Solution Manual for Basic Technical Mathematics, 11th Edition

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SOLUTIONSMANUALMATTHEWG.HUDELSONBASICTECHNICALMATHEMATICSANDBASICTECHNICALMATHEMATICS WITHCALCULUSELEVENTHEDITIONAllyn J. WashingtonDutchess Community CollegeRichard S. EvansCorning Community College

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Solutions Manual forBasic Technical Mathematics andBasic Technical Mathematics with Calculus, 11th EditionChapter 1Basic Algebraic Operations .....................................................................................1Chapter 2Geometry..............................................................................................................104Chapter 3Functions and Graphs ..........................................................................................171Chapter 4The Trigonometric Functions ..............................................................................259Chapter 5Systems of Linear Equations; Determinants........................................................346Chapter 6Factoring and Fractions........................................................................................490Chapter 7Quadratic Equations.............................................................................................581Chapter 8Trigonometric Functions of Any Angle...............................................................666Chapter 9Vectors and Oblique Triangles ............................................................................723Chapter 10Graphs of the Trigonometric Functions...............................................................828Chapter 11Exponents and Radicals .......................................................................................919Chapter 12Complex Numbers .............................................................................................1001Chapter 13Exponential and Logarithmic Functions............................................................1090Chapter 14Additional Types of Equations and Systems of Equations................................1183Chapter 15Equations of Higher Degree...............................................................................1290Chapter 16Matrices; Systems of Linear Equations .............................................................1356Chapter 17Inequalities.........................................................................................................1477Chapter 18Variation ............................................................................................................1598Chapter 19Sequences and the Binomial Theorem...............................................................1634Chapter 20Additional Topics in Trigonometry ...................................................................1696

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Chapter 21Plane Analytic Geometry...................................................................................1812Chapter 22Introduction to Statistics ....................................................................................2052Chapter 23The Derivative ...................................................................................................2126Chapter 24Applications of the Derivative ...........................................................................2310Chapter 25Integration ..........................................................................................................2487Chapter 26Applications of Integration ................................................................................2572Chapter 27Differentiation of Transcendental Functions .....................................................2701Chapter 28Methods of Integration.......................................................................................2839Chapter 29Partial Derivatives and Double Integrals ...........................................................2991Chapter 30Expansion of Functions in Series.......................................................................3058Chapter 31Differential Equations........................................................................................3181

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1Chapter 1Basic Algebraic Operations1.1 Numbers1.The numbers –7 and 12 are integers. They are also rational numbers since they can be written as71−and121.2.The absolute value of –6 is 6, and the absolute value of –7 is 7. We write these as66−=and77−=.3.64−< −; –6 is to the left of –4.–7–6 –5 –4 –3–2 –1014.The reciprocal of32is12213 / 233=×=.5.3 is an integer, rational31, and real.4−is imaginary.6.73is irrational (because7is an irrational number) and real.6−is an integer, rational16−, and real.7.6π−is irrational (becauseπis an irrational number) and real.18is rational and real.8.6−−is imaginary.2332.33100−−=is rational and real.9.33=33−=22ππ−=10.0.8570.857−=22=191944−=

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2Chapter 1Basic Algebraic Operations11.68<; 6 is to the left of 8.345678910 1112.75>; 7 is to the right of 5.345678910 1113.3.1416;π <(3.1415926)π…is to the left of 3.1416.(π)(3.1416)3.1415923.141614.40−<; –4 is to the left of 0.–6 –5–4 –3 –2–101215.43−< − −; –4 is to the left of3− −,( )()333− −= −= −.–6 –5–4 –3 –2–101216.21.42;−> −(()21.414.1.414),2−= −…= −… −is to the right of –1.42.–1.44–1.43–1.42–1.41–1.4017.232;0.666334−> −−= −…is to the right of5340.7−= −.–0.8–0.7–0.6–0.5–0.418.0.60.2−<; –0.6 is to the left of 0.2.–0.6 –0.5 –0.4 –0.3 –0.2 –0.100.1 0.2

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Section 1.1Numbers319.The reciprocal of 3 is13.The reciprocal of413is4343−−= −.The reciprocalof1isybby by=.20.The reciprocal of113is331/ 31−−= −= −.The reciprocal of1140.25is441/ 41===.The reciprocal of 2xis12x.21.Find 2.5,1232.4;0.75;31.732...54−= −−= −=125−34−32.5–4–3–2–10123422.Find721.4141232.333...;0.707; 223.146.28;6.4732219π…−= −−= −= −=×… ==.37−22−2π12319–3–2–10123456723.An absolute value is not always positive,00=which is not positive.24.Since2172.17100−= −, it is rational.25.The reciprocal of the reciprocal of any positive or negative number is the number itself.The reciprocal ofnis1n; the reciprocal of11is11/1nnnn=⋅=.26.Any repeating decimal is rational, so2.72is rational. It turns out that302.7211=.27.It is true that any nonterminating, nonrepeating decimal is an irrational number.

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4Chapter 1Basic Algebraic Operations28.No,baba−=−, as shown below.If0,a>thenaa=.Ifba>and0a>, thenbb=.If ba>then0ba−>, thenbaba−=−.Therefore,bababa−=−=−.The two sides of the expression are equivalent, one side is not less than the other.29.List these numbers from smallest to largest:1, 9,3.14,52.236,88,33,3.1π−==−=− −= −−.–3.13− −-15π8−9–4–3–2–10123456789So, from smallest to largest, they are3.1,3 ,1,5,,8 , 9π−− −−−.30.List these numbers from smallest to largest:10.20,103.16...,66,4, 0.25,3.14...5π=−= −− −= −−−=.–6–410−150.25π−–6–5–4–3–2–101234567So, from smallest to largest, they are16 ,4,10,,0.25,5π− −−−−.31.Ifaandbare positive integers andba>, then(a)ba−is a positive integer.(b)ab−is a negative integer.(c)baba−+, the numerator and denominator are both positive, but the numerator is less than the denominator, so theanswer is a positive rational number than is less than 1.32.Ifaandbare positive integers, then(a)a+bis a positive integer(b)/abis a positive rational number(c)ab×is a positive integer33.(a)Is the absolute value of a positive or a negative integer always an integer?xx=, so the absolute value of a positive integer is an integer.-xx=, so the absolute value of a negative integer is an integer.(b)Is the reciprocal of a positive or negative integer always a rational number?Ifxis a positive or negative integer, then the reciprocal ofxis1x. Since both 1 andxare integers, the reciprocalis a rational number.

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Section 1.1Numbers534.(a)Is the absolute value of a positive or negative rational number rational?xx=, so ifxis a positive or negative rational number, the absolute value of it is also a rational number.(b)Is the reciprocal of a positive or negative rational number a rational number?A rational number is a number that can be expressed as a fraction where both the numerator and denominator areintegers and the denominator is not zero. So a rational numberintegerintegerabhas a reciprocal of1integerintegerintegerintegerbaab=, which is also a rational number if integerais not zero.35.(a)If0x>, thenxis a positive number located to the right of zero on the number line.x–4–3–2–101234(b)If4x< −, thenxis a negative number located to the left of –4 on the number line.x–6–5–4–3–2–101236.(a)If1x<, then11x−<<.x–4–3–2–101234(b)2x>, then2 or2xx< −>.xx–4–3–2–10123437.If11, thenxx>is a positive number less than 1. Or101x<<.1x–4–3–2–10123438.If0x<, thenxis a positive number greater than zero.x–4–3–2–10123439.1abjab+=+−is a real number when1−is eliminated, which is whenb= 0. Soabj+is a real number for allreal values ofaandb= 0.

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6Chapter 1Basic Algebraic Operations40.The variables arewandt.The constants arec, 0.1, and 1.41.12111TCCC=+. FindTC, where10.0040 FC=and20.0010 FC=.1110.00400.0010TC=+11(0.0040)1(0.0010)0.00400.0010TC+=×0.00400.00100.00000400.00400.00100.0050TC×==+0.00080 FTC=42.100100VV=200200VV−=200100VV−>43.bits1000 byteskilobytesbytes1 kilobyteaNn=××1000bitsNan=44.xyLlength of base in mthe shortened length in centimetres.100length of base in cm100 , all dimensions in cm100xyxyLxLxy===+==−45.Yes,20 C30 C−°> −°because30 C−°is found to the left of20 C−°on the number line.30−℃20−℃100−℃℃10℃46.For4 A,12IR<>Ω.

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Section 1.2Fundamental Operations of Algebra71.2 Fundamental Operations of Algebra1.()()162216416420−× −=− −=+=2.()( )()1852335686146−+− −=+− −=+=−3.()1251124224822( 1)62−−−+=+= −+ −= −−−−4.7642is undefined000×==×, not indeterminate.5.()58583+ −=−= −6.()474711−+ −= −−= −7.396−+=or alternatively()()399366−+= +−= +=8.18213−= −or alternatively1821(2118)(3)3−= −−= −= −9.()191619163−− −= −+= −10.()8108102−− −= −+=11.()74(74)28−= −×= −12.( )9 327−= −13.()75(75)35−−= +×=14.933−= −15.6(2010)6(10)6020333−−−−===−−−16.28282847(56)7( 1)7−−−=== −−−−−17.()()()2 458540−−= −−=18.()()()34616272−−−−== −

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8Chapter 1Basic Algebraic Operations19.()()2 2710251010101−÷=−÷= −÷= −20.6464646482 48242(4)8−−−−====−−−−−−21.162( 4)8( 4)32÷−=−= −22.205( 4)4( 4)16−÷−= −−=23.9210989817−−−= −− −= −−= −24.()()()7757020−÷−=÷ −=25.17710 is undefined770−=−26.(77)(2)0(2)0 is indeterminate(77)( 1)0( 1)0−==−−−27.()83481220−−=+=28.208420218−+÷= −+= −29.()826124124162−−+=+ −=+=−30.|2 |2122−== −−−31.()()1083(1050)10( 8)( 3)( 40)80( 3)( 40)240( 40)6−−÷−=−−÷ −= −−÷ −=÷ −= −32.7575211( 2)22− −−===−−33.()242449(49)1236243( 5)2−−=+×= −+=+ −−34.184|6 |18462662831311−− −−−−−=−= −−= −−= −−−−

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Section 1.2Fundamental Operations of Algebra935.()()()141473 687322 23211473(2)27767766−−−−−= −−−−−−= −−−−= −− −−= −+−= −36.()673|9 |(73)293212914−−−+−−= +×+−−=+−=37.3 |92( 3) |3 |96 |11093 |3 |9991−−−−+=−−−=−= −= −38.()201240( 15)240600360is undefined989898980−−−−+===− −−39.()()6 77 6=demonstrates the commutative law of multiplication.40.6886+=+demonstrates the commutative law of addition.41.()( )( )6 316 36 1+=+demonstrates the distributive law.42.()4 5(45)ππ×=×demonstrates the associative law of multiplication.43.()()359359++=++demonstrates the associative law of addition.44.()( )()8 328 38 2−=−demonstrates the distributive law.45.()5395(39)××=××demonstrates the associative law of multiplication.46.()3677(36)××=××demonstrates the commutative law of multiplication.47.()abab−+ −= −−, which is expression (d).48.()babaab− −=+=+, which is expression (a).49.()babaab−− −= −+=−, which is expression (b).

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10Chapter 1Basic Algebraic Operations50.()ababba−− −= −+=−, which is expression (c).51.Since| 5( 2) | | 52 | | 7 |7− −=+==and|5( 2) | |52 | |3 |3−− −= −+= −=,| 5( 2) | |5( 2) |− −> −− −.52.Since|3|7 || |37 | |10 |10− − −= −−= −=and||3 |7 | | 37 | |4 |4−−=−= −=,|3|7 ||||3 |7 |−− −>−−.53.(a)The sign of a product of an even number of negative numbers is positive.()Example :3618−−=(b)The sign of a product of an odd number of negative numbers is negative.Example:()()54240−−−= −54.Subtraction is not commutative becausexyyx−≠−. Example:752 does not equal 572−=−= −55.Yes, from the definition in Section 1.1, the absolute value of a positive number is the number itself, and the absolutevalue of a negative number is the corresponding positive number. So for values ofxwhere0x>(positive) or0x=(neutral) thenxx=.Example : 44=.The claim that absolute values of negative numbersxx= −is also true.Example:()ifis6, then666.x−−= − −=56.The incorrect answer was achieved by subtracting before multiplying or dividing which violates the order of operations.2462318239327−÷×≠÷×=×=The correct value is:24623243324915−÷×=−×=−=57.(a)1xy−=is true for values ofxandythat are negative reciprocals of each other or1yx= −, providing that thenumberxin the denominator is not zero. So if12x=, then112y= −and()112112xy−= −−=.(b)1xyxy−=−is true for all values ofxandy, providing thatxy≠to prevent division by zero.58.(a)xyxy+=+is true for values where bothxandyhave the same sign or either are zero:xyxy+=+, when0 and0xy≥≥or when0 and0xy≤≤Example:63639 and63639Also,6( 3)9963639+=+=+=+=−+ −= −=−+ −=+=xyxy+=+is not true however, when x and y have opposite signsxyxy+≠+, whenx0 and0 ; or0 and0yxy><<>.

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Section 1.2Fundamental Operations of Algebra11Example:2161515,2162162715−+= −=−+=+=≠4( 5)11,454591+ −= −=+ −=+=≠(b)In order forxyxy−=+it is necessary that they have opposite signs or either to be zero.Symbolically,xyxy−=+when0 and0xy≥≤; or when0 and0xy≤≥.Example:6( 3)639 and63639− −=+=+ −=+=Example:117181811711718−−= −=−+ −=+=xyxy−=+is not true, however, whenxandyhave the same signs.xyxy−≠+, whenx0 and0; or0 and0yxy>><<.Example:2161515,2162715−==+=≠59.The total change in the price of the stock is0.680.420.06( 0.11)0.020.29−+++ −+= −.60.The difference in altitude is86( 1396)1396861310 m−− −=−=61.The change in the meter energy readingEwould be:()2.1 kW h1.5 kW 3.0 h2.1 kW h4.5 kW h2.4 kW hchangeusedgeneratedchangechangechangeEEEEEE=−=⋅−=⋅−⋅= −⋅62.Assuming that this batting average is for the current season only which is just starting, the number of hits is zero andthe total number of at-bats is also zero giving us anumber of hits0batting averageatbats0==−which is indeterminate, not0.000.63.The average temperature for the week is:7( 3)231( 4)( 6)C77323146C714C2.0C7avgavgavgTTT−+ −++++ −+ −=°−−+++−−=°−=°= −°

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12Chapter 1Basic Algebraic Operations64.The vertical distance from the flare gun is()( )()()()705162535040035040050 mdddd=+ −=+ −=−= −The flare is 50 m below the flare gun.65.The sum of the voltages is()()6V2V8V5V3V6V2V8V5V3V10VsumsumsumVVV=+ −++ −+=−+−+=66.(a)The change in the current for the first interval is the second reading – the first reading22212 lb/in7 lb/in9 lb/inChange= −−= −.(b)The change in the current for the middle intervals is the third reading – the second reading()2222229 lb/in2 lb/in9 lb/in2 lb/in7 lb/inChange= −− −= −+= −.(c)The change in the current for the last interval is the last reading – the third reading()2222236 lb/in9 lb/in6 lb/in9 lb/in3 lb/inChange= −− −= −+=.67.The oil drilled by the first well is100 m200 m300 m+=which equals the depth drilled by the second well200 m100 m300 m+=.100 m200 m200 m100 m+=+demonstrates the commutative law of addition.68.The first tank leaks()L127 h84 Lh=.The second tank leaks()L712hh84L.=127712×=×demonstrates the commutative law of multiplication.69.The total time spent browsing these websites is the total time spent browsing the first site on each day + the total timespent browsing the second site on each dayminutesminutes7 days257 days15dayday175 min105 min280 minORminutes7 days(2515)dayminutes7 days40day280 mintttttt=×+×=+==×+=×=which illustrates the distributive law.

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