s3 = 9 + 12 + 15 = 36 in.

1 1 c) A = bh = (8)(15) = 60 yd 2 2 2

1 1 c) A = bh = (9)(12) = 54 in.2 2 2

a) b 2 + 52 = 132

22.

b2 + 25 = 169

a) c 2 = 102 + 24 2 c 2 = 100 + 576

b2 = 144

c 2 = 676

c = 144 = 12 km

c = 676 = 26 cm

b) P = s1 + s2 + s3 = 5 + 12 + 13 = 30 km

b) P = s1 + s2 + s3 = 10 + 24 + 26 = 60 cm

c) 1 1 A = bh = (5)(12) = 30 km 2 2 2

c) 1 1 A = bh = (10)(24) = 120 cm 2 2 2 24. Area of larger circle:

23. Area of square: (6) = 36 ft 2 2

π (4) = 16π = 50.26548246 cm 2 2

Area of circle: π (3) = 9π = 28.27433388 ft 2

Shaded area:

Area of smaller circle:

π (3) = 9π = 28.27433388 cm 2 2

36 − 28.27433388 = 7.72566612 ≈ 7.73 ft 2

Shaded area: 50.26548246 − 28.27433388 = 21.99114858 ≈ 21.99 cm 2

312

CHAPTER 9 Geometry

25. Use the Pythagorean Theorem to find the length of a side of the shaded square.

26. Area of rectangle 1: 20 (10) = 200 in.2 Area of rectangle 2: 20 (10) = 200 in.2

x 2 = 22 + 22 x2 = 4 + 4

Area of circle: π (5) = 25π = 78.53981634 in.2

x2 = 8

Shaded area: 200 − 78.53981634 ≈ 321.46 in 2

x= 8 Shaded area:

( 8 ) = 8 in.

27. Find area of trapezoid minus area of unshaded triangle. ⎛ 9 + 11⎞⎟ Trapezoid: 18⎜⎜ = 180 ⎜⎝ 2 ⎠⎟⎟ 1 (18)(10) = 90 2 Shaded area: 180 − 90 = 90 yd 2 Triangle:

29. Area of trapezoid: 1 1 (8)(9 + 20) = (8)(29) = 116 in.2 2 2

28. Find area of large semicircle minus area of small semicircle. 1 1 Large: Small: π ( 2.52 ) π (52 ) 2 2 1 1 Shaded area: π ( 25 − 6.25) = π (18.75) 2 2 ≈ 29.45 in 2 30. Area of circle: π (5) = 25π = 78.53981634 m 2 2

Area of rectangle: 8(6) = 48 m 2

Area of circle: π (4) = 16π = 50.26548246 in.2

Shaded area: 78.53981634 − 48 = 30.53981634 ≈ 30.54 m 2

Shaded area: 116 − 50.26548246 = 65.73451754 ≈ 65.73 in.2

31. Radius of larger circle:

12 = 6 mm 2

32. Radius of circles:

Area of large circle:

Area of 4 circles:

π (6) = 36π = 113.0973355 mm 2

Radius of each smaller circle:

6 = 3 mm 2

π (3) = 9π = 28.27433388 mm 2 2

6 = 3 mm 2

Area of each smaller circle:

Area of rectangle: 24 (6) = 144 mm 2 Shaded area: 144 − 4(28.27433388) = 30.90266448 ≈ 30.90 mm 2

π (3) = 9π = 28.2743388 mm 2 2

Shaded area: 113.0973355 − 28.2743388 − 28.2743388 = 56.5486579 ≈ 56.55 mm 2

33.

1 9 = x 72 9 x = 72

35.

34.

72 = 8 yd 2 9

1 9 = 14.7 x x = 14.7 (9) = 132.3 ft 2

36.

1 9 = x 153 9 x = 153 153 x= = 17 yd 2 9 1 9 = 15.2 x x = 15.2 (9) = 136.8 ft 2

SECTION 9.3

37.

1 10, 000 = 5 x

38.

x = 5(10, 000) = 50, 000 cm 2 39.

1 10, 000 = 0.25 x x = 0.25(10, 000) = 2500 cm 2

1 10,000 = x 8625 10,000 x = 8625 8625 x= = 0.8625 m 2 10, 000

40.

41. Area of living/dining room: 25(22) = 550 ft 2 a) 550 (9.99) = $5494.50

1 10,000 = x 608 10, 000 x = 608 608 x= = 0.0608 m 2 10,000

42. Area of living/dining room: 25(22) = 550 ft 2 a) 550 (2.29) = $1259.50

b) 550 (11.99) = $6594.50 43. Area of kitchen: 12 (14) = 168 ft

313

b) 550 (4.29) = $2359.50 2

Area of first floor bathroom: 6(10) = 60 ft 2 Area of second floor bathroom: 8(14) = 112 ft 2

44. Total area: 168 + 60 + 112 = 340 ft 2 (See Exercise 43.) Cost: 340 ($2.59) = $880.60

Area of kitchen and both bathrooms: 340 ft 2 Cost: 340 ($8.99) = $3056.60 45. Area of bedroom 1: 10 (14) = 140 ft 2

46. Area of all three bedrooms: 480 ft 2 (See Exercise 45.)

Area of bedroom 2: 10 (20) = 200 ft 2

Cost: 480 ($6.99) = $3355.20

Area of bedroom 3: 10 (14) = 140 ft 2 Total area: 140 + 200 + 140 = 480 ft 2 Cost: 480 ($7.99) = $3835.20

48. Area of entire lawn if all grass: 200 (100) = 20,000 ft 2

47. Area of entire lawn if all grass: 400 (300) = 120,000 ft 2 1 (50)(100 + 150) = 6250 ft 2 2 Area of goldfish pond:

Area of house:

π (20) = 400π = 1256.637061 ft 2 2

Area of privacy hedge: 200 (20) = 4000 ft

Area of lawn: 120, 000 − 6250 −1256.637061− 4000 − 2100 −1000 = 105,393.3629 ft 2 = = 11, 710.37366 yd

Area of shed: 10 (8) = 80 ft 2 Area of house: 50 (25) = 1250 ft 2 Area of drive: 30 (10) = 300 ft 2

Area of garage: 70 (30) = 2100 ft 2 Area of driveway: 40 (25) = 1000 ft

Area of patio: 40 (10) = 400 ft 2

105,393.3629 9

Area of pool: π (12) = 144π = 452.3893421 ft 2 2

Area of lawn: 20,000 − 400 − 80 −1250 − 300 − 452.3893421 = 17,517.61066 ft 2 =

17,517.61066 9

= 1946.401184 yd 2 Cost: 1946.401184 ($0.02) = $38.92802368 ≈ $38.93

Cost: 11,710.37366 ($0.02) = $234.2074732 ≈ $234.21

314

CHAPTER 9 Geometry

49. a) Perimeter = 2(94) + 2(50) = 288 ft b) Area = (94)(50) = 4700; 4700 tiles

50. a) Total area: (3.5)(6) + (2.5)(8) + (3)(11.5) = 75.5 ft 2 b) Total cost: (75.5)(75) = $5662.50

51. Let a = height on the wall that the ladder reaches

a + 20 = 29 2

52. Let a = horizontal distance from dock to boat a 2 + 92 = 412

a 2 + 400 = 841

a 2 + 81 = 1681

a 2 = 441

a 2 = 1600

a = 441 = 21 ft

a = 1600 = 40 ft 54.

53. Let d be the distance. d = 37 + 310 2

43 in.

d 2 = 97494 d = 97494 ≈ 312 ft

21 in. x

x + 21 = 432 2

x 2 + 441 = 1849 x 2 = 1408 x = 1408 = 37.52332608 ≈ 37.52 in. 56. a) A = bh

55. a) A = s 2

b) A = 2b (2h ) = 4bh

b) A = ( 2 s) = 4 s 2 2

c) The area of the square in part b) is four times larger than the area of the square in part a). 1 1 57. s = (a + b + c) = (8 + 6 + 10) = 12 2 2

A = 12 (12 − 8)(12 − 6)(12 −10) = 12 (4)(6)(2) = 576 = 24 cm 2

c) The area of the parallelogram in part b) is four times larger than the area of the parallelogram in part a).

58. a) A = a 2 b) A = ab c) A = ab d) A = b 2 e) ( a + b) = a 2 + ab + ab + b2 = a 2 + 2ab + b2 2

Exercise Set 9.4 In this section, we use the π key on the calculator to determine answers in calculations involving π . If you use 3.14 for π , your answers may vary slightly.

1. Volume 2. Surface 3. Platonic 4. Prism 5. Right 6. 2

SECTION 9.4

a) V = lwh = (8)( 2)(4) = 64 ft 3

b) SA = 2lw + 2wh + 2lh

SA = 2(8)(6) + 2(6)(25) + 2(8)(25) = 796 mm 2

a) V = s 3 ; V = 23 = 8 yd3

10.

b) SA = 6s 2 ; SA = 6 ( 22 ) = 24 yd 2 11.

13.

a) V = lwh = (8)(6)(25) = 1200 mm3 b) SA = 2lw + 2wh + 2lh

SA = 2(8)(2) + 2(2)(4) + 2(8)(4) = 112 ft 2

a) V = s 3 ; V = 133 = 2197 ft 3 b) SA = 6s 2 ; SA = 6 (132 ) = 1014 ft 2

a) V = π r 2 h = π ( 22 )(12) = 48π

12.

a) V = π r 2 h = π (62 )(24) = 864π

V ≈ 150.80 in.3

V ≈ 2714.34 in.3

b) SA = 2π r 2 + 2πrh

b) SA = 2π r 2 + 2π rh

SA = 2π ( 22 ) + 2π (2)(12) = 56π

SA = 2π (62 ) + 2π (6)(24) = 360π

SA ≈ 175.93 in 2

SA ≈ 1130.97 in 2

1 1 a) V = π r 2 h = π (32 )(14) = 42π 3 3

14.

V ≈ 131.95 cm 3

b) SA = π r 2 + π r r 2 + h 2

(

315

) (

SA = π 32 + 3 32 + 142 = π 9 + 3 205

10 = 5 ft 2 1 1 V = π r 2 h = π (52 )( 24) = 200π 3 3

a) r =

V ≈ 628.32 ft 3

)

b) SA = π r 2 + πr r 2 + h 2

(

SA ≈ 163.22 cm 2

) (

SA = π 52 + 5 52 + 242 = π 25 + 5 601 SA ≈ 463.63 ft 2

9 = 4.5 cm 2 4 V = πr 3 3 4 4 V = π (4.53 ) = π (91.125) ≈ 381.70 cm3 3 3

15. a) r =

b) SA = 4π r

16.

4 a) V = π r 3 3 4 4 V = π (73 ) = π (343) ≈ 1436.76 mi3 3 3 b) SA = 4π r 2 SA = 4π (62 ) = 4π (49) ≈ 615.75 mi 2

SA = 4π ( 4.52 ) = 4π (20.25) ≈ 254.47 cm2

17.

1 1 Area of the base: B = bh = (10)(10) = 50 m 2 2 2 V = Bh = 50 (15) = 750 m3

19.

18. Area of the base: 1 1 B = h (b1 + b2 ) = (10)(8 + 12) = 100 in. 2 2 2 V = Bh = 100 (24) = 2400 in.3

Area of the base: B = s 2 = 12 2 = 144 cm 2 1 1 V = Bh = (144)(15) = 720 cm3 3 3

20.

Area of the base: 1 1 B = bh = (9)(15) = 67.5 ft 2 2 2 1 1 V = Bh = (67.5)(13) = 292.5 ft 3 3 3

)

316

CHAPTER 9 Geometry

21. V = vol. of large prism − vol. of small prism V = (8)(8)(16) − (4)(4)(16) = (64 −16)(16)

22.

V = vol. cone − vol. of cylinder 1 V = π (2.52 )(11) − π (22 )( 4) 3 ⎛ (6.25)(11) ⎞ − (4)(4)⎟⎟⎟ ≈ 21.73 in 3 V = π ⎜⎜ ⎜⎝ ⎠ 3

24.

V = volume of rect. solid − volume of sphere

V = (48)(16) = 768 ft 3

23. V = 2(volume of one small trough) depth of trough = 7 1 area of triangular ends = (4) (7) = 14 2 V = 2 (14)(14) = 392 ft 3 25.

V = volume of cylinder − volume of 3 spheres ⎡4 2 3⎤ = π (3.5) ( 20.8) − 3 ⎢ π (3.45) ⎥ ⎢⎣ 3 ⎥⎦

= 254.8π −164.2545π = 90.5455π

4 = 6 (6)(6) − π (33 ) ≈ 102.90 mm3 3

26.

V = vol. of large cylinder − vol. of small cylinder

= π (1.5) (5) − π (0.5) (5) = 11.25π −1.25π = 10π 2

V ≈ 284.46 cm 3

V ≈ 31.42 m 3

V = volume of rect. solid − volume of pyramid 1 = 3(3)(4) − (32 )(4) = 36 −12 = 24 ft 3 3

28.

29.

3 yd 3 = 3(27) = 81 ft 3

30.

7.25 yd 3 = 7.25( 27) = 195.75 ft 3

31.

153 ft 3 =

153 = 5.6 ≈ 5.67 yd 3 27

32.

2457 ft 3 =

33.

0.56 m3 = 0.56 (1, 000, 000) = 560, 000 cm3

34.

17.6 m 3 = 17.6 (1,000,000) = 17, 600,000 cm 3

35.

7,500,000 cm3 =

36.

7, 300,000 cm3 =

37.

⎛9⎞ a) V = lwh = ( 20)(15)⎜⎜ ⎟⎟⎟ = 225 ft 3 ⎜⎝12 ⎠

27.

7,500, 000 = 7.5 m 3 1,000,000

V = volume of prism − volume of rect. solid

1 = (6)(8)(11) − 3(4)(11) 2 = 264 −132 = 132 in.3 2457 = 91 yd 3 27

38. a) V = 46 ( 25)(25) = 28,750 in.3 b) (1 ft ) = (12 in.)(12 in.)(12 in.) = 1728 in.3 3

b) Cost = ($11)( 225) = $2475

28,750 in.3 = 39. SA = 2lw + 2 wh + 2lh SA = 2(142)(125) + 2(125)(10) + 2(10)(142) SA = 40,840 mm2

7,300,000 = 7.3 m3 1,000,000

40.

28,750 = 16.63773148 ≈ 16.64 ft 3 1728

20 = 10 2 r = SA = 4πr 2 r=

SA = 4π (102 ) = 400π SA ≈ 1256.64 in 2

41.

V = 12 (4)(3) = 144 in. 3

42.

144 in.3 = 144 (0.01736) = 2.49984 ≈ 2.50 qt

Tubs: V = π r 2 h = π (3) (5) = 45π 2

= 141.3716694 ≈ 141.37 in.3

Boxes: V = s 3 = (5) = 125 in.3 3

SECTION 9.4

43. a) V = 80 (50)(30) = 120, 000 cm 3 b) 120,000 mA c) 120, 000 mA =

317

1 1 2 44. V = Bh = (720) ( 480) = 82,944,000 ft 3 3 3

120,000 = 120 A 1000

3.875 = 1.9375 in. 2 Volume of each cylinder:

4 1 = ft 12 3 ⎛1 ⎞ V = lwh = 9 (18)⎜⎜ ⎟⎟⎟ = 54 ft 3 ⎜⎝ 3 ⎠

45. r =

46. a) 4 in. =

πr 2 h = π (1.9375) (3) 2

54 = 2 yd 3 27

= 11.26171875π = 35.37973289 Total volume: 8(35.37973289) = 283.0378631 ≈ 283.04 in.3 47. a) 5.5 ft = 5.5(12) = 66 in.

b) 2 ($42) = $84 48. a) Round pan: ⎛9⎞ A = πr 2 = π ⎜⎜ ⎟⎟⎟ = 20.25π ⎜⎝ 2 ⎠ 2

2.5 = 1.25 in. 2

V = πr 2 h = π (1.25) (66) = 103.125π

= 63.61725124 ≈ 63.62 in.2

= 323.9767424 ≈ 323.98 in.3 323.98 = 0.187488426 ≈ 0.19 ft 3 b) 1728

b) Round pan:

Rectangular pan: A = lw = 7 (9) = 63 in.2

V = πr 2 h ≈ 63.62 ( 2) = 127.24 in.3 Rectangular pan: V = lwh = 7 (9)( 2) = 126 in.3

1 1 ⎛ 3⎞ 49. V = π r 2 h = π ⎜⎜ ⎟⎟⎟ (6) = 4.5π 3 3 ⎜⎝ 2 ⎠ 2

= 14.13716694 ≈ 14.14 in.3

c) Round pan 50. a) B = area of trapezoid 1 = (9)(8 + 12) = 90 in.2 2 4 ft = 4 (12) = 48 in.

V = Bh = 90 (48) = 4320 in.3 b) 1 ft 3 = (12)(12)(12) = 1728 in.3 4320 = 2.5 ft 3 1728 52. 12 −16 + x = 2 −4 + x = 2 x = 6 faces 4320 in.3 =

51. 8 − x + 4 = 2 12 − x = 2 −x = −10 x = 10 edges 53. x − 8 + 4 = 2 x−4 = 2 x = 6 vertices 55. 7 −12 + x = 2 −5 + x = 2 x = 7 faces

54. 11− x + 5 = 2 16 − x = 2 x = 14 edges 56. x −10 + 4 = 2 x −6 = 2 x = 8 vertices

318

CHAPTER 9 Geometry

12,756.3 = 6378.15 km 2 3474.8 rM = = 1737.4 km 2 rE =

57.

4 d) VE = π (6378.153 ) ≈ 1.09×1012 km 3 3 4 e) VM = π (1737.43 ) ≈ 2.20×1010 km3 3 1.09 ×1012 VE f) = ≈ 50 VM 2.20×1010

a) SAE = 4π (6378.152 ) ≈ 5.11×108 km2 b) SAM = 4π (1737.42 ) ≈ 3.79 ×107 km 2 c)

SAE 5.11×108 = ≈ 13 SAM 3.79×107

58. Let r = the radius of one of the cans of orange juice The length of the box = 6r and the width of the box = 4r Volume of box − volume of cans: lwh − 6( π r 2 h) = (6r )(4r ) h − 6π r 2 h = 24r 2 h − 6π r 2 h = 6r 2 h (4 − π ) Percent of the volume of the interior of the box that is not occupied by the cans: 6r 2 h ( 4 − π ) 6r 2 ( 4 − π ) 4 − π = = = 0.2146018366 ≈ 21.46% lwh 4 (6r )(4r ) 59. a) – e) Answers will vary. f) If we double the length of each edge of a cube, the new volume will be eight times the original volume. 60. a) – e) Answers will vary. f) If we double the radius of a sphere, the new volume will be eight times the original volume. 61. a) Find the volume of each numbered region. Since the length of each side is a + b , the sum of the volumes of each region will equal ( a + b) . 3

b) V1 = a (a )(a ) = a 3

V2 = a ( a )(b) = a 2b

V3 = a (a )(b) = a 2 b

V5 = a (a )(b) = a 2 b

V6 = a (b)(b) = ab2

V7 = b (b)(b) = b 3

c) The volume of the piece not shown is ab2 . 62. a) 5.5 ft = 5.5(12) = 66 in. V = Bh = 5(66) = 330 in.3 b) Radius of cylinder:

0.75 = 0.375 in. 2

Volume of cylinder: π r 2 h = π (0.375) (66) = 9.28125π = 29.15790682 in.3 2

Volume of hollow noodle: 330 − 29.15790682 = 300.8420932 ≈ 300.84 in.3

Exercise Set 9.5 1. Rigid 2. Reflection 3. Axis 4. Translation 5. Vector 6. Rotation

V4 = a (b)(b) = ab2

SECTION 9.5

7. Center 8. Rotation 9. Glide 10. Symmetry 11. Reflective 12. Rotational 13. Tessellation 14. Tessellating

319

320

CHAPTER 9 Geometry

47. a)

48. a)

b) Yes c) Yes 49. a)

b) No c) No

b) Yes c) Yes 50. a)

b) No c) No

SECTION 9.5 51. a)

b) No c) No d)

52. a)

b) Yes c) Yes d)

e) Yes f) Yes

53. a) – c)

d) No. Any 90° rotation will result in the figure being in a different position than the starting position. 54. a)

321

322

CHAPTER 9 Geometry b) No. Any reflection about any horizontal line will result in the figure being in a different position than the starting position. c) No. Any 90° rotation will result in the figure being in a different position than the starting position. d) No. Any 180° rotation will result in the figure being in a different position than the starting position.

55. a) – b)

56. 57. 58. 59. 60. about a

c) No d) The order in which the translation and the reflection are performed is important. The figure obtained in part b) is the glide reflection. Answers will vary. Answers will vary. a) Answers will vary. b) A regular octagon cannot be used as a tessellating shape. a) Answers will vary. b) A regular pentagon cannot be used as a tessellating shape. Although answers will vary depending on the font, the following capital letters have reflective symmetry

horizontal line drawn through the center of the letter: B, C, D, E, H, I, K, O, X. 61. Although answers will vary depending on the font, the following capital letters have reflective symmetry about a vertical line drawn through the center of the letter: A, H, I, M, O, T, U, V, W, X, Y. 62. Although answers will vary depending on the font, the following capital letters have 180° rotational symmetry about a point in the center of the letter: H, I, N, O, S, X, Z.

Exercise Set 9.6 1. Rubber 2. Möbius 3. Klein 4. Four 5. Jordan 6. Genus 7. One 8. Two 9. 7 – Green; 1, 3, 5 – Yellow; 2, 4, 6 – Red. AWV 10. 1, 7 – Red; 4, 6, 8 – Green; 2, 3, 5 – Blue. AWV 11. 1, 2, 4, 5, 7, 9 – Red; 3, 6, 8 – Green. AWV

13. TX, KS, MS, KY, SC, FL – Red OK, LA, TN – Green MO, GA, VA – Blue AR, AL, NC – Yellow, AWV

12. 1, 4, 6 – Red; 2,3 – Yellow; 7 – Green; 5 – Blue. AWV 14. CA, WA, MT, UT – Red OR, WY, AZ – Green ID, NM – Blue NV, CO – Yellow . AWV

SECTION 9.7

15. YT, NU, AB, ON – Red NT, QC – Blue BC, SK – Green MB – Yellow. AWV 17. Outside; a straight line from point A to a point clearly outside the curve crosses the curve an even number of times. 19. Outside; a straight line from point A to a point clearly outside the curve crosses the curve an even number of times. 21. Inside; a straight line from point C to a point clearly outside the curve crosses the curve an odd number of times. 23. 1 24. 0 27. Larger than 5 28. 0 31. 0 32. 0 35. a) - d) Answers will 36. One vary. 39. Two 40. The smaller one is a Möbius strip; the larger one is not.

323

16. BCS, SON, DGO, NLE – Red BCA, CHH, ZAC, TMP – Blue SIN, COA – Green NAY, SLP – Yellow. AWV 18. Inside; a straight line from point B to a point clearly outside the curve crosses the curve an odd number of times. 20. Inside; a straight line from point B to a point clearly outside the curve crosses the curve an odd number of times. 22. Outside; a straight line from point D to a point clearly outside the curve crosses the curve an even number of times. 25. 5 26. 5 29. 5 30. 1 33. larger than 5 34. 0 37. One 38. One

41. a) No, it has an inside and an outside. b) Two c) Two d) Two strips, one inside the other

42. No, it does not. 43. Answers will vary. 44. Answers will vary. 45. Answers will vary.

Exercise Set 9.7 1. Parallel 2. No 3. Two 4. Plane 5. Sphere 6. Pseudosphere 7. Geodesic 8. Chaos 9.

324

CHAPTER 9 Geometry

10.

11.

12.

13. a)

b) Infinite. c) Finite since it covers a finite or closed area. 14. a)

Step

Perimeter

⎛ 4 ⎞0 3⎜⎜ ⎟⎟⎟ = 3(1) = 3 ⎜⎝ 3 ⎠

⎛ 4 ⎞1 ⎛4⎞ 3⎜⎜ ⎟⎟⎟ = 3⎜⎜ ⎟⎟⎟ = 4 ⎝⎜ 3 ⎠ ⎝⎜ 3 ⎠

⎛ 4 ⎞2 ⎛16 ⎞ 16 3⎜⎜ ⎟⎟⎟ = 3⎜⎜ ⎟⎟⎟ = ⎜⎝ 3 ⎠ ⎝⎜ 9 ⎠ 3

⎛ 4 ⎞3 ⎛ 64 ⎞ 64 3⎜⎜ ⎟⎟⎟ = 3⎜⎜ ⎟⎟⎟ = ⎝⎜ 3 ⎠ ⎝⎜ 27 ⎠ 9

⎛ 4 ⎞4 ⎛ 256 ⎞⎟ 256 3⎜⎜ ⎟⎟⎟ = 3⎜⎜ ⎟= ⎜⎝ 3 ⎠ ⎝⎜ 81 ⎠⎟ 27

⎛ 4 ⎞5 ⎛1024 ⎞⎟ 1024 3⎜⎜ ⎟⎟⎟ = 3⎜⎜ ⎟= ⎝⎜ 3 ⎠ ⎝⎜ 243 ⎠⎟ 81

4 multiplied by the previous perimeter. 3 c) The area is finite because it encloses a finite region. The perimeter is infinite. b) At each stage, the perimeter is

REVIEW EXERCISES

325

15. Each type of geometry can be used in its own frame of reference. 16. Spherical - elliptical geometry; flat - Euclidean geometry; saddle-shaped - hyperbolic geometry 17. Coastlines, trees, mountains, galaxies, polymers, rivers, weather patterns, brains, lungs, blood supply 18. a) Euclidean - The sum of the measures of the angles of a triangle is 180°. b) Hyperbolic - The sum of the measures of the angles of a triangle is less than 180°. c) Elliptical - The sum of the measures of the angles of a triangle is greater than 180°. 19. Benoit Mandelbrot – first to use the word fractal to describe shapes that had several common characteristics, including some form of “self-similarity” 20. G.F. Bernhard Riemann - discovered elliptical geometry 21. Nikolay Ivanovich Lobachevsky - discovered hyperbolic geometry 22. Carl Friedrich Gauss - discovered hyperbolic geometry 23. Janos Bolyai - discovered hyperbolic geometry 24. Girolamo Saccheri - proved many theorems of hyperbolic geometry

Review Exercises In the Review Exercises and Chapter Test questions, the π key on the calculator is used to determine answers in calculations involving π . If you use 3.14 for π , your answers may vary slightly.

1. 3.

JJJG BF BFC

2. 4.

AD HJJG BH

{F }

{}

7. 9.

90°− 35.4° = 54.6° Let x = BC BC AC = B ′C A′ C 12 x = 4 3.4 4 x = 40.8 40.8 = 10.2 in. x = 4

180°−100.5° = 79.5° Let x = A′ B ′ A′ B ′ A′ C = AB AC x 4 = 12 6 12 x = 24 24 x = = 2 in. 12

11.

m(ABC = m(A′ B ′C

10.

12.

m(ABC = m(A′ B ′C

m(A′ B ′C = 180°− 88° = 92°

Thus, m(ABC = 92° m(BAC = 180°− 30°− 92° = 58°

Thus, m(ABC = 92°

326

CHAPTER 9 Geometry

13. m(1 = 45° m(6 = 180D −115D = 65D m( 2 = m(1 + m( 6 = 110D m(3 = m( 2 = 110D m( 4 = m(6 = 65D m(5 = 180D − m( 4 = 180D − 65D = 115D 14.

n=8

15.

(n − 2)180° = (8 − 2)180° = 6(180°) = 1080°

16.

18.

b) P = 2l + 2w = 2(11) + 2(9) = 40 mi

1 1 a) A = h (b1 + b2 ) = (2)(4 + 9) = 13 in.2 2 2 b) P = 3.2 + 4 + 3.2 + 9 = 19.4 in.

17.

1 1 a) A = bh = (3)(4) = 6 km 2 2 2

19.

a) A = bh = 12 (7) = 84 in.2 b) P = 2(9) + 2(12) = 42 in. a) A = π r 2 = π (7) ≈ 153.94 ft 2 2

b) C = 2π r = 2π (7) = 14π ≈ 43.98 ft

b) P = 3 + 4 + 32 + 42 = 7 + 25 = 12 km 20.

a) A = lw = 11(9) = 99 mi 2

A = area of rectangle − 3(area of one circle)

21. Shaded area is the area of an 12 by 12 square

Length of rectangle = 3(diameter of circle)

minus the four corner squares (each 3 by 3)

= 3(10) = 30

and minus the area of a circle of diameter 6.

Area of rectangle: (10)(30) = 300

Shaded area = (12)(12) − 4(3)(3) − π (32 )

Area of circle: π (52 ) = 25π

= 108 − 9π ≈ 79.73 yd 2

Shaded area: 300 − 3(25π ) ≈ 64.38 m 2 22.

A = lw = 14 (16) = 224 ft 2 Cost: 224 ($5.25) = $1176

23. a) V = lwh = 10 (3)(4) = 120 cm3

24.

b) SA = 2lw + 2 wh + 2lh

a) V = πr 2 h = π (62 )(18) = 648π ≈ 2035.75 in 3 b) SA = 2π r 2 + 2π rh

SA = 2(10)(3) + 2(3)(4) + 2(10)(4) = 164 cm

SA = 2π (62 ) + 2π (6)(18) = 288π SA ≈ 904.78 in 2

REVIEW EXERCISES

25.

12 = 6 mm 2 1 1 V = π r 2 h = π (62 )(16) = 192π 3 3

a) r =

4 3 πr 3 4 4 V = π (73 ) = π (343) ≈ 1436.76 yd3 3 3

a) V =

26.

V ≈ 603.19 mm 3

b) SA = 4π r 2 SA = 4π (72 ) = 4π (49) ≈ 615.75 yd 2

b) SA = π r 2 + π r r 2 + h 2

(

) (

SA = π 62 + 6 62 + 162 = π 36 + 6 292

)

SA ≈ 435.20 mm 2 28.

1 1 27. B = bh = (9)(12) = 54 m 2 2 2

If h represents the height of the triangle which is the base of the pyramid, then

V = Bh = 54 (8) = 432 m 3

h 2 + 32 = 52 h 2 + 9 = 25 h 2 = 16 h = 16 = 4 ft 1 1 B = bh = (6)(4) = 12 ft 2 2 2 1 1 V = Bh = (12)(7) = 28 ft 3 3 3

29.

V = volume of cylinder − volume of cone 1 2 2 = π ( 2) (9) − π (2) (9) = 36π −12π = 24π 3 = 75.39822369 ≈ 75.40 cm

31.

V = vol. of large sphere − vol. of small sphere

4 4 3 3 π (5) − π ( 2.5) = 166.6667π − 20.8333π 3 3

= 145.8334π ≈ 458.15 in.3 31. b) Weight: 67.88(62.4) + 375 = 4610.7 lb

h 2 + 12 = 32 h2 + 1 = 9

Yes, it will support the trough filled with water.

h2 = 8 h = 8 1 1 8 (2 + 4) = 8.485281374 ft 2 A = h (b1 + b2 ) = 2 2

( )

30. =

327

c) ( 4610.7 − 375) = 4235.7 lb of water 4235.7 = 510.3253 ≈ 510.3 gal 8.3

a) V = Bh = 8.485281374 (8) = 67.88225099 ≈ 67.88 ft 3

328

CHAPTER 9 Geometry

41. Yes 42. No 43. No 44. Yes 45. a) – d) Answers will vary. 46. 1, 3, 7 – Red; 2, 6, 8 – Blue; 4,5 – Green. AWV 47. Outside; a straight line from point A to a point clearly outside the curve crosses the curve an even number of times. 48. Euclidean: Given a line and a point not on the line, one and only one line can be drawn parallel to the given line through the given point. Elliptical: Given a line and a point not on the line, no line can be drawn through the given point parallel to the given line. Hyperbolic: Given a line and a point not on the line, two or more lines can be drawn through the given point parallel to the given line. 49.

50.

CHAPTER TEST

Chapter Test 1.

{ D}

HJJG AC

5. 7.

90°− 74.9° = 15.1° One angle of the triangle is 50° (by vertical angles) and 180° - 115° = 65°. Thus, the measure of angle x = 180° - 50° - 65° = 65°.

6. 8.

180°−10.4° = 169.6° n=5

Let x = B ′C ′ B ′C ′

10. a)

(n − 2)180° = (5 − 2)180° = 3(180°) = 540°

A′ C ′ AC BC 5 x = 13 7 13x = 35 35 = 2.692307692 ≈ 2.69 cm x = 13

14 = 7 cm 2 4 4 a) V = π r 3 = π (73 ) ≈ 1436.76 cm 3 3 3

b) SA = 4π r

+BCD

x 2 = 25 b) c)

x = 25 = 5 in. P = 5 + 13 + 12 = 30 in. 1 1 A = bh = (5)(12) = 30 in.2 2 2

12. Shaded volume =

volume of prism − volume of cylinder Volume of prism: V = lwh = (6)(4)(3) = 72 m 3 Volume of cylinder: V = π r 2 h = π (12 ) 4 = 4π m3

Shaded volume = 72 − 4π ≈ 59.43 m3

SA = 4π (72 ) = 4π (49) ≈ 615.75 cm2

13. B = lw = 4 (7) = 28 ft 2

x 2 + 122 = 132 x 2 + 144 = 169

11. r =

329

14.

1 1 V = Bh = (28)(12) = 112 ft 3 3 3

15.

16.

17.

18. a) No b) Yes