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Solution Manual for Elementary and Intermediate Algebra: Functions and Authentic Applications, 3rd Edition

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    S OLUTIONS M ANUAL DIACRI T ECH E LEMENTARY & I NTERMEDIATE A LGEBRA : F UNCTIONS AND A UTHENTIC A PPLICATIONS T HIRD E DITION Jay Lehmann College of San Mateo

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    iii Contents Chapter 1 Introduction to Modeling 1.1 Variables and Constants 1 1.2 Scatterplots 3 1.3 Exact Linear Relationships 5 1.4 Approximate Linear Relationships 8 Chapter 1 Review Exercises 12 Chapter 1 Test 15 Chapter 2 Operations and Expressions 2.1 Expressions 18 2.2 Operations with Fractions 19 2.3 Absolute Value and Adding Real Numbers 22 2.4 Change in a Quantity and Subtracting Real Numbers 25 2.5 Ratios, Percents, and Multiplying and Dividing Real Numbers 28 2.6 Exponents and Order of Operations 31 Chapter 2 Review Exercises 34 Chapter 2 Test 38 Cumulative Review of Chapters 1 and 2 40 Chapter 3 Using Slope to Graph Linear Equations 3.1 Graphing Equations of the Forms y = mx + b and x = a 43 3.2 Graphing Linear Models; Unit Analysis 49 3.3 Slope of a Line 55 3.4 Using Slope to Graph Linear Equations 58 3.5 Rate of Change 65 Chapter 3 Review Exercises 71 Chapter 3 Test 77 Chapter 4 Simplifying Expressions and Solving Equations 4.1 Simplifying Expressions 81 4.2 Simplifying More Expressions 83 4.3 Solving Linear Equations in One Variable 85 4.4 Solving More Linear Equations in One Variable 90 4.5 Comparing Expressions and Equations 98 4.6 Formulas 101 Chapter 4 Review Exercises 106 Chapter 4 Test 112 Cumulative Review of Chapters 1–4 115

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    iv Chapter 5 Linear Functions and Linear Inequalities in One Variable 5.1 Graphing Linear Equations 120 5.2 Functions 129 5.3 Function Notation 131 5.4 Finding Linear Equations 134 5.5 Finding Equations of Linear Models 141 5.6 Using Function Notation with Linear Models to Make Estimates and Predictions 147 5.7 Solving Linear Inequalities in One Variable 154 Chapter 5 Review Exercises 160 Chapter 5 Test 170 Chapter 6 Systems of Linear Equations and Systems of Linear Inequalities 6.1 Using Graphs and Tables to Solve Systems 175 6.2 Using Substitution to Solve Systems 181 6.3 Using Elimination to Solve Systems 187 6.4 Using Systems to Model Data 196 6.5 Perimeter, Value, Interest, and Mixture Problems 202 6.6 Linear Inequalities in Two Variables; Systems of Linear Inequalities in Two Variables 211 Chapter 6 Review Exercises 219 Chapter 6 Test 233 Cumulative Review of Chapters 1–6 240 Chapter 7 Polynomial Functions and Properties of Exponents 7.1 Adding and Subtracting Polynomial Expressions and Functions 249 7.2 Multiplying Polynomial Expressions and Functions 252 7.3 Powers of Polynomials; Product of Binomial Conjugates 256 7.4 Properties of Exponents 260 7.5 Dividing Polynomials: Long Division and Synthetic Division 262 Chapter 7 Review Exercises 267 Chapter 7 Test 271 Making Sure You’re Ready for Intermediate Algebra: A Review of Chapters 1–7 272 Chapter 8 Factoring Polynomials and Solving Polynomial Equations 8.1 Factoring Trinomials of the Form x 2 + bx + c and Differences of Two Squares 280 8.2 Factoring Out the GCF; Factoring by Grouping 284 8.3 Factoring Trinomials of the Form ax 2 + bx + c 287 8.4 Sums and Differences of Cubes; A Factoring Strategy 298 8.5 Using Factoring to Solve Polynomial Equations 302 8.6 Using Factoring to Make Predictions with Quadratic Models 305 Chapter 8 Review Exercises 310 Chapter 8 Test 316 Making Sure You’re Ready for Intermediate Algebra: A Review of Chapters 1–8 318

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    v Chapter 9 Quadratic Functions 9.1 Graphing Quadratic Functions in Vertex Form 327 9.2 Graphing Quadratic Functions in Standard Form 333 9.3 Simplifying Radical Expressions 342 9.4 Using the Square Root Property to Solve Quadratic Equations 345 9.5 Solving Quadratic Equations by Completing the Square 350 9.6 Using the Quadratic Formula to Solve Quadratic Equations 357 9.7 Solving Systems of Linear Equations in Three Variables; Finding Quadratic Functions 367 9.8 Finding Quadratic Models 374 9.9 Modeling with Quadratic Functions 380 Chapter 9 Review Exercises 385 Chapter 9 Test 394 Cumulative Review of Chapters 1–9 399 Chapter 10 Exponential Functions 10.1 Integer Exponents 409 10.2 Rational Exponents 413 10.3 Graphing Exponential Functions 417 10.4 Finding Equations of Exponential Functions 422 10.5 Using Exponential Functions to Model Data 428 Chapter 10 Review Exercises 435 Chapter 10 Test 440 Chapter 11 Logarithmic Functions 11.1 Composite Functions 444 11.2 Inverse Functions 449 11.3 Logarithmic Functions 456 11.4 Properties of Logarithms 459 11.5 Using the Power Property with Exponential Models to Make Predictions 464 11.6 More Properties of Logarithms 472 11.7 Natural Logarithms 476 Chapter 11 Review Exercises 481 Chapter 11 Test 488 Cumulative Review of Chapters 1–11 491 Chapter 12 Rational Functions 12.1 Finding the Domains of Rational Functions and Simplifying Rational Expressions 502 12.2 Multiplying and Dividing Rational Expressions; Converting Units 507 12.3 Adding and Subtracting Rational Expressions 512 12.4 Simplifying Complex Rational Expressions 519 12.5 Solving Rational Equations 525 12.6 Modeling with Rational Functions 536 12.7 Proportions; Similar Triangles 541 12.8 Variation 543 Chapter 12 Review Exercises 547 Chapter 12 Test 558

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    vi Chapter 13 Radical Functions 13.1 Simplifying Radical Expressions 563 13.2 Adding, Subtracting, and Multiplying Radical Expressions 566 13.3 Rationalizing Denominators and Simplifying Quotients of Radical Expressions 570 13.4 Graphing and Combining Square Root Functions 575 13.5 Solving Radical Equations 580 13.6 Modeling with Square Root Functions 587 Chapter 13 Review Exercises 594 Chapter 13 Test 600 Chapter 14 Sequences and Series 14.1 Arithmetic Sequences 604 14.2 Geometric Sequences 606 14.3 Arithmetic Series 611 14.4 Geometric Series 614 Chapter 14 Review Exercises 616 Chapter 14 Test 619 Cumulative Review of Chapters 1–14 621 Chapter 15 Additional Topics 15.1 Absolute Value: Equations and Inequalities 633 15.2 Performing Operations with Complex Numbers 637 15.3 Pythagorean Theorem, Distance Formula, and Circles 641 15.4 Ellipses and Hyperbolas 648 15.5 Solving Nonlinear Systems of Equations 657

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    Chapter 1 Introduction to Modeling Homework 1.1 2. In 2015, Chris Davis hit 47 home runs. 4. In 2011, about 60 percent of children aged 5– 18 participated in organized physical activity. 6. In 2015, 11.1 percent of American workers were in unions. 8. The temperature is 10 F − ° . That is the temperature is 10 degrees below 0 (in Fahrenheit). 10. The statement 13 t = represents the year 2018 (13 years after 2005). 12. The statement 2 t = − represents the year 2008 (2 years before 2010). 14. Answers may vary. Example: Let t be the amount of time (in hours) that a student prepares for an exam. Then t can represent the numbers 0 and 4, but t cannot represent the numbers 1 − and 3 − . 16. Answers may vary. Example: Let n be the number of students enrolled in an algebra class. Then n can represent the numbers 15 and 28, but n cannot represent the numbers 20 − and 0.5. 18. Answers may vary. Example: Let T be the temperature (in degrees Fahrenheit) in an oven. Then T can represent the numbers 300 and 450, but T cannot represent the numbers 300 − and 450 − . 20. Answers may vary. Example: Let v be the value (in thousands of dollars) of a new home. Then v can represent the numbers 100 and 250, but v cannot represent the numbers 100 − and 250 − . 22. a. Answers may vary. Example: 9 inches 4 inches 12 inches 3 inches 18 inches 2 inches b. In the described situation, the symbols W and L are variables. Their values can change. c. In the described situation, the symbol A is a constant. Its value is fixed at 36 square inches. 24. a. Answers may vary. Example: 4 inches 4 inches 5 inches 3 inches 6 inches 2 inches b. In the described situation, the symbols W and L are variables. Their values can change. c. In the described situation, the symbol P is a constant. Its value is fixed at 16 feet. 26. a. Answers may vary. Example: 2 inches 1 inch 4 inches 2 inches 6 inches 3 inches b. In the described situation, the symbols W , L , and A are all variables. All of their values can change. c. In the described situation, none of the symbols are constants. All of their values can change. 28. a. Answers may vary. Example: 5 cm 5 cm 5 cm 7 cm 5 cm 10 cm

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    2 ISM: Elementary and Intermediate Algebra b. In the described situation, the symbols L and A are variables. Their values can change. c. In the described situation, the symbol W is a constant. Its value is fixed at 5 cm. 30. −8−7−6−5−4−3−2−1 0 1 2 3 4 5 6 7 8 32. −3 −2 −1 0 1 2 3 1 4 5 4 9 4 − 0 1 2 3 4 0.9 1.5 2.3 2.7 3.4 −3 −2 −1 0 1 2 −2.4 −0.7 0.2 0.9 38. The counting numbers between 1 and 5 are 2, 3, and 4. 1 2 3 4 5 40. The integers between 4 − and 4, inclusive, are 4, 3, 2, 1, − − − − 0, 1, 2, 3, and 4. −5 −4 −3 −2 −1 0 1 2 3 4 5 42. The integers between 6 − and 3, inclusive, are 6, 5, 4, 3, 2, 1, − − − − − − 0, 1, 2, and 3. −7 −6 −5 −4 −3 −2 −1 0 1 2 3 4 44. The positive integers between 4 − and 4 are 1, 2, and 3. 0 1 2 3 4 46. The integers in the list are 4, − 0, and 3. 48. The rational numbers in the list are 9.7, 4, − − 0, 3 , 5 and 3. 50. The real numbers in the list are 9.7, 4, − − 0, 3 , 5 7, 3, and . π 52. Answers may vary. Example: 1, 5, and 12 54. Answers may vary. Example: 1, 2, − − and 3 − 56. Answers may vary. Example: 1 3 , , 2 4 and 7 9 58. Answers may vary. Example: 2, 5, and π 60. Answers may vary. Example: 2, 5, and π 62. 2 0 1 5 4 12 2.4 5 5 + + + + = = The average number of songs downloaded per visit is 2.4 songs. 0 1 2 3 4 5 n Number of songs Average: 2.4 songs 64. 79 82 75 77 76 389 77.8 5 5 + + + + = = The average percentage of flights in a year that are on time is 77.8% per year. 73 74 75 76 77 78 79 80 81 82 83 p Percent Average: 77.8% 66. 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 c Per person consumption 68. −10 −8 −6 −4 −2 0 2 4 6 p Annual profit (in millions) −5 3 34. 36.

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    Chapter 1: Introduction to Modeling 3 70. a. 6.1 7.2 7.6 7.7 28.6 7.15 4 4 + + + = = The average sales are about $7.15 million per year. b. Car sales increased from 2011 to 2014. Sales went up each year. c. The increases in car sales decreased from 2011 to 2014. The decreases were: 2011 to 2012 7.2 6.1 1.1 2012 to 2013 7.6 7.2 0.4 2013 to 2014 7.7 7.6 0.1 − = − = − = Years Decrease 72. a. 1 7 19 67 276 370 74 5 5 + + + + = = The average number of cities where Uber operates is about 74. b. The number of cities where Uber operates increased from 2010 to 2014. The number of cities where Uber operates went up each year. c. The increases in the number of cities where Uber operates increased from 2010 to 2014. The increases were 2010 to 2011 7 1 6 2011 to 2012 19 7 8 2012 to 2013 67 19 48 2013 to 2014 276 67 209 − = − = − = − = Years Increase 74. No. Answers may vary. Example: The numbers 2 and 5 are not “between 2 and 5.” The numbers between 2 and 5 are simply 3 and 4. 76. Two consecutive integers are 1 unit apart on the number line. Two consecutive even integers are 2 units apart on the number line. Two consecutive odd integers are 2 units apart on the number line. 78. Answers may vary. Example: 90 points; the fifth score did not change the average, so it must be the same as the average. 80. Answers may vary. Example: Negative quantities are graphed to the left of 0 on the number line. Homework 1.2 2–16 even. y x −5 5 5 (−1, 3) −5 (2, 3) (1, 0) (3,−4) (−3.5, 1.5) (−5,−2) (−2.4,−4.1) (0,−4) 18. The y -coordinate is 4 − . 20. Presumably, the longer a person works for a company, the higher his or her salary will be. So, the salary s depends on the number of years t . Thus, t is the independent variable and s is the dependent variable. 22. As a student’s GPA increases, the percentage of college that would accept him or her would increase. So, the percentage p depends on the GPA g . Thus, g is the independent variable and p is the dependent variable. 24. As the age of men increases, the percentage with gray hair also increases. So, the percentage p depends on the age a . Thus, a is the independent variable and p is the dependent variable. 26. The longer the potato has been out of the oven, the cooler it will be (until it is cooled completely). So, the temperature of the potato F depends on the number of minutes t since it was removed from the oven. Thus, t is the

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    4 ISM: Elementary and Intermediate Algebra independent variable and F is the dependent variable. 28. The percentage p of people who own computers will change by age a . Thus, p is the independent variable and a is the dependent variable. 30. The total cost depends on the number of pens purchased. So, n is the independent variable and c is the dependent variable. The ordered pair (5, 10) means that n = 5 and c = 10. The cost of buying 5 pens is $10. 32. The percentage of Internet users who use social networking sites depends on the number of social networking sites. So, n is the independent variable and p is the dependent variable. The ordered pair (4,5) means that n = 4 and p = 5. So, 5% of Internet users use 4 social networking sites. 34. The number of ads in millions blocked by Google depends on the year. So , t is the independent variable and n is the dependent variable. The ordered pair (5, 780) means that t = 5 and n = 780. In 2010 5 2015 + = , 780 million ads were blocked by Google. 36. The percentage of Americans who are satisfied with the size and influence of major corporations depends on the year. So, t is the independent variable and p is the dependent variable. The ordered pair ( ) 1,35 − means that t = 1 − and p = 35. In 2015 1 2014 − = , 35% of Americans felt satisfied with the size and influence of major corporations. 38. y x −6 12 18 6 6 12 40. Point A is 2 units to the left of the origin and 4 units down. Thus, its coordinates are ( 2, 4) − − . Point B is 3 units to the left of the origin on the x -axis. Thus, its coordinates are ( 3, 0) − . Point C is 5 units to the left of the origin and 4 units up. Thus, its coordinates are ( 5, 4) − . Point D is 4 units to the right of the origin and 2 units up. Thus, its coordinates are (4, 2) . Point E is 3 units below the origin on the y -axis. Thus, its coordinates are (0, 3) − . Point F is 3 units to the right of the origin and 2 units down. Thus, its coordinates are (3, 2) − . 42. a. b. Answers may vary. Example: The average life span of a $100 bill is greater than the average life span of a $10 bill since $100 bills are used less often than $10 bills. c. Answers may vary. Example: Each year, many more $10 bills are printed than $100 bills since there is a larger demand for bills with lower value that tend to be used on a more regular basis. 44. a. b. The total airline fuel cost was the least in 2015. In 2015, the total airline fuel cost was $181 billion. c. The total airline fuel cost was the greatest in 2013. In 2013, the total airline fuel cost was $230 billion. d. No. Answers may vary. Example: The total fuel cost may not be the greatest in the same year as the average price per barrel of crude oil since total fuel cost depends on many factors, including the types of planes that were being utilized by airlines as well as flight paths (distances various fleets flew). Such factors would mean more barrels of crude oil were being used.

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    Chapter 1: Introduction to Modeling 5 46. a. b. The average U.S. hourly pay increased. In each year, the average hourly pay was greater than the previous year. c. The five-year increase in the average U.S. hourly pay did not follow a pattern of increase or decrease. 1995 to 2000 14.03 11.67 2.36 2000 to 2005 16.15 14.03 2.12 2005 to 2010 19.06 16.15 2.91 2010 to 2015 21.05 19.06 1.99 − = − = − = − = Years Change From 1995 to 2005, the change in average U.S. hourly pay decreased. However, from 2005 to 2010, the change in average hourly pay increased. 48. a. p a 20 40 60 80 40 10 30 20 Years Percent b. The highest point in the scatterplot is (21.0, 34). Answers may vary Example: It means that the 18–24 age group has the highest percentage who are ordering more takeout food than they did two years ago. c. The lowest point in the scatterplot is (70.0, 7). Answers may vary. Example: It means that the “over 64” age group has the lowest percentage who are ordering more takeout food than they did two years ago. d. The heights of the points decrease from left to right. Answers may vary. Example: Younger age groups have higher percentages than older age groups who are ordering more takeout food than they did two years ago. 50. a. p a 20 40 60 80 80 20 60 40 Years Percentage b. The 18–34 age group has the most faith in single men raising children on their own c. The “over 64” age group has the least faith in single men raising children on their own. d. The percentages of adults of various age groups who approve of single men raising children on their own are decreasing as the age group increases. Therefore, the student's opinion based on the data is correct. 52. a. Robin Ventura’s number of career grand slams is 18. b. The player who holds the record for the greatest number of career grand slams is Alex Rodriguez with 25. c. The player who hit exactly 21 grand slams is Manny Ramirez. 54. The ordered pairs selected and scatterplots may vary. The points will lie on the same horizontal line. Answers may vary. 56. There are an infinite number of possibilities for the positions of the other two vertices. Answers may vary. Example: (2, 3) and (7, 3); (2, 2) and (7, 2); (2, 10) and (7, 10); (2, 2) − and (7, 2) − . 58. All points on a coordinate system with an x -coordinate of 0 make up the y -axis. 60. Answers may vary. Homework 1.3 2. The line contains the point (4, 1) − , so 1 y = − when 4 x = .

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    6 ISM: Elementary and Intermediate Algebra 4. The line contains the point ( 6, 4) − , so 6 x = − when 4 y = . 6. The line and the y -axis intersect at (0, 1) , so the y -intercept is (0, 1) . 8. The line contains the point (6, 1) , so 1 y = when 6 x = . 10. The line contains the point (3, 0) , so 3 x = when 0 y = . 12. The line and the x -axis intersect at (3, 0) , so the x -intercept is (3, 0) . 14. a–b. y x −8 8 20 4 4 12 −4 16 c. The line contains the point (4, 14) , so 14 y = when 4 x = . d. The line contains the point (8, 17) , so 8 x = when 17 y = . e. The line and the y -axis intersect at (0, 10) , so the y -intercept is (0, 10) . f. The line and the x -axis intersect at ( 10, 0) − , so the x -intercept is ( 10, 0) − . 16. a. The line contains the point (3, 1500), so B = 1500 when t = 3. This means the balance 3 months after the account was opened was $1500. b. The line contains the point (5, 500), so t = 5 when B = 500. This means that 5 months after the account was opened, the balance was $500. c. The line and the B -axis intersect at (0, 3000), so B = 3000 when t = 0. This means that the beginning balance of the account was $3000. d. The line and the t -axis intersect at (6, 0), so t = 6 when B = 0. This means that the account will be empty after 6 months. 18. Yes. y x 20. a. y x 2 4 6 8 10 8 2 6 4 10 b. No, there is not a linear relationship between x and y . The data points do not lie close to one line. 22. a. p t 5 10 15 20 160 40 120 80 Hours Dollars b. The line contains the point (7, 56), so p = 56 when t = 7. This means the student’s pay for working 7 hours is $56. c. The line contains the point (12, 96), so t = 12 when p = 96. This means that the student must work 12 hours to earn $96.

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    Chapter 1: Introduction to Modeling 7 24. a. s t 2 4 6 8 40 10 30 20 Years Thousands of dollars b. The line contains the point (5, 30), so s = 30 when t = 5. We estimate that person’s salary will be $30 thousand after he has worked 5 years at the company. c. The line contains the point (7, 34), so t = 7 when s = 34. We estimate that the person will have worked 7 years at the company when his salary is $34 thousand. d. The line and the s -axis intersect at (0, 20), so s = 20 when t = 0. This means that the person’s beginning salary at the company was $20 thousand. 26. a. b. The line contains the point (10, 2), so t = 10 when p = 2. This means that the company’s annual profit will be $2 million in the year 2005 + 10 = 2015. c. The line and the p -axis intersect at (0, 22), so p = 22 when t = 0. This means that the company’s annual profit was $22 million in the year 2005. d. The line and the t -axis intersect at (11, 0), so t = 11 when p = 0. This means that the company’s annual profit will be $0 in the year 2005 + 11 = 2016. 28. a. v t 2 4 6 8 10 20 5 15 10 Years Thousands of dollars b. The line contains the point (8, 4), so v = 4 when t = 8. This means that the car will be worth $4 thousand when it is 8 years old. c. The line contains the point (6, 8), so v = 8 when t = 6. This means the value of the car will be $8 thousand when it is 6 years old. d. The line and the v -axis intersect at (0, 20), so v = 20 when t = 0. This means that the value of the car was $20 thousand when new. e. The line and the t -axis intersect at (10, 0), so t = 10 when v = 0. This means that the car will have no value after 10 years. 30. a. b. The line contains the point (3, 21), so v = 21 when t = 3. This means the value of the stock was $21 in the year 2008. c. The line contains the point (10, 35), so t = 10 when v = 35. This means the value of the stock will be $35 in 2005 + 10 = 2015. d. The line and the v -axis intersect at (0, 15), so v = 15 when t = 0. This means that the value of the stock was $15 in the year 2005. p t 2 4 6 8 10 20 5 15 10 Years since 2000 Millions of dollars Years since 2005 v t 2 4 6 8 40 10 30 20 Dollars Years since 2000 Years since 2005

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    8 ISM: Elementary and Intermediate Algebra 32. a. a t 2 4 6 8 1600 400 1200 800 Feet Minutes b. The line contains the point (5, 800), so a = 800 when t = 5. This means the altitude of the balloon is 800 feet after air has been released for 5 minutes. c. The line contains the point (9, 0), so t = 9 when a = 0. This means that it will take 9 minutes for the balloon to reach the ground. d. The prediction in part (c) will be an overestimate. A faster decent the last 400 feet means it will take less time to reach the ground than predicted. 34. a. b. No, there is not a linear relationship between t and p . The data points do not lie close to one line. 36. No. The 3 − is the x -coordinate of ordered pair ( ) 3, 4 − , not the x -intercept. 38. No. The x -coordinate of a y -intercept must be 0. The y -intercept might be (0, 5), but not (5, 0). 40. Yes. Any line that passes through the origin (0, 0) will have an x -intercept that is the same as the y -intercept. Answers may vary. Example: y x −5 5 5 −5 (0, 1) 42. Answers may vary. Example: y x −5 5 5 (0, 1) −5 44. Answers may vary. Example: Every point on the y -axis has an x -coordinate of 0. So the y -intercept of a line must have an x -coordinate of 0. 46. Answers may vary. Example: A linear model is a line that describes the relationship between two quantities in an authentic situation. Homework 1.4 Throughout this section, answers may vary. 2. a. y x −8 8 8 4 4 −8 −4 b. The variables are approximately linearly related. c. Draw a line that comes close to the points. See the graph in part (a). d. Approximately ( 1, 2.6) − e. Approximately ( 6.2, 3) − − f. Approximately (0, 3.7) 1 2 3 4 5 6 7 8 9 10 11 12 4 8 12 16 20 24 t p Years since 2000 Percent delayed

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    Chapter 1: Introduction to Modeling 9 g. Approximately ( 3.4, 0) − 4. a. The variables x and y are approximately linearly related. The points in the scatterplot lie close to a line. b. Draw a line that comes close to the points to create the linear model as shown. c. A pizza with 30 carbohydrates has about 290 calories. We performed interpolation because we used a part of the model whose x -coordinates are between the x- coordinates of two data points. d. A pizza with 450 calories has about 51 carbohydrates. We performed interpolation because we used a part of the model whose x -coordinates are between the x- coordinates of two data points. e. The line in the scatterplot goes up from left to right. Answers may vary. Example: A line going up from left to right makes sense because it shows that an increase in the number of carbohydrates results in an increase in the number of calories. 6. a. b. The variables t and p are approximately linearly related. The points in the scatterplot lie close to a line. c. Draw a line that comes close to the points to create the linear model as shown. d. According to the model, 31% of Americans said there should be a ban on possession of handguns around 2014. e. According to the model, about 27% of Americans will say there should be a ban on possession of handguns in 2021. 8. a. Note in the table below the difference in t - values when calculating t based on years since 1990 as in problem #7 ( 1 t ) and years since 1985 as in problem #8 ( 2 t ). A model using 2 t values interpolates n between t =7 and t = 30. A model using 1 t values interpolates n between t = 2 and t = 25. 1 2 2 7 4.9 5 10 4.6 10 15 3.5 15 20 3.1 20 25 2.0 25 30 2.1 t t n b. Draw a line that comes close to the points to create the linear model. See the graph in part (a). c. According to the model, the number of collisions in 2007 ( t = 22) is about 2.8 thousand. We performed interpolation because we used a part of the model whose

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    10 ISM: Elementary and Intermediate Algebra t -coordinates are between the t- coordinates of two data points. The response here is the same as in #7c. The value of t is different than in #7c, but the year itself has not changed and therefore the value of n does not change. d. According to the model, there will be 1.0 thousand collisions by around 2020 ( t = 35). We performed extrapolation because we used a part of the model whose t - coordinates are not between the t- coordinates of two data points. That is, the final data value occurs at t = 30, which is smaller than the value of t used to predict collisions in 2020.The response here is the same as in #8c. The value of n (1.0 thousand) has not changed. Since n correlates to a specific year (2020), the year remains the same so the data value remains unchanged. e. The t -intercept (the point where the model intersects the t -axis) is (43, 0). This means, according to the model, there will be no collisions in 2028. Model breakdown has likely occurred at this point and all years that follow since it is unlikely that in any year there would be no collisions at all (and certainly not a negative number of collisions). Note that the t -intercept is different than in #7e. It has shifted to the right by 5 compared with the value in #7e. The value of t must change in order for the data point “year 2028” and “0 collisions” to remain unchanged. The value of t consistently increases or decreases by 5 because we are calculating the values of t based on time periods that are 5 years apart. Note that the actual data does not change (that is, no matter the value of t , the years still correspond to specific numbers of collisions). Only the reference points for calculating t changes: (43, 0) represents 0 collisions in 2028 just as (38, 0) did in #7e. 10.a. b. Draw a line that comes close to the points to create the linear model. See the graph in part (a). c. According to the model, the boiling point of water at Mt. Everest’s peak (8850 meters) is about 160 F ° . d. According to the model, the elevation at which a temperature of 98.6 F ° would feel like boiling water would be around 19 thousand meters. We performed extrapolation because we used a part of the model whose E -coordinates are not between the E- coordinates of two data points. e. It takes more time to hard-boil eggs at higher altitudes. The heat that cooks the eggs comes from the boiling water. Since the temperature at which water boils is lower, water vaporizes sooner, meaning there is less actual water retaining heat inside the cooking pot. Therefore, it actually takes longer to hard-boil an egg at a higher altitude. 12. a. b. Draw a line that comes close to the points to create the linear model. See the graph in part (a).

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