Solution Manual for College Algebra, 10th Edition

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’SSOLUTIONSMANUALTIMBRITTJackson State Community CollegeCOLLEGEALGEBRATENTHEDITIONMichael SullivanChicago State University

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iiiTable of ContentsChapter RReviewR.1 Real Numbers.............................................................................................................................1R.2 Algebra Essentials......................................................................................................................5R.3 Geometry Essentials.................................................................................................................11R.4 Polynomials..............................................................................................................................16R.5 Factoring Polynomials .............................................................................................................23R.6 Synthetic Division....................................................................................................................28R.7 Rational Expressions................................................................................................................29R.8nth Roots; Rational Exponents.................................................................................................39Chapter 1Equations and Inequalities1.1 Linear Equations.......................................................................................................................481.2 Quadratic Equations .................................................................................................................641.3 Complex Numbers; Quadratic Equations in the Complex Number System ............................821.4 Radical Equations; Equations Quadratic in Form; Factorable Equations ................................881.5 Solving Inequalities ................................................................................................................1081.6 Equations and Inequalities Involving Absolute Value ...........................................................1191.7 Problem Solving: Interest, Mixture, Uniform Motion, and Constant Rate Job Applications 128Chapter Review .............................................................................................................................135Chapter Test...................................................................................................................................144Chapter Projects.............................................................................................................................146Chapter 2Graphs2.1 The Distance and Midpoint Formulas ....................................................................................1472.2 Graphs of Equations in Two Variables; Intercepts; Symmetry ..............................................1592.3 Lines .......................................................................................................................................1712.4 Circles .....................................................................................................................................1872.5 Variation .................................................................................................................................199Chapter Review .............................................................................................................................204Chapter Test...................................................................................................................................210Cumulative Review .......................................................................................................................212Chapter Project ..............................................................................................................................214Chapter 3Functions and Their Graphs3.1 Functions ................................................................................................................................2153.2 The Graph of a Function.........................................................................................................2303.3 Properties of Functions...........................................................................................................2393.4 Library of Functions; Piecewise-defined Functions ...............................................................2553.5 Graphing Techniques: Transformations .................................................................................2673.6 Mathematical Models: Building Functions ............................................................................284Chapter Review .............................................................................................................................290Chapter Test...................................................................................................................................297Cumulative Review .......................................................................................................................300Chapter Projects.............................................................................................................................304

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ivChapter 4Linear and Quadratic Functions4.1 Properties of Linear Functions and Linear Models ................................................................3064.2 Building Linear Functions from Data.....................................................................................3174.3 Quadratic Functions and Their Properties ..............................................................................3224.4 Build Quadratic Models from Verbal Descriptions and from Data .......................................3434.5 Inequalities Involving Quadratic Functions ...........................................................................350Chapter Review .............................................................................................................................369Chapter Test...................................................................................................................................377Cumulative Review .......................................................................................................................379Chapter Projects.............................................................................................................................382Chapter 5Polynomial and Rational Functions5.1 Polynomial Functions and Models .........................................................................................3855.2 Properties of Rational Functions ............................................................................................4075.3 The Graph of a Rational Function ..........................................................................................4165.4 Polynomial and Rational Inequalities.....................................................................................4695.5 The Real Zeros of a Polynomial Function..............................................................................4905.6 Complex Zeros; Fundamental Theorem of Algebra...............................................................521Chapter Review .............................................................................................................................530Chapter Test...................................................................................................................................544Cumulative Review .......................................................................................................................549Chapter Projects.............................................................................................................................553Chapter 6Exponential and Logarithmic Functions6.1 Composite Functions ..............................................................................................................5556.2 One-to-One Functions; Inverse Functions..............................................................................5726.3 Exponential Functions ............................................................................................................5916.4 Logarithmic Functions............................................................................................................6116.5 Properties of Logarithms ........................................................................................................6316.6 Logarithmic and Exponential Equations ................................................................................6406.7 Financial Models ....................................................................................................................6596.8 Exponential Growth and Decay Models; Newton’s Law; Logistic Growthand Decay Models................................................................................................................6676.9 Building Exponential, Logarithmic, and Logistic Models from Data ....................................676Chapter Review .............................................................................................................................681Chapter Test...................................................................................................................................693Cumulative Review .......................................................................................................................697Chapter Projects.............................................................................................................................701

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vChapter 7Analytic Geometry7.2 The Parabola ...........................................................................................................................7037.3 The Ellipse ..............................................................................................................................7177.4 The Hyperbola ........................................................................................................................733Chapter Review .............................................................................................................................752Chapter Test...................................................................................................................................756Cumulative Review .......................................................................................................................759Chapter Projects.............................................................................................................................761Chapter 8Systems of Equations and Inequalities8.1 Systems of Linear Equations: Substitution and Elimination ..................................................7638.2 Systems of Linear Equations: Matrices ..................................................................................7848.3 Systems of Linear Equations: Determinants ..........................................................................8088.4 Matrix Algebra .......................................................................................................................8228.5 Partial Fraction Decomposition ..............................................................................................8408.6 Systems of Nonlinear Equations.............................................................................................8578.7 Systems of Inequalities...........................................................................................................8858.8 Linear Programming...............................................................................................................899Chapter Review .............................................................................................................................911Chapter Test...................................................................................................................................926Cumulative Review .......................................................................................................................935Chapter Projects.............................................................................................................................938Chapter 9Sequences; Induction; the Binomial Theorem9.1 Sequences ...............................................................................................................................9419.2 Arithmetic Sequences .............................................................................................................9509.3 Geometric Sequences; Geometric Series................................................................................9579.4 Mathematical Induction ..........................................................................................................9689.5 The Binomial Theorem...........................................................................................................976Chapter Review .............................................................................................................................981Chapter Test...................................................................................................................................986Cumulative Review .......................................................................................................................989Chapter Projects.............................................................................................................................991Chapter 10Counting and Probability10.1 Counting ...............................................................................................................................99310.2 Permutations and Combinations...........................................................................................99510.3 Probability ............................................................................................................................999Chapter Review ...........................................................................................................................1005Chapter Test.................................................................................................................................1006Cumulative Review .....................................................................................................................1008Chapter Projects...........................................................................................................................1010AppendixGraphing UtilitiesSection 1 The Viewing Rectangle ..............................................................................................1013Section 2 Using a Graphing Utility to Graph Equations ............................................................1014Section 3 Using a Graphing Utility to Locate Intercepts and Check for Symmetry ..................1018Section 5 Square Screens............................................................................................................1020

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1Chapter RReviewSection R.11.rational2.45 63430331+⋅−=+−=3.Distributive4.()536x+=5.a6.b7.True8.False; The Zero-Product Property states that if aproduct equals 0, then at least one of the factorsmust equal 0.9.False; 6 is the Greatest Common Factor of 12and 18. The Least Common Multiple is thesmallest value that both numbers will divideevenly. The LCM for 12 and 18 is 36.10.True11.{}{}{}1, 3, 4,5, 92, 4, 6, 7,81, 2,3, 4, 5, 6, 7,8, 9AB∪=∪=12.{}{}{}1, 3, 4,5, 91, 3, 4, 61, 3, 4, 5, 6, 9AC∪=∪=13.{}{}{}1, 3, 4,5, 92, 4, 6, 7,84AB∩=∩=14.{}{}{}1, 3, 4,5, 91, 3, 4, 61, 3, 4AC∩=∩=15.{}{}(){}{}{}{}()1, 3, 4,5, 92, 4, 6, 7,81,3, 4, 61, 2,3, 4,5, 6, 7,8,91,3, 4, 61, 3, 4, 6ABC∪∩=∪∩=∩=16.{}{}(){}{ }{}{}()1, 3, 4,5, 92, 4, 6, 7,81,3, 4, 641, 3, 4, 61,3, 4, 6ABC∩∪=∩∪=∪=17.{}0, 2, 6, 7, 8A=18.{}0, 2, 5, 7, 8, 9C=19.{}{}{ }{}1, 3, 4, 5, 92, 4, 6, 7, 840, 1, 2, 3, 5, 6, 7, 8, 9AB∩=∩==20.{}{}{}{}2, 4, 6, 7, 81, 3, 4, 61, 2, 3, 4, 6, 7, 80, 5, 9BC∪=∪==21.{}{}{}0, 2, 6, 7, 80, 1, 3, 5, 90, 1, 2, 3, 5, 6, 7, 8, 9AB∪=∪=22.{}{}{}0, 1, 3, 5, 90, 2, 5, 7, 8, 90, 5, 9BC∩=∩=23.a.{}2,5b.{}6, 2,5−c.{}16,,1.333...1.3, 2,52−−= −d.{}πe.{}16,,1.333...1.3,, 2,52π−−= −24.a.{}1b.{}0,1c.{}5 , 2.060606...2.06,1.25, 0,13−=d.{}5

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Chapter R:Review2e.{}5 , 2.060606...2.06,1.25, 0,1,53−=25.a.{}1b.{}0,1c.{}1 110,1,,,23 4d.Nonee.{}1 110,1,,,23 426.a.Noneb.{}1−c.{}1.3,1.2,1.1,1−−−−d.Nonee.{}1.3,1.2,1.1,1−−−−27.a.Noneb.Nonec.Noned.{}12,,21,2ππ++e.{}12,,21,2ππ++28.a.Noneb.Nonec.{}110.32+d.{}2,2π−+e.{}12,2,10.32π−++29.a.18.953b.18.95230.a.25.861b.25.86131.a.28.653b.28.65332.a.99.052b.99.05233.a.0.063b.0.06234.a.0.054b.0.05335.a.9.999b.9.99836.a.1.001b.1.00037.a.0.429b.0.42838.a.0.556b.0.55539.a.34.733b.34.73340.a.16.200b.16.20041.325+=42.5 210⋅=43.23 4x+=⋅44.322y+=+45.312y=+46.24 6x=⋅47.26x−=48.26y−=49.62x=50.26x=51.942527−+=+=52.643235−+=+=53.64 36126−+⋅= −+=54.84 2880−⋅=−=55.458981+−=−=56.834541−−=−=57.1121134333++==58.14132222−−==

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Section R.1:Real Numbers359.()( )63 52326152161711−⋅+⋅−=−+⋅=−= −60.()()[][]2834232836328183210320323⋅−⋅+−=⋅−⋅−=⋅−−=⋅ −−= −−= −61.()()2358 2122161416112111⋅−+⋅−=⋅ −+−= −+−=−=62.()()14 3221122211211−⋅−+=−−+=−= −63.()[][][ ]1062 283210645210252107210144−−⋅+−⋅=−−+⋅=−+⋅=−⋅=−= −64.()()[]25 46342206118618612−⋅−⋅−=−−⋅ −= −− −= −+= −65.()()11532122−==66.()()11549333+==67.48126532+==−68.2421532−−== −−69.3 103 2 535215 3 7⋅⋅⋅==⋅⋅25⋅⋅53⋅277=⋅70.535 359 103 3 5 2⋅⋅==⋅⋅⋅3⋅335⋅⋅162=⋅71.6102 3 5 22325275 5 3 9⋅⋅⋅⋅⋅==⋅⋅⋅5⋅25⋅53⋅⋅4459=⋅72.21 1003 7 4 25325325 3⋅⋅⋅⋅==⋅7 425⋅⋅⋅253⋅28=73.3215823452020++==74.4183113266++==75.59255479653030++==76.81516135151921818++==77.511031318123636++==78.28640461594545++==79.1733532163018909045−−== −= −80.3294514214242−−==81.3298120156060−−==82.631215335147070−−== −83.5185275 9 35918119 2 111127⋅⋅⋅=⋅==⋅⋅39⋅15222 11=⋅⋅84.5215355 7 5572127 3 2235⋅⋅⋅=⋅==⋅⋅57⋅2563 2=⋅⋅

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Chapter R:Review485.1373737101251010101010+⋅+=+===86.24122 22222235635 3 235 3 2315252102102123515151515154 34 345 35 35⋅⋅+⋅=+=+=+⋅⋅⋅⋅+=⋅+=+==⋅⋅===⋅⋅87.33233636232 4814848428123123158888⋅+=⋅+=+=⋅++=+==88.513513 513 513 621623 223 225151422222⋅⋅⋅−=⋅−=−=−⋅⋅−=−===89.()64624xx+=+90.()4 2184xx−=−91.()244xxxx−=−92.()243412xxxx+=+93.31312 3222242422 222 32312222xxxxx⋅−=⋅−⋅=−⋅⋅=−=−⋅94.21213 23333363633 23 231233 22xxxxx⋅+=⋅+⋅=+⋅⋅=+=+⋅95.()()222442868xxxxxxx++=+++=++96.()()22515565xxxxxxx++=+++=++97.()()2221222xxxxxxx−+=+−−=−−98.()()22414434xxxxxxx−+=+−−=−−99.()()228228161016xxxxxxx−−=−−+=−+100.()()224224868xxxxxxx−−=−−+=−+101.()()23232355xxxxxxx+=⋅+⋅=+⋅=⋅=102.23 421214+⋅=+=since multiplication comes before addition in theorder of operations for real numbers.()2345 420+⋅=⋅=since operations inside parentheses come beforemultiplication in the order of operations for realnumbers.103.()()2 3 42 1224⋅==() ()()( )2 32 46848⋅⋅⋅==104.4371257+==+, but434 53 220626132.6251010105⋅+⋅++=====105.Subtraction is not commutative; forexample:231132−= −≠=−.106.Subtraction is not associative; forexample:()()52124521−−=≠=−−.107.Division is not commutative; for example:2332≠.108.Division is not associative; forexample:()1222623÷÷=÷=, but()122212112÷÷=÷=.109.The Symmetric Property implies that if 2 =x,thenx= 2.

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Section R.2:Algebra Essentials5110.From theprinciple of substitution,if5x=, then()()( )( )222552525530xxxxxxx==+=++=111.There are no real numbers that are both rationaland irrational, since an irrational number, bydefinition, is a number that cannot be expressedas the ratio of two integers; that is, not a rationalnumberEvery real number is either a rational number oran irrational number, since the decimal form of areal number either involves an infinitelyrepeating pattern of digits or an infinite, non-repeating string of digits.112.The sum of an irrational number and a rationalnumber must be irrational. Otherwise, theirrational number would then be the difference oftwo rational numbers, and therefore would haveto be rational.113.Answers will vary.114.Since 1 day = 24 hours, we compute12997541.541624=.Now we only need to consider the decimal partof the answer in terms of a 24 hour day. That is,()()0.54162413≈hours. So it must be 13 hourslater than 12 noon, which makes the time 1 AMCST.115.Answers will vary.Section R.21.variable2.origin3.strict4.base; exponent (or power)5.31.234567810×6.d7.b8.True.9.True10.False; the absolute value of a real number isnonnegative.00=which is not a positivenumber.11.False; a number in scientific notation isexpressed as the product of a number, x,110x≤<or101x−<≤ −, and a power of 10.12.False; to multiply two expressions with the samebase, retain the base andaddthe exponents.13.−2.5−152100.253414.1332−22−1.502315.102>16.56<17.12−> −18.532−< −19.3.14π >20.21.41>21.10.52=22.10.333>23.20.673<24.10.254=

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Chapter R:Review625.0x>26.0z<27.2x<28.5y> −29.1x≤30.2x≥31.Graph on the number line:2x≥ −0−232.Graph on the number line:4x<4033.Graph on the number line:1x> −−1034.Graph on the number line:7x≤0735.(,)(0,1)1011d C Dd==−==36.(,)(0,3)3033d C Ad=−=−−=−=37.(,)(1,3)3122d D Ed==−==38.(,)(0,3)3033d C Ed==−==39.(,)( 3,3)3( 3)66d A Ed=−=− −==40.(,)(1,1)1122d D Bd=−=−−=−=41.222 3264xy+= −+⋅= −+=42.33(2)3633xy+=−+= −+= −43.525(2)(3)230228xy+=−+= −+= −44.22(2)(2)(3)462xxy−+= −−+ −=−= −45.2(2)4242355xxy−−===−−−−46.23112355xyxy−++=== −−−−−47.3(2)2(3)66320022355xyy−+−++====++48.2(2)343237333xy−−−−−=== −49.3(2)11xy+=+ −==50.3(2)55xy−=− −==51.32325xy+=+ −=+=52.32321xy−=− −=−=53.33133xx===54.22122yy−=== −−−55.454(3)5(2)12102222xy−=−−=+==56.323(3)2(2)9455xy+=+−=−==57.454(3)5(2)1210121022xy−=−−=− −=−==58.323 3223 32 29413xy+=+−=⋅+⋅=+=59.21xx−Part (c) must be excluded. The value0x=mustbe excluded from the domain because it causesdivision by 0.

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Section R.2:Algebra Essentials760.21xx+Part (c) must be excluded. The value0x=mustbe excluded from the domain because it causesdivision by 0.61.2(3)(3)9xxxxx=−+−Part (a) ,3x=, must be excluded because itcauses the denominator to be 0.62.29xx+None of the given values are excluded. Thedomain is all real numbers.63.221xx+None of the given values are excluded. Thedomain is all real numbers.64.332(1)(1)1xxxxx=−+−Parts (b) and (d) must be excluded. The values1, and1xx== −must be excluded from thedomain because they cause division by 0.65.223510510(1)(1)xxxxx xxxx+−+−=−+−Parts (b), (c), and (d) must be excluded. Thevalues0,1, and1xxx=== −must be excludedfrom the domain because they cause division by0.66.22329191(1)xxxxxxx x−−+−−+=++Part (c) must be excluded. The value0x=mustbe excluded from the domain because it causesdivision by 0.67.45x−5x=must be exluded because it makes thedenominator equal 0.{}Domain5x x=≠68.64x−+4x= −must be excluded sine it makes thedenominator equal 0.{}Domain4x x=≠ −69.4xx+4x= −must be excluded sine it makes thedenominator equal 0.{}Domain4x x=≠ −70.26xx−−6x=must be excluded sine it makes thedenominator equal 0.{}Domain6x x=≠71.555(32)(3232)(0)0 C999CF=−=−==°72.555(32)(21232)(180)100 C999CF=−=−==°73.555(32)(7732)(45)25 C999CF=−=−==°74.55(32)(432)995 ( 36)920 CCF=−=−−=−= −°75.2(4)(4)(4)16−= −−=76.224(4)16−= −= −77.22114164−==78.22114164−−= −= −79.64642211333393−− +−⋅====80.2323144444−− +⋅===81.()()()121223339−−−−===

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Chapter R:Review882.()()()313132228−−−−===83.22555==84.23666==85.()2444−= −=86.()2333−= −=87.()()2232368864xxx==88.()122211444xxx−−== −−89.()()()42222121422xx yxyx yy−−−=⋅==90.()()333113333yxyxyxyx−−−=⋅==91.232 134114x yxxyx yyxy−−−===92.22 11 231231xyxyxyx yx y−− −−−−===93.3424222334 1232 13113(2)()839898989xy zx y zx y zx y zxyzx yzx zy−−−−−−=−=−== −94.21211344241 11621624()428481212xy zxyzx yx yxyzxyzx y z−−−−−− −− −−−−−====95.2221222122233441643439yxxxxxyyyy−−−−====96.()()3332222223326363255666562161255yxxyxyxxyy−−−−====97.()()12 22241xxyy−=== −−98.()()13133322yxyx−−−−−===99.()()222221415xy+=+ −=+=100.() ()2222214 14x y=−=⋅=101.()()()()2222124xy=⋅ −=−=102.()()()( )2222111xy+=+ −==103.222xx===104.()22xx==105.()()222221415xy+=+ −=+=106.2221213xyxy+=+=+ −=+=107.1122yx−==108.()211xy=−=

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Section R.2:Algebra Essentials9109.If2,x=323223542 23 25 24161210410xxx−+−=⋅−⋅+⋅−=−+−=If1,x=323223542 13 15 1423540xxx−+−=⋅−⋅+⋅−=−+−=110.If1,x=32324324 13 11243128xxx+−+=⋅+⋅−+=+−+=If2,x=32324324 23 22232122244xxx+−+=⋅+⋅−+=+−+=111.4444(666)666381222(222)===112.()333333331(0.1) (20)2 101012101028=⋅⋅=⋅⋅==113.6(8.2)304, 006.671≈114.5(3.7)693.440≈115.3(6.1)0.004−≈116.5(2.2)0.019−≈117.6(2.8)481.890−≈118.6(2.8)481.890−≈ −119.4(8.11)0.000−−≈120.4(8.11)0.000−−≈ −121.2454.24.54210=×122.132.143.21410=×123.20.0131.310−=×124.30.004214.2110−=×125.432,1553.215510=×126.421, 2102.12110=×127.40.0004234.2310−=×128.20.05145.1410−=×129.46.151061,500×=130.39.7109700×=131.31.214100.001214−×=132.49.88100.000988−×=133.81.110110, 000, 000×=134.24.11210411.2×=135.28.1100.081−×=136.16.453100.6453−×=137.Alw=138.()2Plw=+139.Cdπ=140.12Abh=141.234Ax=142.3Px=143.343Vrπ=144.24Srπ=

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Chapter R:Review10145.3Vx=146.26Sx=147.a.If1000,x=4000240002(1000)40002000$6000Cx=+=+=+=The cost of producing 1000 watches is$6000.b.If2000,x=4000240002(2000)40004000$8000Cx=+=+=+=The cost of producing 2000 watches is$8000.148.210801202560325$98+−+−−−=His balance at the end of the month was $98.149.We want the difference betweenxand 4 to be atleast 6 units. Since we don’t care whether thevalue forxis larger or smaller than 4, we takethe absolute value of the difference. We want theinequality to be non-strict since we are dealingwith an ‘at least’ situation. Thus, we have46x−≥150.We want the difference betweenxand 2 to bemore than 5 units. Since we don’t care whetherthe value forxis larger or smaller than 2, wetake the absolute value of the difference. Wewant the inequality to be strict since we aredealing with a ‘more than’ situation. Thus, wehave25x−>151.a.110108110225x−=−=−=≤108 volts is acceptable.b.110104110665x−=−=−=>104 volts isnotacceptable.152.a.220214220668x−=−=−=≤214 volts is acceptable.b.22020922011118x−=−=−=>209 volts isnotacceptable.153.a.32.99930.0010.0010.01x−=−=−=≤A radius of 2.999 centimeters is acceptable.b.32.8930.110.110.01x−=−=−=≤/A radius of 2.89 centimeters isnotacceptable.154.a.98.69798.61.61.61.5x−=−=−=≥97˚F is unhealthy.b.98.610098.61.41.41.5x−=−==<100˚F isnotunhealthy.155.The distance from Earth to the Moon is about8410400, 000, 000×=meters.156.The height of Mt. Everest is about388488.84810=×meters.157.The wavelength of visible light is about75100.0000005−×=meters.158.The diameter of an atom is about101100.0000000001−×=meters.159.The smallest commercial copper wire has adiameter of about40.0005510−=×inches.160.The smallest motor ever made is less than20.05510−=×centimeters wide.

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Section R.3:Geometry Essentials11161.()() ()()511221.86106102.4103.6510186, 000 60 60 24 365××××⋅⋅⋅⋅=1012586.5696105.86569610=×=×There are about125.910×miles in one light-year.162.72593, 000, 0009.310510186, 0001.8610500 seconds8 min. 20 sec.×==××=≈It takes about 8 minutes 20 seconds for a beamof light to reach Earth from the Sun.163.10.333333 ...0.3333=>13is larger by approximately0.0003333 ...164.230.666666 ...0.666=>23is larger by approximately 0.0006666 ...165.No. For any positive numbera, the value2aissmaller and therefore closer to 0.166.We are given that2110x<<. This implies that110x<<. Since103.162x<≈and3.142xπ>≈, the number could be 3.15 or 3.16(which are between 1 and 10 as required). Thenumber could also be 3.14 since numbers such as3.146 which lie betweenπand10wouldequal 3.14 when truncated to two decimal places.167.Answers will vary.168.Answers will vary.5 < 8 is a true statement because 5 is further tothe left than 8 on a real number line.Section R.31.right; hypotenuse2.12Abh=3.2Crπ=4.similar5.c6.b7.True.8.True.22268366410010+=+==9.False; the surface area of a sphere of radiusrisgiven by24Vrπ=.10.True. The lengths of the corresponding sides areequal.11.True. Two corresponding angles are equal.12.False. The sides are not proportional.13.222225,12,5122514416913abcabc===+=+=+==14.222226,8,68366410010abcabc===+=+=+==15.2222210,24,102410057667626abcabc===+=+=+==16.222224,3,43169255abcabc===+=+=+==

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