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Solution Manual for College Algebra in Context with Applications for the Managerial, Life, and Social Sciences, 6th Edition

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    ’ S S OLUTIONS M ANUAL D EANA J. T HORNOCK R ICHMOND C OLLEGE A LGEBRA IN C ONTEXT WITH A PPLICATIONS FOR THE M ANAGERIAL , L IFE , AND S OCIAL S CIENCES S IXTH E DITION Ronald Harshbarger University of South Carolina — Beaufort Lisa S. Yocco East Georgia State College

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    TABLE OF CONTENTS Chapter 1 Functions, Graphs, and Models; Linear Functions Algebra Toolbox 1 1.1 Functions and Models 4 1.2 Graphs of Functions 10 1.3 Linear Functions 19 1.4 Equations of Lines 25 Preparing for Calculus 35 Chapter 1 Skills Check 36 Chapter 1 Review 39 Group Activities/Extended Applications 41 Chapter 2 Linear Models, Equations, and Inequalities Algebra Toolbox 43 2.1 Algebraic and Graphical Solution of Linear Equations 47 2.2 Fitting Lines to Data Points: Modeling Linear Functions 59 2.3 Systems of Linear Equations in Two Variables 66 2.4 Solutions of Linear Inequalities 78 Preparing for Calculus 87 Chapter 2 Skills Check 88 Chapter 2 Review 92 Group Activities/Extended Applications 96 Chapter 3 Quadratic, Piecewise-Defined, and Power Functions Algebra Toolbox 97 3.1 Quadratic Functions; Parabolas 101 3.2 Solving Quadratic Equations 112 3.3 Power and Root Functions 125 3.4 Piecewise-Defined Functions and Absolute Value Functions 131 3.5 Quadratic and Power Models 137

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    Preparing for Calculus 145 Chapter 3 Skills Check 147 Chapter 3 Review 153 Group Activities/Extended Applications 157 Chapter 4 Additional Topics with Functions Algebra Toolbox 161 4.1 Transformations of Graphs and Symmetry 163 4.2 Combining Functions; Composite Functions 173 4.3 One-to-One and Inverse Functions 182 4.4 Additional Equations and Inequalities 189 Preparing for Calculus 203 Chapter 4 Skills Check 204 Chapter 4 Review 208 Group Activities/Extended Applications 211 Chapter 5 Exponential and Logarithmic Functions Algebra Toolbox 215 5.1 Exponential Functions 218 5.2 Logarithmic Functions; Properties of Logarithms 225 5.3 Exponential and Logarithmic Equations 232 5.4 Exponential and Logarithmic Models 243 5.5 Exponential Functions and Investing 251 5.6 Annuities; Loan Repayment 258 5.7 Logistic and Gompertz Functions 263 Preparing for Calculus 269 Chapter 5 Skills Check 270 Chapter 5 Review 272 Group Activities/Extended Applications 278

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    Chapter 6 Higher-Degree Polynomial and Rational Functions Algebra Toolbox 279 6.1 Higher-Degree Polynomial Functions 283 6.2 Modeling with Cubic and Quartic Functions 292 6.3 Solution of Polynomial Equations 299 6.4 Polynomial Equations Continued; 307 Fundamental Theorem of Algebra 6.5 Rational Functions and Rational Equations 316 6.6 Polynomial and Rational Inequalities 328 Preparing for Calculus 337 Chapter 6 Skills Check 339 Chapter 6 Review 346 Group Activities/Extended Applications 351 Chapter 7 Systems of Equations and Matrices Algebra Toolbox 355 7.1 Systems of Linear Equations in Three Variables 357 7.2 Matrix Solution of Systems of Linear Equations 371 7.3 Matrix Operations 385 7.4 Inverse Matrices; Matrix Equations 392 7.5 Determinants and Cramer’s Rule 407 7.6 Systems of Nonlinear Equations 419 Chapter 7 Skills Check 431 Chapter 7 Review 441 Group Activities/Extended Applications 447

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    Chapter 8 Special Topics in Algebra 8.1 Systems of Inequalities 449 8.2 Linear Programming: Graphical Methods 472 8.3 Sequences and Discrete Functions 485 8.4 Series 489 8.5 The Binomial Theorem 496 8.6 Conic Sections: Circles and Parabolas 499 8.7 Conic Sections: Ellipses and Hyperbolas 510 Chapter 8 Skills Check 522 Chapter 8 Review 533 Group Activities/Extended Applications 538

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    Algebra Toolbox 1 CHAPTER 1 Functions, Graphs, and Models; Linear Functions Toolbox Exercises 1. { } { } 1,2,3,4,5,6,7,8 and , 9 x x N x ∈ < Remember that x N ∈ means that x is a natural number. 2. Yes 3. No. A is not a subset of B . A contains elements 2, 7, and 10, which are not in B . 4. No. { } 1,2,3,4, . . . , N = therefore 1 . 2 N ∉ 5. Yes. Every integer can be written as a fraction with the denominator equal to 1. 6. Yes. Irrational numbers are by definition numbers that are not rational. 7. a. { } 3,0, A B π ∩ = − b. 2 15 5, 3, ,0, ,4,5,7, ,8 3 2 A B π   ∩ = − − −     8. Integers, rational numbers. The integers are a subset of the rational numbers. Exercise 5. 9. Rational numbers 10. Irrational numbers 11. ( ) ( ) 6 16 22 − + − = − 12. ( ) 22 13 22 13 35 − − = + = 13. ( ) 56 34 56 34 22 − − − = − + = − 14. ( ) ( ) ( ) ( ) 12 6 3 6 14 18 3 6 14 15 6 14 21 14 35 − + − − − − + − = − + − − = − − − = − − = − 15. ( )( )( ) 7 3 2 42 − − = 16. ( ) ( )( ) 18 2 16 3 9 48 39 ÷ − + − − = − + = 17. 35 42 5 7 2 7 6 − + = − + = − − 18. [ ] [ ] [ ] [ ] ( ) ( 8)( 3) 5( 2) 3( 4) ( 5) 24 10 12 5 14 7 2 − − + − ÷ − − − = − ÷ − + = ÷ − = − 19. ( )( ) 3 0 0 − = 20. 0 0 6 = − 21. 5 is undefined. 0 Not possible; division by 0 is undefined.

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    2 CHAPTER 1 Functions, Graphs, and Models; Linear Functions 22. a. ( ) 8 6 6 8 − + = + − b. ( )( ) ( )( ) 9 3 3 9 − = − 23. a. ( ) ( ) 8 2 3 8 2 3 + − + = + − + b. ( )( )( ) ( )( ) ( ) 7 3 4 7 3 4 − − = − −     24. a. ( ) ( ) ( ) 6 12 5 6 12 6 5 72 30 + = + = + b. ( ) x y x y − + = − − 25. This is an example of both the commutative and the associative property . ( ) ( ) ( ) 3 7 8 7 3 8 Commutative Property 7 3 8 Associative Property + + = + + = + + 26. a. 10 0 10 − + = − b. ( ) 1 3 3 ⋅ = 27. a. 4 4 0 − + = b. 1 7 1 7 ⋅ = 28. 3 x > − 29. 3 3 x − ≤ ≤ 30. 3 x ≤ 31. ( ] ,7 −∞ 32. ( ] 3,7 33. ( ) , 4 −∞ 34. 35. 5 2 implies 2 5 x x > ≥ ≤ < 36. 37. 2 2 3 4 8 The coefficient of 3 is 3. The coefficient of 4 is 4. The constant term is 8. x x x x − − + − − − − 38. 4 3 4 3 5 7 3 The coefficient of 5 is 5. The coefficient of 7 is 7. The constant term is 3. x x x x + − − 39. ( ) ( ) 2 4 2 4 4 5 8 20 28 a b − = − − = + = 40. ( ) 3 3 2 5 2 7 5 6 1 19 14 5 2 2 y x − + − = − − + − = + − = 41. ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 3 2 0.6 3 3 2( 2) 4 0.6 4 3(0.8) 3 4 4 0.6 4 2.4 3 8 0.6 1.6 24 0.96 23.04 a b b c − + − = − − + − = − − + − = − + = − + = − 42. ( )( ) 1 1 4.5 6.5 14.625 sq ft 2 2 bh = =

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    Algebra Toolbox 3 43. ( ) ( ) 2 2 16 16 9 16 81 1296 feet t = = = 44. ( ) ( ) ( ) ( ) ( ) 4 2 4 3 2 4 4 3 2 2 4 3 2 15 20 6 2 4 12 5 2 4 15 12 20 6 5 3 4 27 20 11 z z z z z z z z z z z z z z z z − + − + + − − = + + + − − + + − − = + − + − 45. ( ) ( ) ( ) 4 3 4 2 4 4 4 3 4 2 4 3 4 2 3 2 2 119 5 3 5 110 2 5 2 3 3 5 119 110 7 2 3 2 9 x y x y x y y y y x y y x x y x y y x + − − − − + + = + − − + − + − + = − − − − 46. ( ) ( ) ( ) 4 4 4 4 4 p d p d p d + = + = + 47. ( ) ( ) ( ) 2 3 7 2 3 2 7 6 14 x y x y x y − − = − − − = − + 48. ( ) ( ) ( ) 8 8 8 a b c a b a c ab ac − + = − − = − − 49. ( ) ( ) 4 3 2 4 4 3 2 6 x y x y x y x y x y − − + = − − − = − 50. ( ) ( ) ( ) ( ) ( ) ( ) 4 2 4 5 2 4 8 4 4 5 5 2 4 8 2 4 5 4 5 4 6 9 5 x y xy y xy x y x y xy y xy x y x x xy xy y y y x xy y − + − − − − = − + − + − + = − + + + − − + = + − 51. ( ) ( ) ( ) ( ) 2 4 4 5 3 8 8 5 3 8 5 8 3 3 5 x yz xyz x xyz x xyz x xyz xyz x x xyz x − − − = − − + = − + − + = − 52. Subtraction property 7 11 7 7 11 7 4 x x x + = + − = − = 53. Addition property 4 10 4 4 10 4 6 x x x − = − − + = − + = − 54. Division property 5 20 5 20 5 5 4 x x x = − − = = − 55. Division property 2 18 2 18 2 2 9 x x x − = − = − − = − 56. Multiplication property ( ) 3 6 6 6 3 6 18 y y y =   =     = 57. Multiplication property ( ) 8 4 4 4 8 4 32 p p p − = − −   − = − −     =

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    4 CHAPTER 1 Functions, Graphs, and Models; Linear Functions 58. 59. 60. 61. 62. Skills Check 1.1 1. Use Table A. a. –5 is an x -value and therefore is an input of the function ( ). f x b. ( 5) f − is an output of the function. c. The domain is the set of all inputs. { } 9, 7, 5,6,12,17,20 D = − − − The range is the set of all outputs. { } 4,5,6,7,9,10 R = d. Each input x of the function f yields exactly one output ( ). y f x = 2. Use Table B. a. 0 is an x -value and therefore is an input of the function ( ). g x b. (7) g is an output of the function. c. The domain is the set of all inputs. { } 4, 1,0,1,3,7,12 D = − − The range is the set of all outputs. { } 3,5,7,8,9,10,15 R = d. Each input x of the function g yields exactly one output ( ). y g x = 3. ( 9) 5 (17) 9 y f y f = − = = = 4. ( 4) 5 (3) 8 y g y g = − = = = 5. No. In Table A, x is not a function of y . If y is considered the input variable, one input will correspond with more than one output. Namely, if 9, y = then 12 x = or 17. x = 6. Yes. Each input y produces exactly one output x .

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    1.2 Graphs of Functions 5 7. a. (2) 1, f = − since 2 x = in the table corresponds with ( ) 1. f x = − b. 2 (2) 10 3(2) 10 3(4) 10 12 2 f = − = − = − = − c. (2) 3, f = − since (2, 3) − is a point on the graph. 8. a. ( 1) 5, f − = since ( ) 1,5 − is a point on the graph. b. ( 1) 8, f − = − since 1 x = − in the table corresponds with ( ) 8. f x = − c. 2 ( 1) ( 1) 3( 1) 8 1 3 8 6 f − = − + − + = − + = 9. 0 2 2 4 x y − − 10. 4 1 4 3 x y − − 11. Recall that ( ) 5 8. R x x = + a. ( 3) 5( 3) 8 15 8 7 R − = − + = − + = − b. ( 1) 5( 1) 8 5 8 3 R − = − + = − + = c. (2) 5(2) 8 10 8 18 R = + = + = 12. Recall that 2 ( ) 16 2 . C s s = − a. 2 (3) 16 2(3) 16 2(9) 16 18 2 C = − = − = − = − b. 2 ( 2) 16 2( 2) 16 2(4) 16 8 8 C − = − − = − = − = c. 2 ( 1) 16 2( 1) 16 2(1) 16 2 14 C − = − − = − = − = 13. Yes. Each input corresponds with one output. { } { } 1,0,1,2,3 ; 8, 1,2,5,7 D R = − = − − 14. No. Each input x does not match with exactly one output y . Specifically, if 2, x = then 3 or 4. y y = − = 15. No. The graph fails the vertical line test. Each input does not match with exactly one output. 16. Yes. The graph passes the vertical line test. Each input matches with exactly one output. 17. Yes. The graph passes the vertical line test. Each input matches with exactly one output.. 18. No. The graph fails the vertical line test. Each input does not match with exactly one output. 19. No. If 3, x = then 5 or 7. y y = = One value of x , 3, gives two values of y . 20. Yes. Each input x yields exactly one output y . 21. a. This is not a function. If 4, x = then 12 y = or 8. y = b. This is a function. Each input yields exactly one output. 22. a. Yes. Each input yields exactly one output. b. This is not a function. If 3, x = then 4 y = or 6. y = 23. a. This is not a function. If 2, x = then 3 y = or 4. y = b. This is a function. Each input yields exactly one output. 24. a. This is a function. Each input yields exactly one output. b. This is not a function. If 3, x = − then 3 y = or 5. y = −

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    6 CHAPTER 1 Functions, Graphs, and Models; Linear Functions 25. The domain is the set of all inputs. { } 3, 2, 1,1,3,4 D = − − − The range is the set of all outputs. { } 8, 4,2,4,6 R = − − 26. The domain is the set of all inputs. { } 6, 4, 2,0,2,4 D = − − − The range is the set of all outputs. { } 5, 2,0,1,4,6 R = − − 27. Consider y as a function of x. Domain is the set of all inputs x . [ ] 10, 8 D = − Range is the set of all outputs y . [ ] 12, 2 R = − 28. Consider y as a function of x. Domain is the set of all inputs x . [ ] 4, 3 D = − Range is the set of all outputs y . [ ] 1, 4 R = − 29. Consider y as a function of x . Domain is the set of all inputs x . ( ) , D = −∞ ∞ Range is the set of all outputs y . [ ) 4, R = − ∞ 30. Consider y as a function of x . Domain is the set of all inputs x . ( ] ,3 D = −∞ Range is the set of all outputs y . [ ) 0, R = ∞ 31. The input is the number of years after 2000. The year 2015 is 15 years after 2000; 15. x = The year 2022 is 22 years after 2000; 22. x = 32. The input is the number of years after 1990. The year 1990 is 0 years after 1990; 0. x = The year 2015 is 25 years after 1990; 25. x = 33. No. If 0, x = then 2 2 2 (0) 4 4 2. y y y + =  =  = ± One input of 0 corresponds with two outputs of –2 and 2. Therefore, the equation is not a function. 34. Yes. Each input for x corresponds with exactly one output for y . 35. 9 32, 5 C F = + where F is the Fahrenheit temperature and C is the Celsius temperature. 36. 2 , C r π = where C is the circumference and r is the radius. 37. Convert F temperature to C temperature using the following steps. a. Subtract 32. b. Multiply by 5. c. Divide by 9. 38. Represent the function D using the following steps. a. Square E . b. Multiply by 3. c. Subtract 5. Exercises 1.1 39. No. There are two different outputs for 20. Y = 40. Yes. Each input corresponds with exactly one output. 41. a. Yes. Each input (year) corresponds with exactly one output (revenue). b. {2014,2015,2016,2017,2018,2019, 2020,2021,2022,2023,2024} D = {207.3,240.3,290.1,349.3,414.0,454.2, 475.1,497.0,515.7,532.2,544.8} R = 42. T , temperature, is a function of m , the number of minutes after the power outage. Each value for m corresponds with exactly one value for T . The graph of the equation passes the vertical line test.

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    1.2 Graphs of Functions 7 43. a. Yes. Each input (barcode) corresponds with exactly one output (price). b. No. An input (price) could correspond with more than one output (barcode). Numerous items can have the same price but different barcodes. 44. a. Yes. Each input (piano key) corresponds with exactly one output (note). The domain is the set of all inputs and there are 12 keys on the piano, so there are 12 elements in the domain of the function. b. Yes. Each input (note) corresponds with exactly one output (piano key). The range is the set of all outputs and there are 12 keys, so there are 12 elements in the range of the function. 45. Each input x (in years) corresponds with one output V , the value of the property. 46. Yes. Each input d (depth) corresponds with exactly one output p (pressure). 47. a. Yes. Each input (day) corresponds with exactly one output (weight). b. The domain is the first 14 days of May. { } 1,2,3,4,5,6,7,8,9,10,11,12,13,14 D = c. The range is the list of weights. { } 171,172,173,174,175,176,177,178 R = d. Highest weights were on May 1 and 3. e. Lowest weight was on May 14. f. Longest period during which his weight decreased is 3 days, May 8 to 11. 48. a. No. This is not a function. One input of 75 corresponds with two outputs of 70 and 81. b. Yes. This is a function. Each input (average score on final exam) corresponds with exactly one output (average score on math placement test). 49. a. (3) 1096.78 P = If the car is financed over 3 years, the payment is $1096.78. b. (5) 42,580.80 C = The total cost if financed over 5 years is $42,580.80. c. (4) 41,014.08 C = If the total cost is $41,014.08, then the car has been financed over 4 years d. (5) 42,580.80; C = (3) 39,484.08 C = The savings would be $42,580.80 $39,484.08 $3096.72. − = 50. a. (103,000) 20 f = They must make payments for 20 years. b. (120,000) 30 f = It will take the couple 30 years to pay off a $120,000 mortgage at 7.5%. c. (3 40,000) (120,000) 30 f f ⋅ = = d. If 40,000 A = then ( ) (40,000) 5. f A f = = e. (3 40,000) (120,000) 30 3 (40,000) 3 5 15 The expressions are not equal. f f f ⋅ = = ⋅ = ⋅ = 51. a. Approximately 79.3 million b. (2050) 91.5 f = Approximately 91.5 million women are projected to be in the workforce in 2050. c. {1950,1960,1970,1980,1990,2000, 2010,2020,2030,2040,2050} D = d. As the year increases, the number of women in the workforce also increases. 52. a. In 2020, t = the ratio is about 3 to 1. b. (2005) 4 f = . In 2005, the projected ratio of the working-age population to the elderly population is 4 to 1. c. { } 1995,2000,2005,2010,2015, 2020,2025,2030 D = d. As the years increase, the projected ratio of the working-age population to the elderly population decreases.

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    8 CHAPTER 1 Functions, Graphs, and Models; Linear Functions 53. a. (25,000) $14.15 (75,000) $42.44 f f = = b. (50,000) $12.83 (100,000) $25.67 g g = = c. If ( ) 28.29, 50,000. f x x = = d. Yes 54. a. (2030) 39, 244 f = b. In 2030, the population of females under the age of 18 is projected to be 39,244,000. c. The function is increasing. 55. a. In 2020, 5.7 million U.S. citizens age 65 and older are expected to have Alzheimer’s disease. b. (2030) 7.7. f = In 2030, 7.7 million U.S. citizens age 65 and older are expected to have Alzheimer’s disease. c. 2040; (2040) 11 f = d. Since 2000, the number of U.S citizens 65 and older that are expected to have Alzheimer’s disease is increasing. 56. a. Yes. Each year, t , corresponds with exactly one number of jobs, N . b. (2025) 12.9 (million) f = c. In 2025, there will be 12.9 million jobs. d. 2035; (2035) 11.8 f = 57. a. (2030) 15.8; f = 15.8% b. (2040) 17.1. f = The projected percent of the U.S. population that is foreign-born in 2040 is 17.1%. c. (2050) 18.2. f = This population is projected to be 18.2% in the year 2050. d. { } 2020,2030,2040,2050,2060 D = e. The projected population of the U.S. population that is foreign-born is increasing. 58. a. (1990) 3.4 f = In 1990, there were 3.4 workers for each retiree. b. 2030; (2030) 2 f = c. As time passes, the number of workers decreases. Funding for Social Security in the future is problematic. Workers may need to pay a larger portion of salaries to fund benefits to retirees. 59. a. ( ) ( ) 200 32 200 6400 R = = The revenue from the sale of 200 hats is $6400. b. ( ) ( ) 2500 32 2500 $80,000 R = = 60. a. ( ) ( ) 200 4000 12 200 6400 C = + = The cost of producing 200 hats is $6400. b. ( ) ( ) 2500 4000 12 2500 $34,000 C = + = 61. a. ( ) ( ) 1000 0.857 1000 19.35 857 19.35 876.35 f = + = + = The charge for 1000 kilowatt hours is $876.35. b. ( ) ( ) 1500 0.857 1500 19.35 1285.50 19.35 1304.85 f = + = + = The charge for 1500 kilowatt hours is $1304.85. 62. a. ( ) 2 500 450(500) 0.1(500) 2000 225,000 25,000 2000 198,000 P = − − = − − = The profit from the production and sale of 500 iPod players is $198,000. b. ( ) 2 4000 450(4000) 0.1(4000) 2000 1,800,000 1,600,000 2000 198,000 P = − − = − − = The profit from the production and sale of 4000 iPod players is $198,000.

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    1.2 Graphs of Functions 9 63. a. ( ) ( ) ( ) 2 100 32 100 0.1 100 1000 3200 1000 1000 1200 P = − − = − − = Daily profit from producing and selling 100 Blue Chief bicycles is $1200. b. ( ) ( ) ( ) 2 160 32 160 0.1 160 1000 5120 2560 1000 1560 P = − − = − − = Daily profit from producing and selling 160 Blue Chief bicycles is $1560. 64. a. 2 (1) 6 96(1) 16(1) 6 96 16 86 h = + − = + − = Height of ball after one second is 86 feet. b. 2 (3) 6 96(3) 16(3) 6 288 144 150 h = + − = + − = Height of ball after 3 seconds is 150 feet. c. Test 2 : t = 2 (2) 6 96(2) 16(2) 6 192 64 134 h = + − = + − = Test 4. t = 2 (4) 6 96(4) 16(4) 6 384 256 134 h = + − = + − = Test 5. t = 2 (5) 6 96(5) 16(5) 6 480 400 86 h = + − = + − = It appears that the ball reaches a high point after 3 seconds and begins to fall. At 1 and 5 seconds, and again at 2 and 4 seconds, the respective heights are equal. 65. a. 0.3 0.7 0 0.7 0.3 0.7 0.3 0.7 0.7 3 7 n n n n + = = − − = = − The domain of ( ) R n is all real numbers except 3 7 − or 3 3 , , . 7 7     −∞ − − ∞          b. In the context of the problem, n represents the factor for increasing the number of questions on a test. Therefore, it makes sense that n is positive, 0. n > 66. a. Yes. Each value of s corresponds with exactly one value of K c . b. In order for the value of K c to be real, the radicand, 4 1, s + must be greater than or equal to zero. 4 1 0 4 1 1 4 s s s + ≥ ≥ − ≥ − The domain is all real numbers greater than or equal to 1 4 − or, 1 , . 4   − ∞     c. s represents wind speed, and cannot be less than zero. The domain is restricted based on the context of the problem. The actual domain in context is [ ) 0, ∞ . 67. a. Since p is a percentage, 0 100. p ≤ ≤ In the given function, the denominator, 100 , p − cannot equal zero, so 100. p ≠ The domain is 0 100 p ≤ < or [ ) 0, 100 . b. ( ) ( ) ( ) ( ) 237,000 60 60 355,500 100 60 237,000 90 90 2,133,000 100 90 C C = = − = = − 68. a. Since the square root expression is in the denominator of the function, the radicand must be positive. 2 1 0 2 1 1 2 p p p + > > − > − The domain of q is 1 , . 2   − ∞     In the context of the problem, p represents the price of a product and cannot be negative. [ ) 0, D = ∞ In context, q represents the quantity of the product demanded by consumers. It cannot be 0 and can be no larger than 100. ( ] 0,100 R =

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    10 CHAPTER 1 Functions, Graphs, and Models; Linear Functions 69. a. ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 2 12 12 108 4(12) 144 108 48 144 60 8640 18 18 108 4(18) 324 108 72 324 36 11,664 V V = − = − = = = − = − = = b. First, x represents a side length so 0. x > Second, to satisfy postal restrictions, the length (longest side) plus the girth must be less than or equal to 108 inches. Length + Girth 108 Length 4 108 4 108 Length 108 Length 4 Length 27 4 x x x x ≤ + ≤ ≤ − − ≤ ≤ − x is greatest if length is minimized, so let Length 0 = and find x . 0 27 and 27 4 x x ≤ − ≤ The conditions on x are 0 27. x < ≤ If 27, x = the length would be zero and the package would not exist. In context, 0 27 x < < and the corresponding domain for the function ( ) V x is ( ) 0,27 . c. x Volume 10 6,800 15 10,800 20 11,200 21 10,584 19 11,552 18 11,664 17 11,560 The table shows a maximum volume of 11,664 cubic inches when 18. x = length girth 108, + = so length 4 x + = ( ) length 4 18 108, + = and length 36. = The dimensions that maximize the volume of the box are 18 inches by 18 inches by 36 inches. 70. a. 2 (0) 4.9(0) 98(0) 2 2 S = − + + = The initial height of the bullet is 2 feet. b. 2 (9) 4.9(9) 98(9) 2 487.1 S = − + + = 2 (0) 4.9(10) 98(10) 2 492 S = − + + = 2 (11) 4.9(11) 98(11) 2 487.1 S = − + + = c. The bullet appears to reach a maximum height at 10 seconds, then begins to fall. t Height 9 487.1 9.5 490.78 10 492 10.5 490.78 11 487.1 Skills Check 1.2 1. a. 3 3 2 1 0 1 2 3 27 8 1 0 1 2 3 x y x − − − = − − − b. c. The graph in part (a) is generated by plotting points from the table and matches the calculator graph in part (b).

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