(Solution Manual) College Algebra in Context with Applications for the Managerial, Life, and Social Sciences

Harshbarger, Yocco, 5

Math & Statistics

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5th Edition

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Solution Manual for College Algebra in Context with Applications for the Managerial, Life, and Social Sciences, The downloadable file consists of 721 pages, and the total size of the file is 8.68 MB. Purchase now only for highly discounted price.

Author(s): Harshbarger, Yocco

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CHAPTER 1 Toolbox 1 Copyright ©2017 Pearson Education, Inc. CHAPTER 1 Functions, Graphs, and Models; Linear Functions Toolbox Exercises 1. { 1 ,2,3 ,4,5 ,6,7,8 } and {9 ,}xx x N<∈ Remember that xN∈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, 1.2N∉ 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. Integers. However, note that this set of integers could also be considered as a set of rational numbers. See question 5. 8. Rational numbers 9. Irrational numbers 10. 3 x>− 11. 33x −≤ ≤ 12. 3x≤ 13. ( ],7−∞ 14. (]3, 7 15. (),4−∞ 16. 17. Note that 5 2 implies 2 5 xx>≥ ≤< , therefore: 18. 19. 20. 2 CHAPTER 1 Functions, Graphs, and Models; Linear Functions Copyright ©2017 Pearson Education, Inc. 21. 22. 23. 24. Yes, it is a polynomial with degree 4. 25. No, it is not a polynomial since there is a variable in a denominator. 26. No, it is not a polynomial since there is a variable inside a radical. 27. Yes, it is a polynomial with degree 6. 28. The x2 term has a coefficient of –3. The x term has a coefficient of –4. The constant term is 8. 29. The x4 term has a coefficient of 5. The x3 term has a coefficient of 7. The constant term is –3. 30. () ( ) () ( ) ( ) () ( )42 4 32 44 3 2 2 43 215 20 6 2 4 12 5 24 1 5 1 2 20 6 5 3 4 27 20 11zz z z zz zz z z z z zz z z−+ − ++ −− =+ + + − − + +−− =+− +− 31. () () ( )43 4 24 44 3 42 43 423 2 2 119 5 3 5 110 25 2 3 3 5 119 110 72 32 9xy x y xy y yy x yy xx yx yyx+− − − −++ =+− − + −+ −+ =− −− − 32. ()44 4pdp d+= + 33. ()23 7 6 1 4x yx y −− = − + 34. ()8 8ab c ab ac−+ =− − 35. () ( )43 2 4432 16 6xyx y x yxy xy xy−− + =−−−=−=− 36. () () ( ) () ( ) ( )42 4 5 2 4 844 55 24 82 4 5 454 69 5xy x y yx y x y xyx yyx yxy x xx y x y y y y xx yy−+ − − − − =−+ −+ −+ =−+ + + − − + =+ − 37. () ( ) () ( )24 4 5 3 88 53 85 8 3 35x yz xyz x xyz x xyz x xyz xyz x x xyz x−− − =− −+ =−+ − + =− CHAPTER 1 Toolbox 3 Copyright ©2017 Pearson Education, Inc. 38. 23 23 25343 1 2xya x a x y  39.  32 2 34 23 3 222 43 48 6xy x y xz z xyx y z x y z   40.   2 272 3 2 3 1 4 2 1 21 1 2 1xx x x x x x      41.  2 2 2(3 ) 3 3 339 69kk k kk k kk     42.  22 224 7 4 7 16 28 28 49 16 49x yx y x x y x y y xy      43. 212 512 5xxx x 44. 28241 2xxx x 4 CHAPTER 1 Functions, Graphs, and Models; Linear Functions Copyright ©2017 Pearson Education, Inc. Section 1.1 Skills Check 1. Using Table A a. –5 is an x-value and therefore is an input into the function ( ) fx. b. ()5f− represents an output from the function. c. The domain is the set of all inputs. D:  9, 7, 5,6,12,17,20 . The range is the set of all outputs. R:   4,5,6,7,9,10 d. Each input x into the function f yields exactly one output (). yf x= 2. Using Table B a. 0 is an x-value and therefore is an input into the function ( ). gx b. ()7g represents an output from the function. c. The domain is the set of all inputs. D:  4, 1,0,1,3,7,12 . The range is the set of all outputs. R:   3,5,7,8,9,10,15 d. Each input x into the function g yields exactly one output () yg x= . 3. ()(9 ) 5 17 9f f−= = 4. (4 ) 5 (3) 8gg−= = 5. No. In the given table, x is not a function of y. If y is considered the input variable, one input will correspond with more than one output. Specifically, if 9 y=, then 12x=or 17x=. 6. Yes. Each input y produces exactly one output x. 7. a. (2) 1f =− , since x = 2 in the table corresponds with ( ) 1 fx =− . b. 2(2) 10 3(2) 10 3(4) 10 12 2f =−=−=−=− c. (2) 3f =− , since (2, 3) − is a point on the graph. 8. a. ()15f−= , since ( 1,5) − is a point on the graph. b. ()18f−= − , since x = −1 in the table corresponds with f(x) = −8. c. ()21 ( 1) 3( 1) 8 138 6f−= − + − + =−+ = 9. 02 24x y −− 10. 41 43x y −− CHAPTER 1 Section 1.1 5 Copyright ©2017 Pearson Education, Inc. 11. Recall that () 5 8Rxx=+ . a. (3 ) 5 (3 ) 8 1 5 8 7R−=−+ = − + = − b. (1 ) 5 (1 ) 8 5 8 3R−=−+= − += c. (2) 5(2) 8 10 8 18R =+ = + = 12. Recall that 2() 1 6 2Cs s =− . a. 2(3) 16 2(3) 16 2(9) 16 18 2C =− =−=−=− b. 2(2 ) 1 6 2 (2 ) 16 2(4) 16 8 8C−= −− =− =−= c. 2(1 ) 1 6 2 (1 ) 16 2(1) 16 214C −=− − =− =−= 13. Yes. Each input corresponds with exactly one output. The domain is  1,0,1,2,3 . The range is   8, 1, 2,5,7 . 14. No. Each input x does not match with exactly one output y. Specifically, if 2x= then 3 or 4 yy=− = . 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 3x=, then 5 or 7 yy== . One input yields two outputs. The relation is not a function. 20. Yes. Each input x yields exactly one output y. 21. a. Not a function. If 4 x=, then 12 y= or 8y=. b. Yes. Each input yields exactly one output. 22. a. Yes. Each input yields exactly one output. b. Not a function. If 3 x=, then 4 y=or 6y=. 23. a. Not a function. If 2 x=, then 3 y=or 4y=. b. Function. Each input yields exactly one output. 24. a. Function. Each input yields exactly one output. b. Not a function. If 3 x=− , then 3 y=or 5 y=− . 6 CHAPTER 1 Functions, Graphs, and Models; Linear Functions Copyright ©2017 Pearson Education, Inc. 25. The domain is the set of all inputs. D:  3, 2, 1,1, 3, 4 . The range is the set of all outputs. R:   8, 4,2,4,6 26. The domain is the set of all inputs. D:  6, 4, 2,0,2,4 . The range is the set of all outputs. R:   5, 2,0,1,4,6 27. Considering y as a function of x, the domain is the set of all inputs, x. Therefore the domain is D: []10, 8− . The range is the set of all outputs, y. Therefore, the range is R:[]12, 2− . 28. Considering y as a function of x, the domain is the set of all inputs, x. Therefore the domain is D: []4, 3− . The range is the set of all outputs, y. Therefore, the range is R:[]1, 4− . 29. Considering y as a function of x, the domain is the set of all inputs, x. Therefore the domain is D: ( , )   . The range is the set of all outputs, y. Therefore, the range is R:[ 4, ). 30. Considering y as a function of x, the domain is the set of all inputs, x. Therefore the domain is D: ( ,3]  . The range is the set of all outputs, y. Therefore, the range is R:[0, ). 31. The input is the number of years after 2000, therefore 2015 is 15 years after 2000 and the input would be 15.x Similarly, 2022 is 22 years after 2000, and the input would be 22.x 32. The input is the number of years after 1990, therefore 1990 is 0 years after 1990 and the input would be 0. x Similarly, 2015 is 25 years after 1990, and the input would be 25.x Therefore, the input would be 0 to 25. 33. No. If 0 x=, then 22 2(0) 4 4 2 yyy+= ⇒ = ⇒ = ± . So, one input of 0 corresponds with 2 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. 932,5CF where F is the Fahrenheit temperature and C is the Celsius temperature. 36. 2Cr π= , where C is the circumference and r is the radius. 37. C is found by using the steps; a. subtract 32; b. multiply by 5; c. divide by 9. 38. D is found by squaring E, multiplying the result by 3, and subtracting 5. Section 1.1 Exercises 39. a. Yes. Each input ( a, age in years) corresponds with exactly one output ( p, life insurance premium). The independent variable is a, age in years, and the dependent variable is p, life insurance premium. b. No. One input of $11.81 corresponds with six outputs ( a, age in years). 40. Yes. Each input ( d, degree) corresponds with exactly one output ( M, mean earnings). The independent variable is d, highest degree for females, and the dependent variable is M, mean earnings in dollars. 41. Yes. Each input ( y, year) corresponds with exactly one output ( p, percent). The independent variable is y, the year, and the dependent variable is p, the percent of Americans who are obese. CHAPTER 1 Section 1.1 7 Copyright ©2017 Pearson Education, Inc. 42. T, temperature, is a function of m, number of minutes after the power outage, since each value for m corresponds with exactly one value for T. The graph of the equation passes the vertical line test, which implies there is one temperature for each value of m, number of minutes after the power outage. 43. a. Yes. Each input (the barcode) corresponds with exactly one output (an item’s price). b. No. Every input (an item’s price) could correspond with more than one output (the barcode). Numerous items can have the same price but different barcodes. 44. a. Yes. Each input (a child’s piano key) corresponds with exactly one output (a musical note). Since the domain is the set of all inputs into the function and there are 12 keys on the child’s piano keyboard, there are 12 elements in the domain of the function. b. Yes. Each input (a note from the child’s piano) corresponds with exactly one output (a piano key). Since the range is the set of all outputs from the function and there are 12 keys on the child’s piano keyboard, there are 12 elements in the range of the function. 45. Each input ( x, years) corresponds with exactly one output ( V, value of the property). The graph of the equation passes the vertical line test. 46. Yes. Each input ( d, depth) corresponds with exactly one output ( p, pressure). The graph of the equation passes the vertical line test. 47. a. Yes. Each input (day of the month) corresponds with exactly one output (weight). b. The domain is the first 14 days of May or D:  1,2,3,4,5,6,7,8,9,10,11,12,13,14 . c. The range is   171,172,173,174,175,176,177,178 . d. The highest weights were on May 1 and May 3. e. The lowest weight was on May 14. f. Three days from May 8 until May 11. 48. a. No. One input of 75 matches with two outputs of 70 and 81. b. Yes. Each input (average score on the final exam) matches with exactly one output (average score on the math placement test). 49. a. (3) $1096.78.P = If the car is financed over three years, the payment is $1096.78. b. (5) $42,580.80.C = c. t = 4. If the total cost is $41,014.08, then the car has been financed over four years, d. Since C(5) = $42,580.80, and C(3) = $39,484.08, the savings would be C(5) – C(3) = $3096.72. 50. a. The couple must make payments for 20 years. ()103,000 20f = b. ()120,000 30.f = It will take the couple 30 years to payoff a $120,000 mortgage at 7.5%. c.   3 40,000 120,000 30ff  d. If 40,000A= then () ( ) 40,000 5 fA f == . 8 CHAPTER 1 Functions, Graphs, and Models; Linear Functions Copyright ©2017 Pearson Education, Inc. e.    3 40,000 120,000 30 34 0 , 0 0 0 3 5 1 5 The expressions are not equal.ff f     51. a. Approximately 22 million b. ()1930 11f =. Approximately 11 million women were in the work force in 1930. c. D:  1930,1940,1950,1960,1970, 1980,1990,2000,2005,2010,2015 d. Increasing. As the year increases, the number of women in the work force also increases. 52. a. When 2020 t= , the ratio is approximately 3 to 1. b. (2005) 4f =. For year 2005, the projected ratio of the working-age population to the elderly is 4 to 1. c. The domain is the set of all possible inputs. In this example, the domain consists of all the years, t, represented in the figure. Specifically, the domain is  1995,2000,2005,2010,2015, 2020,2025,2030 . d. As the years, t, increase, the projected ratio of the working-age population to the elderly decreases. Notice that the bars in the figure grow smaller as the time increases. 53. a. ()1890 26.1f = 2015 29.2f b. ()1940 21.5g = 2013 26.6g c. If 1980x= , then () 24.7 fx = . The median age at first marriage for men in 1980 was 24.7 years. d.  2015 29.2 22.8 1960ff . Therefore, the median age at first marriage for men increased between 1960 and 2015. 54. a. (2030) 39,244f b. The population of females under the age of 18 is projected to be 39,244,000 in the year 2030. c. The function is increasing. The number of females under the age of 18 is projected to keep increasing. 55. a. In 2020, 5.7 million U.S. citizens age 65 and older are expected to have Alzheimer’s disease. b. 2030 7.7f . In 2030, 7.7 million U.S. citizens age 65 and older are expected to have Alzheimer’s disease. c. 2040. 2040 11f. d. The function is increasing. Since 2000, the number of U.S citizen’s 65 and older that are expected to have Alzheimer’s disease in going up. 56. a. Yes. Each year, t, corresponds with exactly one number of U.S. farms, N. b. (1940) 6.3(millions).f = (1940)f represents the number of U.S. farms in millions in the year 1940. c. If ( ) 2.4 ft= , then 1990t= . d. (2012) 2.1f implies that in 2012, there were 2.1 million farms in the U.S. CHAPTER 1 Section 1.1 9 Copyright ©2017 Pearson Education, Inc. 57. a. ()1995 56.0f = . In 1995, the birth rate for U.S. girls ages 15 to 19 was 56.0 per 1000 girls. b. 2005. ()2005 40.5f = . c. The birth rate appears to be at its maximum (59.9) in the year 1990. d. In the years 2005 to 2009, the birth rate appears to have increased until 2007, then decreased. 58. a. ()1990 3.4f = . In 1990 there were 3.4 workers for each retiree. b. 2030. ()2030 2f =. c. As the years increase, the number of workers available to support retirees decreases. Therefore, funding for social security into the future is problematic. Workers will need to pay a larger portion of their salaries to fund payments to retirees. 59. a. () ()200 32 200 6400R == . The revenue generated from selling 200 golf hats is $6400. b. ()()2500 32 2500 $80,000R == 60. a. () ()200 4000 12 200 6400C =+ = . The cost of producing 200 golf hats is $6400. b. () ()2500 4000 12 2500 $34,000C =+ = 61. a. () ()1000 0.857 1000 19.35 857 19.35 876.35f =+ =+ = The monthly charge for using 1000 kilowatt hours is $876.35. b. () ()1500 0.857 1500 19.35 1285.50 19.35 1304.85f =+ =+ = The monthly charge for using 1500 kilowatt hours is $1304.85. 62. a. ()2500 450(500) 0.1(500) 2000 225,000 25,000 2000 198,000P =−− =− − = The profit generated from the production and sale of 500 iPod players is $198,000. b. () ()24000 450(4000) 0.1(4000) 2000 1,800,000 1,600,000 2000 198,000 4000 $198,000P P=−− =−− = = 63. a. () () ()2100 32 100 0.1 100 1000 3200 1000 1000 1200P =− − =−− = The daily profit from the production and sale of 100 Blue Chief bicycles is $1200. b. () () ()2160 32 160 0.1 160 1000 5120 2560 1000 1560P =− − =−−= The daily profit from the production and sale of 160 Blue Chief bicycles is $1560. 64. a. () () ()216 9 6 1 1 6 1 69 61 6 86h =+ − =+ − = The height of the ball after one second is 86 feet. 10 CHAPTER 1 Functions, Graphs, and Models; Linear Functions Copyright ©2017 Pearson Education, Inc. b. () () ()236 9 6 3 1 6 3 6 288 144 150h =+ − =+ − = After three seconds the ball is 150 feet high. c. () () ()2Test 2. 26 9 6 2 1 6 2 6 192 64 134t h= =+ − =+ − = () () ()2Test 4. 46 9 6 4 1 6 4 6 384 256 134t h= =+ − =+ − = () () ()2Test 5. 56 9 6 5 1 6 5 6 480 400 86t h= =+ − =+ − = Since h(1) = 86, and h(2) = 134, and h(3) = 150, and h(4) = 134, and h(5) = 86, it appears that the ball stops climbing after 3 seconds and begins to fall. One might conclude that the ball reaches its maximum height at 3 seconds since at 1 and 5 seconds, and again at 2 and 4 seconds, the respective heights are the same. 65. a. 0.3 0.7 0 0.7 0.3 0.7 0.3 0.7 0.7 3 7n n n n+= =− −= =− Therefore the domain of ( ) Rn is all real numbers except 3 7− or 33,,77⎛⎞ ⎛ ⎞−∞ − − ∞⎜⎟ ⎜ ⎟⎝⎠ ⎝ ⎠U . 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, since each value of s produces exactly one value of Kc. b. Any input into the function must not create a negative number under the radical. Therefore, the radicand, 4 1 s+, must be greater than or equal to zero. Isolating s yields: 41 0 41 1 0 1 41 1 4s s s s+≥ +−≥ − ≥− ≥− Therefore, the domain defined by the equation is all real numbers greater than or equal to 1 4− or, in interval notation, 1,4⎡ ⎞−∞ ⎟ ⎢⎣⎠. c. Since s represents wind speed in the given function, and wind speed cannot be less than zero, the domain of the function is restricted based on the physical context of the problem. Even though the domain implied by the function is 1,4⎡ ⎞−∞ ⎟ ⎢⎣⎠, the actual domain in the given physical context is [)0,∞. 67. a. Since p is a percentage, 0 100 p≤≤ . However in the given function, the denominator, 100 p−, cannot equal zero. Therefore, 100 p≠ . The domain is 0 100 p≤< or, in interval notation, [ ) 0, 100 . CHAPTER 1 Section 1.1 11 Copyright ©2017 Pearson Education, Inc. b. ()() ()()237,000 6060 355,500100 60 237,000 9090 2,133,000100 90C C==− ==− 68. a. Any input into the function must not create a negative number under the square root. Therefore, 2 1 0 p+≥ . Isolating p yields 21 0 21 1 2p p p+≥ ≥−≥− Since the denominator cannot equal zero, 1 2p≠− . Therefore the domain of qis 1,2⎛⎞−∞⎜⎟⎝⎠. b. In the context of the problem, p represents the price of a product. Since the price can not be negative, 0 p≥. The domain is [)0,∞. Also, since q represents the quantity of the product demanded by consumers, 0 q≥. The range is ( ] 0,100 . 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) 11,664V V=− =−== =− =− == b. First, since x represents a side length in the diagram, x must be greater than zero. Second, to satisfy postal restrictions, the length (longest side) plus the girth must be less than or equal to 108 inches. Therefore, Length + Girth 108 Length 4 108 4 108 Length 108 Length 4 Length274x x x x≤ +≤ ≤− −≤≤− Since x is greatest if the longest side is smallest, let the length equal zero to find the largest value for x. 0274 27x x≤−≤ Therefore the conditions on x are 02 7x<≤ . If 27x= , the length would be zero and the package would not exist. Therefore, in the context of the question, 02 7x<< and the corresponding domain for the function ( ) Vx is ()0,27 . c. x Volume 10 6800 12 8640 14 10192 16 11264 18 11664 20 11200 22 9680 The maximum volume occurs when 18x=. Therefore the dimensions that maximize the volume of the box are 18 inches by 18 inches by 72 inches, a total of 108 inches. 70. a. () () ()204 . 9 0 9 8 0 2 2 The initial height of the bullet is 2 meters.S =− + + = 12 CHAPTER 1 Functions, Graphs, and Models; Linear Functions Copyright ©2017 Pearson Education, Inc. b. t Height 9 487.1 9.5 490.78 10 492 10.5 490.78 11 487.1 () ()()9 487.1 10 492 11 487.1SSS= = = c. The bullet seems to reach a maximum height at 10 seconds and then begins to fall. See the table in part b) for further verification, using 9.5 seconds and 10.5 seconds. CHAPTER 1 Section 1.2 13 Copyright ©2017 Pearson Education, Inc. Section 1.2 Skills Check 1. a. x 3yx= (),xy –3 –27 (–3, –27) –2 –8 (–2, –8) –1 –1 (–1, –1) 0 0 (0, 0) 1 1 (1, 1) 2 8 (2, 8) 3 27 (3, 27) b. c. Your hand-drawn graph in part a) by plotting points from the table should match the calculator-drawn graph in part b). 2. a. x 221yx=+ (),xy –3 19 (–3, 19) –2 9 (–2, 9) –1 3 (–1, 3) 0 1 (0, 1) 1 3 (1, 3) 2 9 (2, 9) 3 19 (3, 19) b. c. Your hand-drawn graph in part a) by plotting points from the table should match the calculator-drawn graph in part b). 3. 21 0 1 2 741 2 5x y−− −−− 14 CHAPTER 1 Functions, Graphs, and Models; Linear Functions Copyright ©2017 Pearson Education, Inc. 4. 21 012 97531x y−− −−−−− 5. 42 0 2 4 82 0 2 8x y−− 6. 21 0 1 2 12 3 0 3 12x y−− 7. 01 2 3 4 111 undefined 122x y−− 8. 54 3 2 1 514 undefined 222x y−− − − − −− 9. 25 yx=− , yes there is a turning point in this window CHAPTER 1 Section 1.2 15 Copyright ©2017 Pearson Education, Inc. 10. 24yx=− , yes there is a turning point in this window 11. 323 yx x=− , yes there are 2 turning points in this window 12. 3234 yx x=− + , yes there are 2 turning points in this window 13. ()29/ 1yx=+ , yes there is a turning point in this window 14. ()240 / 4yx=+ , yes there is a turning point in this window 15. a. 20 yx=+ b. 20yx=+ View b) is better. 16. a. 331 3 yx x=−+ 16 CHAPTER 1 Functions, Graphs, and Models; Linear Functions Copyright ©2017 Pearson Education, Inc. b. 331 3 yx x=−+ View b) is better. 17. a. () 20.04 0.1 300xyx−=+ b. () 20.04 0.1 300xyx−=+ View b) is better. 18. a. 220 20 yx x=− + − b. 220 20 yx x=− + − View b) is better. 19. When 8 or 8, 114 xx y=− = = . When 0, 50xy== . Letting y vary from 5 to 100− gives one view. 250 yx=+ The turning point is at (0,50) and is a minimum point. CHAPTER 1 Section 1.2 17 Copyright ©2017 Pearson Education, Inc. 20. When 60, 30 xy=− = . When 0, 30 xy== . When 30, 870 xy=− =− . Letting y vary from 1000 to 0 − gives one view. 260 30 yx x=+ + The turning point is at ( 30, 870) −− and is a minimum point. 21. When 10, 250 xy=− =− . When 10, 850xy== . When 0, 0 xy== . Letting y vary from 200 to 300− gives one view. 3234 5 yx x x=+ − The turning points are at ( 5,175) − , a maximum, and at (3, 81) − , a minimum. 22. When 28, 0 xy== . When 25, 27xy== − . When 31, 27 xy== . Letting y vary from 30 to 30 − gives one view. There is no turning point. 23. Note that answers for the window may vary. 24. 234 120 yx x=− + − Note that answers for the window may vary. 25. t () 5 . 2 1 0 . 5St t=− 12 51.9 16 72.7 28 135.1 43 213.1 18 CHAPTER 1 Functions, Graphs, and Models; Linear Functions Copyright ©2017 Pearson Education, Inc. 26. q 2() 3 5 8fq q q  –8 240 –5 108 24 1616 43 5340 27. 28. 29. a. b. c. Yes. Yes. 30. a. b. c. Yes. Yes. 31. a. ()() ()220 20 5 20 400 100 300f =− =− = b. 20x= implies 20 years after 2000. Therefore the answer to part a) yields the millions of dollars earned in 2020. 32. a. () () ()210 100 10 5 10 10,000 50 9950f =− =− = b. In 2010, 10 x=. Therefore, 9950 thousands of units or 9,950,000 units are produced in 2010. CHAPTER 1 Section 1.2 19 Copyright ©2017 Pearson Education, Inc. Section 1.2 Exercises 33. 34. 35. a. b. Considering the table, S = 148 feet when x is 1 or when x is 3. The height is the same for two different times because the height of the ball increases, reaches a maximum height, and then decreases. c. From the table in part b), it appears the maximum height is 164 feet, occurring 2 seconds into the flight of the ball. 36. a. b. When x = 10, () 600,000 20,000 10 600,000 200,000 $400,000V=−=− = 37. 38. 39. a. 0.79 20.86 0.79(70) 20.86 55.3 20.86 76.16Px   The model predicts that there will be 76.16 million women in the workforce in 2020. 20 CHAPTER 1 Functions, Graphs, and Models; Linear Functions Copyright ©2017 Pearson Education, Inc. b. 80; 2030 – 1950 = 80 0.79 20.86 0.79(80) 20.86 63.2 20.86 84.06Px    The model predicts that there will be 84.06 million (or 84,060,000) women in the workforce in 2030. 40. a. Year 2000 For 2005, 2005 2000 5 For 2016, 2016 2000 16x x x   b. For x = 10. 4320.002(10) 0.051(10) 0.371(10) 1.055(10) 4.234 8.684y     The unemployment rate in the U.S. for 2010 was 8.684%. c. For 2015, x = 15. Therefore, 4320.002(15) 0.051(15) 0.371(15) 1.055(15) 4.234 7.459y     The unemployment rate in the U.S. for 2015 was 7.459%. 41. a. Year 2005 For 2010, 2010 2005 5 For 2016, 2016 2005 11t tt  b. (10)Pf= represents the value of P in 2015 2005 10 2015 .  (10) 1.84 10 60.89 79.29f  79.29 represents the percent of households with Internet access in 2015. c. min max2005 2005 0 2020 2005 15x x  42. a. Year 2000 For 2012, 2012 2000 12 For 2016, 2016 2000 16 For 2020, 2020 2000 20t t t t    b. (18)Sf represents the value of S in 2018 2000 18 2018 . 32(18) 2.4807(18) 52.25(18) 528.68(18) 5192.6 12,246.9f   Approximately 12,247 would be the Federal Tax per Capita in 2018. . c. min max2005 2000 0 2020 2000 20x x  43. a. b. Use the Trace feature, and when x = 130, P = 10,984,200. 44. a. b. Year 1950 For 0, Year 0 1950 1950 For 110, Year 110 1950 2060x xx      CHAPTER 1 Section 1.2 21 Copyright ©2017 Pearson Education, Inc. c. Use the Trace feature, and when x = 88, y = 20.6 years. 45. a. b. 130.7(13) 699.7 2398.8y  In 2023, the balance of federal direct student loans will be 2398.8 billion dollars. c. 130.7(19) 699.7 3183y  In 2029, the balance of federal direct student loans will be 3183 billion dollars. 46. 47. 48. a. 32( ) 2.11 93.0 1370 394Bx x x x b. The tax burden increased. Reading the graph from left to right, as x increases B(x) also increases 49. a. Year 2000 For 2000, 2000 2000 0 For 2030, 2030 2000 30x xx  b. For x = 0. 0.665(0) 23.4 23.4y  For x = 30. 0.665(30) 23.4 43.35y c. 50. 20.0152 0.9514 21.5818yx x=− + a. Since the base year is 1975, 1975-2010 correspond to values of x between 0 and 35. b. Since percentages are between 0 and 100, y must correspond to values between 0 and 100. 22 CHAPTER 1 Functions, Graphs, and Models; Linear Functions Copyright ©2017 Pearson Education, Inc. c. d. e. 2013 corresponds to 2013 1975 38x=−= . When x = 38, y = 7.377. Thus in 2013, approximately 7.38% of high school seniors will have used cocaine. 51. a. 299.9 million, or 299,900,000 b. Years after 2000 Population (millions) 0 275.3 10 299.9 20 324.9 30 351.1 40 377.4 50 403.7 60 432 c. 52. a. b. 53. a. b. CHAPTER 1 Section 1.2 23 Copyright ©2017 Pearson Education, Inc. 54. a. b. c. Yes, the function is a good visual fit to the data. 55. a. In 2026 the crude oil production will be 2.26 billion barrels. b. c. 56. a. b. c. For x = 47. 26.20(47) 104(47) 1110 19,693.8y   In the year 2017, the GDP will be 19,693.8 billion dollars. 24 CHAPTER 1 Functions, Graphs, and Models; Linear Functions Copyright ©2017 Pearson Education, Inc. Section 1.3 Skills Check 1. Recall that linear functions must be in the form ()fxa x b=+ . a. Not linear. The equation has a 2nd degree (squared) term. b. Linear. c. Not linear. The x-term is in the denominator of a fraction. 2. No. A vertical line is not a function. 3. 21 2166 1 2 1 28 4 24 2yymxx− −− −== = = −−− 4. 21 21 4( 1 0 ) 88 14 0 undefinedyymxx−=− −−=− = = Zero in the denominator creates an undefined expression. 5. The given line passes through ()2,0− and ()0,4 . Therefore the slope is ()21 2140 4202 2yymxx− −== = =−− − 6. Since the line is horizontal, the slope of the line is zero. 0m=. 7. a. x-intercept: Let y = 0 and solve for x. 53 ( 0 ) 1 5 51 5 3x x x−= == y-intercept: Let x = 0 and solve for y. 5(0) 3 15 31 5 5y yy−= −= =− x-intercept: (3, 0), y-intercept: (0, –5) b. 8. a. x-intercept: Let y = 0 and solve for x. 5(0) 17 17x x+= = y-intercept: Let x = 0 and solve for y. 05 1 7 51 7 17 5 3.4y y y y+= = = = x-intercept: (17, 0), y-intercept: (0, 3.4) b. CHAPTER 1 Section 1.3 25 Copyright ©2017 Pearson Education, Inc. 9. a. x-intercept: Let y = 0 and solve for x. 3(0) 9 6 096 06 96 669 9 6 31.52x x x xx x x x=− =− += −+ = === y-intercept: Let x = 0 and solve for y. 39 6 ( 0 ) 39 3yy y=−= = x-intercept: (1.5, 0), y-intercept: (0, 3) b. 10. a. x-intercept: Let y = 0 and solve for x. 09 0x x= = y-intercept: Let x = 0 and solve for y. 9(0) 0y y= = x-intercept: (0, 0), y-intercept: (0, 0). Note that the origin, (0, 0), is both an x- and y-intercept. To graph, use the slope, m = 9 = 9 1, or find another point from the equation, like (1,9) or ( 1, 9) −− . b. 11. Horizontal lines have a slope of zero. Vertical lines have an undefined slope. 12. Since the slope is undefined, the line is vertical. 13. a. Positive. The graph is rising. b. Undefined. The line is vertical. 14. a. Negative. The graph is falling. b. Zero. The line is horizontal. 15. a. 4, 8mb== b.