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

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’SSOLUTIONSMANUALDEANAJ.THORNOCKRICHMONDCOLLEGEALGEBRA INCONTEXTWITHAPPLICATIONS FOR THEMANAGERIAL,LIFE,ANDSOCIALSCIENCESSIXTHEDITIONRonald HarshbargerUniversity of South Carolina — BeaufortLisa S. YoccoEast Georgia State College

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TABLE OF CONTENTSChapter 1Functions, Graphs, and Models; Linear FunctionsAlgebra Toolbox11.1 Functions and Models41.2 Graphs of Functions101.3 Linear Functions191.4 Equations of Lines25Preparing for Calculus35Chapter 1 Skills Check36Chapter 1 Review39Group Activities/Extended Applications41Chapter 2Linear Models, Equations, and InequalitiesAlgebra Toolbox432.1 Algebraic and Graphical Solution of Linear Equations472.2 Fitting Lines to Data Points: Modeling Linear Functions592.3 Systems of Linear Equations in Two Variables662.4 Solutions of Linear Inequalities78Preparing for Calculus87Chapter 2 Skills Check88Chapter 2 Review92Group Activities/Extended Applications96Chapter 3Quadratic, Piecewise-Defined, and Power FunctionsAlgebra Toolbox973.1 Quadratic Functions; Parabolas1013.2 Solving Quadratic Equations1123.3 Power and Root Functions1253.4 Piecewise-Defined Functions and Absolute Value Functions1313.5 Quadratic and Power Models137

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Preparing for Calculus145Chapter 3 Skills Check147Chapter 3 Review153Group Activities/Extended Applications157Chapter 4Additional Topics with FunctionsAlgebra Toolbox1614.1 Transformations of Graphs and Symmetry1634.2 Combining Functions; Composite Functions1734.3 One-to-One and Inverse Functions1824.4 Additional Equations and Inequalities189Preparing for Calculus203Chapter 4 Skills Check204Chapter 4 Review208Group Activities/Extended Applications211Chapter 5Exponential and Logarithmic FunctionsAlgebra Toolbox2155.1 Exponential Functions2185.2 Logarithmic Functions; Properties of Logarithms2255.3 Exponential and Logarithmic Equations2325.4 Exponential and Logarithmic Models2435.5 Exponential Functions and Investing2515.6 Annuities; Loan Repayment2585.7 Logistic and Gompertz Functions263Preparing for Calculus269Chapter 5 Skills Check270Chapter 5 Review272Group Activities/Extended Applications278

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Chapter 6Higher-Degree Polynomial and Rational FunctionsAlgebra Toolbox2796.1 Higher-Degree Polynomial Functions2836.2 Modeling with Cubic and Quartic Functions2926.3 Solution of Polynomial Equations2996.4 Polynomial Equations Continued;307Fundamental Theorem of Algebra6.5 Rational Functions and Rational Equations3166.6 Polynomial and Rational Inequalities328Preparing for Calculus337Chapter 6 Skills Check339Chapter 6 Review346Group Activities/Extended Applications351Chapter 7Systems of Equations and MatricesAlgebra Toolbox3557.1 Systems of Linear Equations in Three Variables3577.2 Matrix Solution of Systems of Linear Equations3717.3 Matrix Operations3857.4 Inverse Matrices; Matrix Equations3927.5 Determinants and Cramer’s Rule4077.6 Systems of Nonlinear Equations419Chapter 7 Skills Check431Chapter 7 Review441Group Activities/Extended Applications447

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Chapter 8Special Topics in Algebra8.1 Systems of Inequalities4498.2 Linear Programming: Graphical Methods4728.3 Sequences and Discrete Functions4858.4 Series4898.5 The Binomial Theorem4968.6 Conic Sections: Circles and Parabolas4998.7 Conic Sections: Ellipses and Hyperbolas510Chapter 8 Skills Check522Chapter 8 Review533Group Activities/Extended Applications538

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Algebra Toolbox1CHAPTER 1Functions, Graphs, and Models;Linear FunctionsToolbox Exercises1.{}{}1,2,3,4,5,6,7,8and,9x xNx∈<Remember thatxN∈means thatxis anatural number.2.Yes3.No.Ais not a subset ofB.Acontainselements 2, 7, and 10, which are not inB.4.No.{}1,2,3,4, . . . ,N=therefore1.2N∉5.Yes. Every integer can be written as afraction with the denominator equal to 1.6.Yes. Irrational numbers are by definitionnumbers that are not rational.7.a.{}3,0,ABπ∩=−b.2155,3,,0,,4,5,7,,832ABπ∩=−−−8.Integers, rational numbers. The integers area subset of the rational numbers. Exercise 5.9.Rational numbers10.Irrational numbers11.()()61622−+ −= −12.()2213221335− −=+=13.()5634563422−− −= −+= −14.()()()()126361418361415614211435−+ −− −−+ −= −+−−= −−−= −−= −15.()()()73242−−=16.()()()18216394839÷ −+ −−= −+=17.354257276−+= −+=−−18.[][][][]()( 8)( 3)5( 2)3( 4)( 5)24101251472−−+−÷−− −=−÷ −+=÷ −= −19.()()300−=20.006=−21.5 is undefined.0Not possible; division by 0 is undefined.

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2CHAPTER 1Functions, Graphs, and Models; Linear Functions22.a.()8668−+=+ −b.()( )( )()9339−=−23.a.()()823823+ −+=+ −+b.()( )()()( ) ()734734−−=−−24.a.()()( )6 1256 126 57230+=+=+b.()xyxy−+= −−25.This is an example of both the commutativeand the associative property.()()()378738Commutative Property738Associative Property++=++=++26.a.10010−+= −b.( )133⋅=27.a.440−+=b.1717⋅=28.3x> −29.33x−≤≤30.3x≤31.(],7−∞32.(]3,733.(), 4−∞34.35.52 implies 25xx>≥≤<36.37.22348The coefficient of3is3.The coefficient of4is4.The constant term is 8.xxxx−−+−−−−38.4343573The coefficient of 5is 5.The coefficient of 7is 7.The constant term is3.xxxx+−−39.()()242 44582028ab−=−−=+=40.()3325275611914522yx−+−= −−+−=+−=41.()()()()()()()()3 20.633 2( 2)40.6 43(0.8)3440.6 42.4380.6 1.6240.9623.04abbc−+−=−−+−=−−+−=−+= −+= −42.()()11 4.56.514.625 sq ft22bh==

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Algebra Toolbox343.()()221616 916 811296 feett===44.()()()()()42432443224321520624125241512206534272011zzzzzzzzzzzzzzzz−+−++−−=+++ −−++ −−=+−+−45.()()()4342444342434232211953511025233511911072329xyx yxyyyyx yyxxyx yyx+−−−−++=+−−+−+ −+=−−−−46.()()()44444pdpdpd+=+=+47.()()()2 372 327614xyxyxy−−= −−−= −+48.()()()888a bca bacabac−+= −−= −−49.()()43244326xyxyxyxyxy−−+=−−−=−50.()()()()()()4 2452484455248245454695xyxyyxyxyxyxyyxyxyxxxyxyyyyxxyy−+−−−−=−+−+−+=−+++ −−+=+−51.()()()()244538853858335xyzxyzxxyzxxyzxxyzxyzxxxyzx−−−=−−+=−+ −+=−52.Subtraction property711771174xxx+=+−=−=53.Addition property410441046xxx−= −−+= −+= −54.Division property520520554xxx= −−== −55.Division property218218229xxx−=−=−−= −56.Multiplication property( )3666 3618yyy===57.Multiplication property()84448432ppp−= −−−= −−=

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4CHAPTER 1Functions, Graphs, and Models; Linear Functions58.59.60.61.62.Skills Check 1.11.Use Table A.a.–5 is anx-value and therefore is aninput of the function( ).fxb.( 5)f−is an output of the function.c.The domain is the set of all inputs.{}9,7,5,6,12,17,20D=−−−The range is the set of all outputs.{}4,5,6,7,9,10R=d.Each inputxof the functionfyieldsexactly one output( ).yfx=2.Use Table B.a.0 is anx-value and therefore is aninput of the function( ).gxb.(7)gis an output of the function.c.The domain is the set of all inputs.{}4,1,0,1,3,7,12D=−−The range is the set of all outputs.{}3,5,7,8,9,10,15R=d.Each inputxof the functiongyieldsexactly one output( ).yg x=3.( 9)5(17)9yfyf=−===4.(4)5(3)8ygyg=−===5.No. In Table A,xis not a function ofy. Ifyis considered the input variable, one inputwill correspond with more than one output.Namely, if9,y=then12x=or17.x=6.Yes. Each inputyproduces exactly oneoutputx.

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1.2Graphs of Functions57.a.(2)1,f= −since2x=in the tablecorresponds with( )1.fx= −b.2(2)103(2)103(4)10122f=−=−=−= −c.(2)3,f= −since (2,3)−is a point on thegraph.8.a.( 1)5,f−=since()1,5−is a point onthe graph.b.( 1)8,f−= −since1x= −in the tablecorresponds with( )8.fx= −c.2( 1)( 1)3( 1)81386f−=−+−+=−+=9.0224xy−−10.4143xy−−11.Recall that( )58.R xx=+a.( 3)5( 3)81587R−=−+= −+= −b.( 1)5( 1)8583R−=−+= −+=c.(2)5(2)810818R=+=+=12.Recall that2( )162.C ss=−a.2(3)162(3)162(9)16182C=−=−=−= −b.2( 2)162( 2)162(4)1688C−=−−=−=−=c.2(1)162(1)162(1)16214C−=−−=−=−=13.Yes. Each input corresponds with one output.{}{}1,0,1,2,3 ;8,1,2,5,7DR=−=−−14.No. Each inputxdoes not match withexactly one outputy. Specifically, if2,x=then3 or4.yy= −=15.No. The graph fails the vertical line test.Each input does not match with exactly oneoutput.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 oneoutput.19.No. If3,x=then5 or7.yy==One value ofx, 3, gives two values ofy.20.Yes. Each inputxyields exactly one outputy.21.a.This is not a function.If4,x=then12y=or8.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.If3,x=then4y=or6.y=23.a.This is not a function.If2,x=then3y=or4.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.If3,x= −then3y=or5.y= −

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6CHAPTER 1Functions, Graphs, and Models; Linear Functions25.The domain is the set of all inputs.{}3,2,1,1,3,4D=−−−The range is the set of all outputs.{}8,4,2,4,6R=−−26.The domain is the set of all inputs.{}6,4,2,0,2,4D=−−−The range is the set of all outputs.{}5,2,0,1,4,6R=−−27.Consideryas a function ofx.Domain is the set of all inputsx.[]10, 8D= −Range is the set of all outputsy.[]12, 2R= −28.Consideryas a function ofx.Domain is the set of all inputsx.[]4, 3D= −Range is the set of all outputsy.[]1, 4R= −29.Consideryas a function ofx.Domain is the set of all inputsx.(),D=−∞ ∞Range is the set of all outputsy.[)4,R= −∞30.Consideryas a function ofx.Domain is the set of all inputsx.(],3D=−∞Range is the set of all outputsy.[)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. If0,x=then222(0)442.yyy+=== ±One input of 0 corresponds with two outputsof –2 and 2. Therefore, the equation is not afunction.34.Yes. Each input forxcorresponds withexactly one output fory.35.932,5CF=+whereFis the Fahrenheittemperature andCis the Celsius temperature.36.2,Crπ=whereCis the circumference andris the radius.37.ConvertFtemperature to Ctemperatureusing the following steps.a.Subtract 32.b.Multiply by 5.c.Divide by 9.38.Represent the functionDusing thefollowing steps.a.SquareE.b.Multiply by 3.c.Subtract 5.Exercises 1.139.No. There are two different outputs for20.Y=40.Yes. Each input corresponds with exactlyone output.41.a.Yes. Each input (year) correspondswith 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 ofm, thenumber of minutes after the power outage.Each value formcorresponds with exactlyone value forT. The graph of the equationpasses the vertical line test.

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1.2Graphs of Functions743.a.Yes. Each input (barcode) correspondswith exactly one output (price).b.No. An input (price) could correspondwith more than one output (barcode).Numerous items can have the same pricebut different barcodes.44. a.Yes. Each input (piano key) correspondswith exactly one output (note). Thedomain is the set of all inputs and thereare 12 keys on the piano, so there are 12elements in the domain of the function.b.Yes. Each input (note) corresponds withexactly one output (piano key). The rangeis the set of all outputs and there are 12keys, so there are 12 elements in therange of the function.45.Each inputx(in years) corresponds withone outputV, the value of the property.46.Yes. Each inputd(depth) corresponds withexactly one outputp(pressure).47.a.Yes. Each input (day) corresponds withexactly one output (weight).b.The domain is the first 14 days of May.{}1,2,3,4,5,6,7,8,9,10,11,12,13,14D=c.The range is the list of weights.{}171,172,173,174,175,176,177,178R=d.Highest weights were on May 1 and 3.e.Lowest weight was on May 14.f.Longest period during which his weightdecreased is 3 days, May 8 to 11.48.a.No. This is not a function.One input of 75 corresponds withtwo 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)1096.78P=If the car is financed over 3 years,the payment is $1096.78.b.(5)42,580.80C=The total cost if financed over 5 years is$42,580.80.c.(4)41,014.08C=If the total cost is $41,014.08, then thecar has been financed over 4 yearsd.(5)42,580.80;C=(3)39,484.08C=The savings would be$42,580.80$39,484.08$3096.72.−=50.a.(103,000)20f=They must make payments for 20 years.b.(120,000)30f=It will take the couple 30 years to pay offa $120,000 mortgage at 7.5%.c.(3 40,000)(120,000)30ff⋅==d.If40,000A=then()(40,000)5.fAf==e.(3 40,000)(120,000)303(40,000)3 515The expressions are not equal.fff⋅==⋅=⋅=51.a.Approximately 79.3 millionb.(2050)91.5f=Approximately 91.5million women are projected to be in theworkforce in 2050.c.{1950,1960,1970,1980,1990,2000,2010,2020,2030,2040,2050}D=d.As the year increases, the number ofwomen in the workforce also increases.52.a.In2020,t=the ratio is about 3 to 1.b.(2005)4f=. In 2005, the projectedratio of the working-age populationto the elderly population is 4 to 1.c.{}1995,2000,2005,2010,2015,2020,2025,2030D=d.As the years increase, the projectedratio of the working-age populationto the elderly population decreases.

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8CHAPTER 1Functions, Graphs, and Models; Linear Functions53.a.(25,000)$14.15(75,000)$42.44ff==b.(50,000)$12.83(100,000)$25.67gg==c.If( )28.29,50,000.fxx==d.Yes54.a.(2030)39, 244f=b.In 2030, the population of females under theage 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 65and older are expected to have Alzheimer’sdisease.b.(2030)7.7.f=In 2030, 7.7 million U.S.citizens age 65 and older are expected tohave Alzheimer’s disease.c.2040;(2040)11f=d.Since 2000, the number of U.S citizens65 and older that are expected to haveAlzheimer’s disease is increasing.56. a.Yes. Each year,t, corresponds withexactly one number of jobs,N.b.(2025)12.9 (million)f=c.In 2025, there will be 12.9 million jobs.d.2035;(2035)11.8f=57.a.(2030)15.8;f=15.8%b.(2040)17.1.f=The projected percent ofthe U.S. population that is foreign-bornin 2040 is 17.1%.c.(2050)18.2.f=This population isprojected to be 18.2% in the year 2050.d.{}2020,2030,2040,2050,2060D=e.The projected population of the U.S.population that is foreign-born isincreasing.58.a.(1990)3.4f=In 1990, there were 3.4 workers for eachretiree.b.2030;(2030)2f=c.As time passes, the number of workersdecreases. Funding for Social Security inthe future is problematic. Workers mayneed to pay a larger portion of salaries tofund benefits to retirees.59. a.()()20032 2006400R==The revenue from the sale of 200 hatsis $6400.b.()()250032 2500$80,000R==60.a.()()200400012 2006400C=+=The cost of producing 200 hats is $6400.b.()()2500400012 2500$34,000C=+=61. a.()()10000.857 100019.3585719.35876.35f=+=+=The charge for 1000 kilowatt hoursis $876.35.b.()()15000.857 150019.351285.5019.351304.85f=+=+=The charge for 1500 kilowatt hoursis $1304.85.62. a.()2500450(500)0.1(500)2000225,00025,0002000198,000P=−−=−−=The profit from the production and saleof 500 iPod players is $198,000.b.()24000450(4000)0.1(4000)20001,800,0001,600,0002000198,000P=−−=−−=The profit from the production and saleof 4000 iPod players is $198,000.

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1.2Graphs of Functions963. a.()()()210032 1000.1 10010003200100010001200P=−−=−−=Daily profit from producing and selling100 Blue Chief bicycles is $1200.b.()()()216032 1600.1 16010005120256010001560P=−−=−−=Daily profit from producing and selling160 Blue Chief bicycles is $1560.64. a.2(1)696(1)16(1)6961686h=+−=+−=Height of ball after one second is 86 feet.b.2(3)696(3)16(3)6288144150h=+−=+−=Height of ball after 3 seconds is 150 feet.c.Test2 :t=2(2)696(2)16(2)619264134h=+−=+−=Test4.t=2(4)696(4)16(4)6384256134h=+−=+−=Test5.t=2(5)696(5)16(5)648040086h=+−=+−=It appears that the ball reaches a highpoint after 3 seconds and begins to fall.At 1 and 5 seconds, and again at 2 and 4seconds, the respective heights are equal.65.a.0.30.700.70.30.70.30.70.737nnnn+== −−== −The domain of( )Rnis all real numbersexcept37−or33,,.77−∞ −−∞b.In the context of the problem,nrepresents the factor for increasing thenumber of questions on a test. Therefore,it makes sense thatnis positive,0.n>66. a.Yes. Each value ofscorresponds withexactly one value ofKc.b.In order for the value ofKcto be real,the radicand, 41,s+must be greater thanor equal to zero.4104114sss+≥≥ −≥ −The domain is all real numbers greaterthan or equal to14−or,1 ,.4−∞c.srepresents wind speed, and cannot beless than zero. The domain is restrictedbased on the context of the problem.The actual domain in context is[)0,∞.67. a.Sincepis a percentage, 0100.p≤≤In the given function, the denominator,100,p−cannot equal zero, so100.p≠The domain is 0100p≤<or[)0, 100 .b.()()()()237,000 6060355,50010060237,000 90902,133,00010090CC==−==−68. a.Since the square root expression is in thedenominator of the function, the radicandmust be positive.2102112ppp+>> −> −The domain ofqis1 ,.2−∞In the context of the problem,prepresents the price of a product andcannot be negative.[)0,D=∞In context,qrepresents the quantity ofthe product demanded by consumers.It cannot be 0 and can be no larger than100.(]0,100R=

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10CHAPTER 1Functions, Graphs, and Models; Linear Functions69. a.()() ()()()()() ()()()2212121084(12)144 10848144 60864018181084(18)324 10872324 3611,664VV=−=−===−=−==b.First,xrepresents a side length so0.x>Second, to satisfy postal restrictions, thelength (longest side) plus the girth mustbe less than or equal to 108 inches.Length + Girth108Length41084108Length108Length4Length274xxxx≤+≤≤−−≤≤−xis greatest if length is minimized,so let Length0=and findx.027and274xx≤−≤The conditions onxare 027.x<≤If27,x=the length would be zero andthe package would not exist. In context,027x<<and the corresponding domainfor the function( )V xis()0,27 .c.xVolume106,8001510,8002011,2002110,5841911,5521811,6641711,560The table shows a maximum volumeof 11,664 cubic inches when18.x=lengthgirth108,+=so length4x+=()length4 18108,+=and length36.=The dimensions that maximize thevolume of the box are 18 inches by18 inches by 36 inches.70. a.2(0)4.9(0)98(0)22S= −++=The initial height of the bullet is 2 feet.b.2(9)4.9(9)98(9)2487.1S= −++=2(0)4.9(10)98(10)2492S= −++=2(11)4.9(11)98(11)2487.1S= −++=c.The bullet appears to reach a maximumheight at 10 seconds, then begins to fall.tHeight9487.19.5490.781049210.5490.7811487.1Skills Check 1.21.a.3321012327810123xyx−−−=−−−b.c.The graph in part (a) is generatedby plotting points from the table andmatches the calculator graph in part (b).

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