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Solution Manual for Finite Mathematics and Its Applications, 12th Edition

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    S OLUTIONS M ANUAL S ALVATORE S CIANDRA Niagara County Community College F INITE M ATHEMATICS & I TS A PPLICATIONS T WELFTH E DITION Larry J. Goldstein Goldstein Educational Technologies David I. Schneider University of Maryland Martha J. Siegel Towson University Steven M. Hair Pennsylvania State University

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    Contents Chapter 1: Linear Equations and Straight Lines 1 – 1 Chapter 2: Matrices 2 – 1 Chapter 3: Linear Programming, A Geometric Approach 3 – 1 Chapter 4: The Simplex Method 4 – 1 Chapter 5: Sets and Counting 5 – 1 Chapter 6: Probability 6 – 1 Chapter 7: Probability and Statistics 7 – 1 Chapter 8: Markov Processes 8 – 1 Chapter 9: The Theory of Games 9 – 1 Chapter 10: The Mathematics of Finance 10 – 1 Chapter 11: Logic 11 – 1 Chapter 12: Difference Equations and Mathematical Models 12 – 1

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    1-1 Chapter 1 Exercises 1.1 1. Right 2, up 3 x y (2, 3) 2. Left 1, up 4 x y (–1, 4) 3. Down 2 x y (0, –2) 4. Right 2 x y (2, 0) 5. Left 2, up 1 x y (–2, 1) 6. Left 1, down 5 2 x y –1, – ( ) 5 2 7. Left 20, up 40 x y (–20, 40) 8. Right 25, up 30 x y (25, 30) 9. Point Q is 2 units to the left and 2 units up or ( 2, 2). - 10. Point P is 3 units to the right and 2 units down or (3, 2). - 11. 1 2(1) (3) 2 1 1 3 - + = - + = - so yes the point is on the line. 12. 1 2(2) (6) 1 3 - + = - is false, so no the point is not on the line

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    Chapter 1: Linear Equations and Straight Lines ISM: Finite Math 1-2 13. 1 2 1 3 x y - + = - Substitute the x and y coordinates of the point into the equation: ( ) 1 1 1 ,3 2 3 1 1 1 1 2 2 3 æ ö æ ö ÷ ÷ ç ç  - + = -  - + = - ÷ ÷ ç ç ÷ ÷ ç ç è ø è ø is a false statement. So no the point is not on the line. 14. 1 1 2 ( 1) 1 3 3 æ ö æ ö ÷ ÷ ç ç - + - = - ÷ ÷ ç ç ÷ ÷ ç ç è ø è ø is true so yes the point is on the line. 15. m = 5, b = 8 16. m = –2 and b = –6 17. y = 0 x + 3; m = 0, b = 3 18. 2 2 0; , 0 3 3 y x m b = + = = 19. 14 7 21 7 14 21 2 3 x y y x y x + = = - + = - + 20. 3 3 3 x y y x y x - = - = - + = - 21. 3 5 5 3 x x = = 22. 1 2 – 10 2 3 2 1 10 3 2 3 15 4 x y y x y x + = = + = + 23. 0 4 8 4 8 2 x x x = - + = = x -intercept: (2, 0) y = –4(0) + 8 y = 8 y -intercept: (0, 8) 24. 0 = 5 no solution x -intercept: none When x = 0, y = 5 y -intercept: (0, 5) 25. When y = 0, x = 7 x -intercept: (7, 0) 0 = 7 no solution y -intercept: none 26. 0 = –8 x x = 0 x -intercept: (0, 0) y = –8(0) y = 0 y -intercept: (0, 0) 27. 1 0 – 1 3 x = x = 3 x -intercept: (3, 0) 1 (0) – 1 3 y = y = –1 y -intercept: (0, –1) x y (3, 0) (0, –1) 28. When x = 0, y = 0. When x = 1, y = 2. x y (1, 2) (0, 0)

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    ISM: Finite Math Chapter 1: Linear Equations and Straight Lines 1-3 29. 5 0 2 = no solution x -intercept: none When x = 0, 5 2 y = y -intercept: 5 0, 2 æ ö ÷ ç ÷ ç ÷ ç è ø x y 5 2 0, ( ) 30. The line coincides with the y -axis. x y x = 0 31. 3 4(0) 24 8 x x + = = x -intercept: (8, 0) 3(0) 4 24 6 y y + = = y -intercept: (0, 6) x y (8, 0) (0, 6) 32. 0 3 3 x x + = = x -intercept: (3, 0) 0 3 3 y y + = = y -intercept: (0, 3) x y (0, 3) (3, 0) 33. 5 2 x = - 34. 1 1 (0) 1 2 3 2 x x - = - = - x intercept (–2, 0) 1 1 (0) 1 2 3 3 y y - = - = y intercept (0, 3)

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    Chapter 1: Linear Equations and Straight Lines ISM: Finite Math 1-4 35. 2 3 6 3 2 6 2 2 3 x y y x y x + = = - + = - + a. 4 6 12 6 4 12 2 2 3 x y y x y x + = = - + = - + Yes b. Yes c. 3 3 2 3 3 2 2 2 3 x y y x y x = - = - + = - + 2 – 2 3 y x = + Yes d. 6 2 0 6 2 2 6 x y y x x - - = = - = - + No e. 2 2 2 – – 2 3 3 y x x = = + Yes f. 1 1 x y y x + = = - + No 36. 1 – 5 1 2 1 –5 – 1 2 1 1 – 10 5 x y y x y x = = + = a. 1 2 – 1 5 1 2 1 5 10 5 x y y x y x = - = - + = - No b. 5 2 5 2 1 2 5 5 x y y x y x = + = - = - No c. 2 5 10 0 10 5 2 1 1 2 5 x y y x y x - + = - = - + = - No d. .1( 2) .1 .2 1 1 10 5 y x y x y x = - = - = - Yes e. 10 2 10 2 1 1 10 5 y x y x y x - = - = - = - Yes f. 1 .5 2 5 5 .5 1 1 1 10 5 x y y x y x + = + = - = - Yes

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    ISM: Finite Math Chapter 1: Linear Equations and Straight Lines 1-5 37. a. 3 3 x y y x + = = - + m = –1, b = 3 3 L b. 2 2 2 2 2 2 x y y x y x - = - - = - - = + m = 2, b = 2 1 L c. 3 3 3 3 1 1 3 x y y x y x = + = - = - 1 , 3 m = b = –1 2 L 38. a. No; 5 + 4 ≠ 3 b. No; 2 ≠ 1 – 1 c. Yes; 2(2) = 1 + 3 and 2(4) = 5 + 3 39 . 30 72 y x = + a. When x = 0, y = 72. This is the temperature of the water at time = 0 before the kettle is turned on. b. 30(3) 72 162 y y F = + = o c. Water boils when y = 212 so we have 212 30 72. x = + Solving for x gives 2 3 4 x  minutes or 4 minutes 40 seconds. 40. a. A person born in 1960 has a life expectancy of 70 years. b. 1 75 70 6 1 5 6 x x æ ö ÷ ç = + ÷ ç ÷ ç è ø æ ö ÷ ç = ÷ ç ÷ ç è ø x = 30 1960 + 30 = 1990 c. 1999 1960 39 - = 1 (39) 70 6 6.5 70 76.5 y y y æ ö ÷ ç = + ÷ ç ÷ ç è ø = + = A person born in 1999 has a life expectancy of 76.5 years. 41. a. x -intercept: 1 –33 , 0 3 æ ö ÷ ç ÷ ç ÷ ç è ø y -intercept: (0, 2.5) x y (0, 2.5) –33 , 0 ( ) 1 3 b. In 1960, 2.5 trillion cigarettes were sold. c. 4 = .075 x + 2.5 x = 20 1960 + 20 = 1980 d. 2024 – 1960 = 64 y = .075(64) + 2.5 y = 7.3 7.3 trillion 42. a. x -intercept: (–12.17, 0) y -intercept: (0, 14) b. In 2000 the income from ecotourism was $14,000. c. 20 = 1.15 x + 14 x ≈ 5.22 2000 + 5.22 = 2005.22 The year 2005 should have had $20,000 in ecotourism income.

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    Chapter 1: Linear Equations and Straight Lines ISM: Finite Math 1-6 d. 2022 – 2000 = 22 y = 1.15(22) + 14 y = 39.3 $39,300 43. a. x -intercept: (–32.9, 0) y -intercept: (0, 756) b. In 1999 the car insurance rate for a small car was $756. c. 2007 – 1999 = 8 y = 23(8) + 756 y = 940 $940 d. 1308 23 756 552 23 24 1999 24 2023 x x x = + = = + = The yearly rate will be $1308 in 2023. 44. a. x -intercept: 100 , 0 3 æ ö ÷ ç- ÷ ç ÷ ç è ø y -intercept: (0, 1000) b. 30(2) 1000 60 1000 1060 y y y = + = + = $1060 will be in the account after 2 years. c. 1180 30 1000 180 30 6 x x x = + = = The balance will be $1180 after 6 years. 45. a. In 2000, 4.1% of entering college freshmen intended to major in biology. b. 2014 2000 14 - = 0.2(14) 4.1 6.9 y y = + = 6.9% of college freshmen in 2014 intended to major in biology. c. 5.5 0.2 4.1 1.4 0.2 7 2000 7 2007 x x x = + = = + = In 2007, the percent of college freshmen who intended to major in biology was 5.5. 46. a. In 2004, 6.32% of college freshmen smoked. b. 2014 2004 10 - = 0.46(10) 6.32 1.72 y y = - + = 1.72% of college freshmen smoked in 2014. c. 2.6 0.46 6.32 3.72 0.46 8 2004 8 2012 x x x = - + - = - » + = In 2012, the percent of college freshmen who smoked was 2.6. 47. a. 2011 2004 7 - = 461(7) 16800 20, 027 y y = + = $20,027 was the approximate average tuition in 2011. b. 25000 461 16800 8200 461 17.8 2004 17 2021 x x x = + = » + = In 2021, the approximate average cost of tuition will be more than $25,000.

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    ISM: Finite Math Chapter 1: Linear Equations and Straight Lines 1-7 48. a. 2007 2003 4 - = 667(4) 12403 15071 y y = + = Approximately 15,071 bachelor degrees in mathematics and statistics were awarded in 2007. b. 25000 667 12403 12597 667 18.89 2003 18 2021 x x x = + = » + = In 2021, there will be more than 25,000 bachelor degrees in mathematics and statistics awarded. 49. 8 (0) 8 y mx b m b b = + = + = 0 (16) 8 1 2 m m = + = - 1 8 2 y x = - + 50. 0.9 (0) 0.9 y mx b m b b = + = + = 0 (0.6) 0.9 1.5 m m = + = - 1.5 0.9 y x = - + 51. y = mx + b 5 = m (0) + b b = 5 0 = m (4) + 5 5 – 4 m = 5 – 5 4 y x = + 52. Since the equation is parallel to the y axis, it will be in the form x = a. Therefore the equation will be x = 5. 53. On the x -axis, y = 0. 54. No, because two straight lines (the graphed line and the x -axis) cannot intersect more than once. 55. The equation of a line parallel to the y axis will be in the form x = a . 56. y = b is an equation of a line parallel to the x- axis. 57. 2 3 x y - = - 58. 3 4 x y - + = - 59. 2 5 3 2 3 15 x y x y + = - + = - 60. 5 4 6 24 6 5 x y x y - = - = 61. Since (a,0) and (0,b) are points on the line the slope of the line is (b-0)/(0-a) = -b/a. Since the y intercept is (0,b), the equation of the line is ( / ) y b a x b = - + or . ay bx ab = - + In general form, the equation is bx + ay = ab. 62. If (5, 0) and (0, 6) are on the line, then a = 5 and b = 6. Substituting these values into the equation bx + ay = ab gives 6x + 5y = 30. 63. One possible equation is 9. y x = - 64. One possible equation is 10. y x = + 65. One possible equation is 7. y x = + 66. One possible equation is 6. y x = - 67. One possible equation is 2. y x = + 68. One possible equation is . y x = 69. One possible equation is 9. y x = + 70. One possible equation is 5. y x = -

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    Chapter 1: Linear Equations and Straight Lines ISM: Finite Math 1-8 71. The x – intercept has a y coordinate of 0, therefore the x coordinate of the first equation is: 2 0 2 3 2 2 3 3 x x x = - = = Using this x coordinate in the second equation will find the value of c. 0 4(3) 0 12 12 c c c = - + = - + = 72. The y – intercept has a x coordinate of 0, therefore the y coordinate of the first equation is: 6(0) 3 9 3 9 3 y y y - = - = = - Using this y coordinate in the second equation will find the value of b. 3 4(0) 3 b b - = + - = 73 . a. y = –3x + 6 b . The intercepts are at the points (2, 0) and (0, 6) c . When x = 2, y = 0 74 . a. y = .25x – 2 b. (0, –2) and ( 8,0) are intercepts c. When x = 2, y = –1.5. 75 . a. 3 2 9 3 2 9 2 3 3 y x y x y x - = = + = + b. The intercepts are at the points (–4.5, 0) and (0, 3). c . When x = 2, y = 4.33 or 13 / 3. 76 . a. 2y + 5x = 8. So y = –2.5x + 4. b. The intercepts are (0, 4) and (1.6, 0). c. When x = 2 then y = –1.

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    ISM: Finite Math Chapter 1: Linear Equations and Straight Lines 1-9 77 . 2y + x = 100. When y = 0, x = 100, and when x = 0, y = 50. An appropriate window might be [-10, 110] and [-10,60]. Other answers are possible. 78 . x – 3y = 60. When x = 0, then y = - 20 and when y = 0 x = 60. An appropriate window might be [-40, 100] and [-40 , 20] but other answers are equally correct. Exercises 1.2 1. 2 3 m = 2. y = 0 x – 4 m = 0 3. y – 3 = 5( x + 4) y = 5 x + 23 m = 5 4. 7 5 10 7 – 2 5 x y y x + = = + 7 – 5 m = 5. 6 5 4 4 24 5 4 – 24 5 x y x y y x + = + = = + 4 – 5 m = 6. 1 7 8 8 8 7 8 8 7 x y x y y x - = - = = - 8 7 m = 7. 9 – 4 5 7 – 3 4 m = = 8. 3 –1 3 – (–2) m - = 4 – 5 = 9. 4 – 0 4 5 – 0 5 m = =

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    Chapter 1: Linear Equations and Straight Lines ISM: Finite Math 1-10 10. 17 –17 0 –2 – 4 m = = 11. The slope of a vertical line is undefined. 12. The slope of a vertical line is undefined. 13. 14. 15. 16. 17. –2 –2 1 m = = y – 3 = –2( x – 2) y = –2 x + 7 18. 1 2 1 1 2 m = = 1 –1 ( – 3) 2 1 1 2 2 y x y x = = - 19. 0 – 2 –2 2 –1 m = = y – 0 = –2( x – 2) y = –2 x + 4 20. 1 2 2 – 3 1 – (–1) 4 m = = 3 – 2 ( – 1) 4 3 5 4 4 y x y x = = + 21. 1 1 – –4 4 m = = 1 – 2 ( – 2) 4 1 3 4 2 y x y x = = + 22. 1 3 m = 1 – 3 ( – 5) 3 1 4 3 3 y x y x = = + 23. m = –1 y – 0 = –1( x – 0) y = – x 24. 1 2 1 – 2 – m = = y – (–1) = 2( x – 2) y = 2 x – 5 25. 1 1 2 2 1 – 0 1 m - - - = = = - y – (1) = –2( x – 0) y = –2 x + 1

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    ISM: Finite Math Chapter 1: Linear Equations and Straight Lines 1-11 26. 2 1 3 3 0 – 1 1 m - - - = = = - y – (1) = 3( x – 1) y = 3 x – 2 27. 0 2 2 1 4 – 0 4 2 m - - = = = - - 1 – 0 ( 4) 2 1 2 2 y x y x = + = + 28. 0 1 1 1 3 – 0 3 3 m - - = = = - 1 – 0 ( 3) 3 1 1 3 y x y x = - - = - + 29. m = 0 y – 3 = 0( x – 2) y = 3 30. m = undefined, therefore the equation is of the form x = a. x = 2 31. 3 – 6 ( – 5) 5 3 3 5 y x y x = = + y -intercept: (0, 3) 32. 6 3 3 4 – ( 1) 5 m - = = - 3 – 6 ( – 4) 5 3 12 6 5 5 3 18 5 5 y x y x y x = - = - = + y -intercept: 18 5 (0, ) 33. m = undefined, therefore the equation is of the form x = a. x = 0 34. 4 – 4 0 0 – 1 m = = – 4 0( 0) 4 y x y = - = 35. Let y = cost in dollars. y = 4 x + 2000 36. a. p -intercept: (0, 1200); at $1200 no one will buy the item. b. 0 = –3 q + 1200 q = 400 units q -intercept: (400, 0); even if the item is given away, only 400 will be taken. c. –3; to sell an additional item, the price must be reduced by $3. d. p = –3(350) + 1200 = $150 e. 300 3 1200 300 items q q = - + = f. 37. a. Let x = altitude and y = boiling point. 212 – 202.8 0.00184 0 – 5000 m = = - 212 0.00184( 0) 0.00184 212 y x y x - = - - = - + b. 0.00184 212 0.00184(29029) 212 158.6 F y x y y  » - + » - + » 38. a. 172 – 124 4 80 – 68 m = = 124 4( 68) 4 148 c F c F - = - = -

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    Chapter 1: Linear Equations and Straight Lines ISM: Finite Math 1-12 b. 1 37, 4 F c = + so add 37 to the number of chirps counted in 15 seconds 1 of a minute . 4 æ ö ÷ ç ÷ ç ÷ ç è ø 39. a. Let x = quantity and y = cost. 9500 – 6800 90 50 – 20 m = = 6800 90( 20) 90 5000 y x y x - = - = + b. $5000 c. $90 d. 40. a. y = 40(100) + 2400 = $6400 b. 3600 = 40 x + 2400 x = 30 coats c. y = 40(0) + 2400 = $2400 (0, 2400); even if no coats are made there is a cost for having the ability to make them. d. 40; each additional coat costs $40 to make. 41. a. 100(300) = $30,000 b. 6000 100 60 coats x x = = c. y = 100(0) = 0 (0, 0); if no coats are sold, there is no revenue. d. 100; each additional coat yields an additional $100 in revenue. 42. a. Profit = revenue – cost y = 100 x – (40 x + 2400) y = 60 x – 2400 b. (0, –2400); if no coats are sold, $2400 will be lost. c. 0 = 60 x – 2400 x = 40 (40, 0); the break-even point is 40 coats. Less than 40 coats sold yields a loss, more than 40 yields a profit. d. 60; each additional coat sold yields an additional $60 profit. e. y = 60(80) – 2400 = $2400 f. 6000 = 60 x – 2400 x = 140 coats g. 43. a. b. On February 1, 31 days have elapsed since January 1. The amount of oil y = 30,000 – 400(31) = 17,600 gallons. c. On February 15, 45 days have elapsed since January 1. Therefore, the amount of oil would be y = 30,000- 400(45) = 12,000 gallons. d. The significance of the y-intercept is that amount of oil present initially on January 1. This amount is 30,000 gallons. e. The t-intercept is (75,0) and corresponds to the number of days at which the oil will be depleted.

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    ISM: Finite Math Chapter 1: Linear Equations and Straight Lines 1-13 44. a. b. y = 2.3 – 0.15(15) = $0.05 million $50,000 c. (0, 2.3); $2.3 million is the amount of cash reserves on July 1. d. 0 = 2.3 – 0.15 t 1 15 3 t = 1 15 , 0 ; 3 æ ö ÷ ç ÷ ç ÷ ç è ø the cash reserves will be depleted after 1 15 3 days. e. y = 2.3 – 0.15(3) = $1.85 million f. 0.8 = 2.3 – 0.15 t t = 10 After 10 days, on July 11 45. a. 0.10 220 y x = + b. 0.10(2000) 220 420 y y = + = c. 540 0.10 220 $3200 x x = + = 46. Each unit sold yields a commission of $5. In addition, she receives $60 per week base pay. 47. 1 – , 2 m = b = 0 1 – 2 y x = 48. m = 3, b = –1 y = 3 x – 1 49. 1 – 3 m = 1 – (–2) – ( – 6) 3 1 3 y x y x = = - 50. m = 1 y – 2 = 1( x – 1) y = x + 1 51. 1 2 m = 1 – (–3) ( – 2) 2 1 4 2 y x y x = = - 52. m = –7 y – 0 = –7( x – 5) y = –7 x + 35 53. 2 – 5 m = 2 – 5 – ( – 0) 5 2 5 5 y x y x = = - + 54. m = 0 y – 4 = 0( x – 7) y = 4 55. 3 – (–3) –1 –1 – 5 m = = y – 3 = –1[ x – (–1)] y = – x + 2 56. 2 – 1 1 4 – 2 2 m = = 1 1 ( – 2) 2 1 2 y x y x - = = 57. –1 – (–1) 0 3 – 2 m = = y – (–1) = 0( x – 2) y = –1

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