Skip to content
Textbook Guides

Solution Manual for Basic Technical Mathematics, 11th Edition

Solution Manual for Basic Technical Mathematics, 11th Edition simplifies even the toughest textbook questions with step-by-step solutions and easy explanations.

Preview 16 of 3387 pages

See the first 16 pages exactly as they appear in the document.

  1. Page 1
    Read page 1 as text

    S OLUTIONS M ANUAL M ATTHEW G. H UDELSON B ASIC T ECHNICAL M ATHEMATICS AND B ASIC T ECHNICAL M ATHEMATICS WITH C ALCULUS E LEVENTH E DITION Allyn J. Washington Dutchess Community College Richard S. Evans Corning Community College

  2. Page 2
  3. Page 3
    Read page 3 as text

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

  4. Page 4
    Read page 4 as text

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

  5. Page 5
    Read page 5 as text

    1 Chapter 1 Basic Algebraic Operations 1.1 Numbers 1. The numbers –7 and 12 are integers. They are also rational numbers since they can be written as 7 1 − and 12 1 . 2. The absolute value of –6 is 6, and the absolute value of –7 is 7. We write these as 6 6 − = and 7 7 − = . 3. 6 4 − < − ; –6 is to the left of –4. –7 –6 –5 –4 –3 –2 –1 0 1 4. The reciprocal of 3 2 is 1 2 2 1 3 / 2 3 3 = × = . 5. 3 is an integer, rational 3 1       , and real. 4 − is imaginary. 6. 7 3 is irrational (because 7 is an irrational number) and real. 6 − is an integer, rational 1 6       − , and real. 7. 6 π − is irrational (because π is an irrational number) and real. 1 8 is rational and real. 8. 6 − − is imaginary. 233 2.33 100 − − = is rational and real. 9. 3 3 = 3 3 − = 2 2 π π − = 10. 0.857 0.857 − = 2 2 = 19 19 4 4 − =

  6. Page 6
    Read page 6 as text

    2 Chapter 1 Basic Algebraic Operations 11. 6 8 < ; 6 is to the left of 8. 3 4 5 6 7 8 9 10 11 12. 7 5 > ; 7 is to the right of 5. 3 4 5 6 7 8 9 10 11 13. 3.1416; π < (3.1415926 ) π … is to the left of 3.1416. ( π ) (3.1416) 3.141592 3.1416 14. 4 0 − < ; –4 is to the left of 0. –6 –5 –4 –3 –2 –1 0 1 2 15. 4 3 − < − − ; –4 is to the left of 3 − − , ( ) ( ) 3 3 3 − − = − = − . –6 –5 –4 –3 –2 –1 0 1 2 16. 2 1.42; − > − ( ( ) 2 1.414 . 1.414 ), 2 − = − … = − … − is to the right of –1.42. –1.44 –1.43 –1.42 –1.41 –1.40 17. 2 3 2 ; 0.666 3 3 4 − > − − = − … is to the right of 5 3 4 0.7 − = − . –0.8 –0.7 –0.6 –0.5 –0.4 18. 0.6 0.2 − < ; –0.6 is to the left of 0.2. –0.6 –0.5 –0.4 –0.3 –0.2 –0.1 0 0.1 0.2

  7. Page 7
    Read page 7 as text

    Section 1.1 Numbers 3 19. The reciprocal of 3 is 1 3 . The reciprocal of 4 1 3 is 4 3 4 3 − − = − . The reciprocal of 1 is y b b y b y = . 20. The reciprocal of 1 1 3 is 3 3 1/ 3 1 − − = − = − . The reciprocal of 1 1 4 0.25 is 4 4 1/ 4 1 = = = . The reciprocal of 2 x is 1 2 x . 21. Find 2.5, 12 3 2.4; 0.75; 3 1.732... 5 4 − = − − = − = 12 5 − 3 4 − 3 2.5 –4 –3 –2 –1 0 1 2 3 4 22. Find 7 2 1.414 123 2.333...; 0.707; 2 2 3.14 6.28; 6.47 3 2 2 19 π … − = − − = − = − = × … = = . 3 7 − 2 2 − 2 π 123 19 –3 –2 –1 0 1 2 3 4 5 6 7 23. An absolute value is not always positive, 0 0 = which is not positive. 24. Since 217 2.17 100 − = − , it is rational. 25. The reciprocal of the reciprocal of any positive or negative number is the number itself. The reciprocal of n is 1 n ; the reciprocal of 1 1 is 1 1/ 1 n n n n = ⋅ = . 26. Any repeating decimal is rational, so 2.72 is rational. It turns out that 30 2.72 11 = . 27. It is true that any nonterminating, nonrepeating decimal is an irrational number.

  8. Page 8
    Read page 8 as text

    4 Chapter 1 Basic Algebraic Operations 28. No, b a b a − = − , as shown below. If 0, a > then a a = . If b a > and 0 a > , then b b = . If b a > then 0 b a − > , then b a b a − = − . Therefore, b a b a b a − = − = − . 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, 5 2.236, 8 8, 3 3, 3.1 π − = = − = − − = − − . –3.1 3 − − -1 5 π 8 − 9 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 9 So, from smallest to largest, they are 3.1, 3 , 1, 5, , 8 , 9 π − − − − − . 30. List these numbers from smallest to largest: 1 0.20, 10 3.16..., 6 6, 4, 0.25, 3.14... 5 π = − = − − − = − − − = . –6 –4 10 − 1 5 0.25 π − –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 So, from smallest to largest, they are 1 6 , 4, 10, , 0.25, 5 π − − − − − . 31. If a and b are positive integers and b a > , then (a) b a − is a positive integer. (b) a b − is a negative integer. (c) b a b a − + , the numerator and denominator are both positive, but the numerator is less than the denominator, so the answer is a positive rational number than is less than 1. 32. If a and b are positive integers, then (a) a + b is a positive integer (b) / a b is a positive rational number (c) a b × is a positive integer 33. (a) Is the absolute value of a positive or a negative integer always an integer? x x = , so the absolute value of a positive integer is an integer. - x x = , 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? If x is a positive or negative integer, then the reciprocal of x is 1 x . Since both 1 and x are integers, the reciprocal is a rational number.

  9. Page 9
    Read page 9 as text

    Section 1.1 Numbers 5 34. (a) Is the absolute value of a positive or negative rational number rational? x x = , so if x is 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 are integers and the denominator is not zero. So a rational number integer integer a b has a reciprocal of 1 integer integer integer integer b a a b = , which is also a rational number if integer a is not zero. 35. (a) If 0 x > , then x is a positive number located to the right of zero on the number line. x –4 –3 –2 –1 0 1 2 3 4 (b) If 4 x < − , then x is a negative number located to the left of –4 on the number line. x –6 –5 –4 –3 –2 –1 0 1 2 36. (a) If 1 x < , then 1 1 x − < < . x –4 –3 –2 –1 0 1 2 3 4 (b) 2 x > , then 2 or 2 x x < − > . x x –4 –3 –2 –1 0 1 2 3 4 37. If 1 1, then x x > is a positive number less than 1. Or 1 0 1 x < < . 1 x –4 –3 –2 –1 0 1 2 3 4 38. If 0 x < , then x is a positive number greater than zero. x –4 –3 –2 –1 0 1 2 3 4 39. 1 a bj a b + = + − is a real number when 1 − is eliminated, which is when b = 0. So a bj + is a real number for all real values of a and b = 0.

  10. Page 10
    Read page 10 as text

    6 Chapter 1 Basic Algebraic Operations 40. The variables are w and t . The constants are c , 0.1, and 1. 41. 1 2 1 1 1 T C C C = + . Find T C , where 1 0.0040 F C = and 2 0.0010 F C = . 1 1 1 0.0040 0.0010 T C = + 1 1(0.0040) 1(0.0010) 0.0040 0.0010 T C + = × 0.0040 0.0010 0.0000040 0.0040 0.0010 0.0050 T C × = = + 0.00080 F T C = 42. 100 100 V V = 200 200 V V − = 200 100 V V − > 43. bits 1000 bytes kilobytes bytes 1 kilobyte a N n = × × 1000 bits N an = 44. x y L length of base in m the shortened length in centimetres. 100 length of base in cm 100 , all dimensions in cm 100 x y x y L x L x y = = = + = = − 45. Yes, 20 C 30 C − ° > − ° because 30 C − ° is found to the left of 20 C − ° on the number line. 30 − ℃ 20 − ℃ 10 0 − ℃ ℃ 10 ℃ 46. For 4 A, 12 I R < > Ω .

  11. Page 11
    Read page 11 as text

    Section 1.2 Fundamental Operations of Algebra 7 1.2 Fundamental Operations of Algebra 1. ( ) ( ) 16 2 2 16 4 16 4 20 − × − = − − = + = 2. ( )( ) ( ) 18 5 2 3 3 5 6 8 6 14 6 − + − − = + − − = + = − 3. ( ) 12 5 1 12 4 2 2 4 8 2 2( 1) 6 2 − − − + = + = − + − = − − − − 4. 7 6 42 is undefined 0 0 0 × = = × , not indeterminate. 5. ( ) 5 8 5 8 3 + − = − = − 6. ( ) 4 7 4 7 11 − + − = − − = − 7. 3 9 6 − + = or alternatively ( ) ( ) 3 9 9 3 6 6 − + = + − = + = 8. 18 21 3 − = − or alternatively 18 21 (21 18) (3) 3 − = − − = − = − 9. ( ) 19 16 19 16 3 − − − = − + = − 10. ( ) 8 10 8 10 2 − − − = − + = 11. ( ) 7 4 (7 4) 28 − = − × = − 12. ( ) 9 3 27 − = − 13. ( ) 7 5 (7 5) 35 − − = + × = 14. 9 3 3 − = − 15. 6(20 10) 6(10) 60 20 3 3 3 − − − − = = = − − − 16. 28 28 28 4 7(5 6) 7( 1) 7 − − − = = = − − − − − 17. ( )( ) ( ) 2 4 5 8 5 40 − − = − − = 18. ( )( ) ( ) 3 4 6 1 6 2 72 − − − − = = −

  12. Page 12
    Read page 12 as text

    8 Chapter 1 Basic Algebraic Operations 19. ( ) ( ) 2 2 7 10 2 5 10 10 10 1 − ÷ = − ÷ = − ÷ = − 20. 64 64 64 64 8 2 4 8 2 4 2(4) 8 − − − − = = = = − − − − − − 21. 16 2( 4) 8( 4) 32 ÷ − = − = − 22. 20 5( 4) 4( 4) 16 − ÷ − = − − = 23. 9 2 10 9 8 9 8 17 − − − = − − − = − − = − 24. ( ) ( ) ( ) 7 7 5 7 0 2 0 − ÷ − = ÷ − = 25. 17 7 10 is undefined 7 7 0 − = − 26. (7 7)(2) 0(2) 0 is indeterminate (7 7)( 1) 0( 1) 0 − = = − − − 27. ( ) 8 3 4 8 12 20 − − = + = 28. 20 8 4 20 2 18 − + ÷ = − + = − 29. ( ) 8 2 6 12 4 12 4 16 2 − − + = + − = + = − 30. | 2 | 2 1 2 2 − = = − − − 31. ( ) ( ) 10 8 3 (10 50) 10( 8)( 3) ( 40) 80( 3) ( 40) 240 ( 40) 6 − − ÷ − = − − ÷ − = − − ÷ − = ÷ − = − 32. 7 5 7 5 2 1 1( 2) 2 2 − − − = = = − − 33. ( ) 24 24 4 9 (4 9) 12 36 24 3 ( 5) 2 − − = + × = − + = + − − 34. 18 4 | 6 | 18 4 6 2 6 6 2 8 3 1 3 1 1 − − − − − − − = − = − − = − − = − − − −

  13. Page 13
    Read page 13 as text

    Section 1.2 Fundamental Operations of Algebra 9 35. ( ) ( ) ( ) 14 14 7 3 6 8 7 3 2 2 2 3 2 1 14 7 3(2) 2 7 7 6 7 7 6 6 − − − − − = − − − − − − = − − − − = − − − − = − + − = − 36. ( ) 6 7 3 | 9 | (7 3) 2 9 3 21 2 9 14 − − − + − − = + × + − − = + − = 37. 3 | 9 2( 3) | 3 | 9 6 | 1 10 9 3 | 3 | 9 9 9 1 − − − − + = − − − = − = − = − 38. ( ) 20 12 40( 15) 240 600 360 is undefined 98 98 98 98 0 − − − − + = = = − − − 39. ( ) ( ) 6 7 7 6 = demonstrates the commutative law of multiplication. 40. 6 8 8 6 + = + demonstrates the commutative law of addition. 41. ( ) ( ) ( ) 6 3 1 6 3 6 1 + = + demonstrates the distributive law. 42. ( ) 4 5 (4 5) π π × = × demonstrates the associative law of multiplication. 43. ( ) ( ) 3 5 9 3 5 9 + + = + + demonstrates the associative law of addition. 44. ( ) ( ) ( ) 8 3 2 8 3 8 2 − = − demonstrates the distributive law. 45. ( ) 5 3 9 5 (3 9) × × = × × demonstrates the associative law of multiplication. 46. ( ) 3 6 7 7 (3 6) × × = × × demonstrates the commutative law of multiplication. 47. ( ) a b a b − + − = − − , which is expression (d). 48. ( ) b a b a a b − − = + = + , which is expression (a). 49. ( ) b a b a a b − − − = − + = − , which is expression (b).

  14. Page 14
    Read page 14 as text

    10 Chapter 1 Basic Algebraic Operations 50. ( ) a b a b b a − − − = − + = − , which is expression (c). 51. Since | 5 ( 2) | | 5 2 | | 7 | 7 − − = + = = and | 5 ( 2) | | 5 2 | | 3 | 3 − − − = − + = − = , | 5 ( 2) | | 5 ( 2) | − − > − − − . 52. Since | 3 | 7 || | 3 7 | | 10 | 10 − − − = − − = − = and || 3 | 7 | | 3 7 | | 4 | 4 − − = − = − = , | 3 | 7 || || 3 | 7 | − − − > − − . 53. (a) The sign of a product of an even number of negative numbers is positive. ( ) Example : 3 6 18 − − = (b) The sign of a product of an odd number of negative numbers is negative. Example: ( )( ) 5 4 2 40 − − − = − 54. Subtraction is not commutative because x y y x − ≠ − . Example: 7 5 2 does not equal 5 7 2 − = − = − 55. Yes, from the definition in Section 1.1, the absolute value of a positive number is the number itself, and the absolute value of a negative number is the corresponding positive number. So for values of x where 0 x > (positive) or 0 x = (neutral) then x x = . Example : 4 4 = . The claim that absolute values of negative numbers x x = − is also true. Example: ( ) if is 6, then 6 6 6. x − − = − − = 56. The incorrect answer was achieved by subtracting before multiplying or dividing which violates the order of operations. 24 6 2 3 18 2 3 9 3 27 − ÷ × ≠ ÷ × = × = The correct value is: 24 6 2 3 24 3 3 24 9 15 − ÷ × = − × = − = 57. (a) 1 xy − = is true for values of x and y that are negative reciprocals of each other or 1 y x = − , providing that the number x in the denominator is not zero. So if 12 x = , then 1 12 y = − and ( ) 1 12 1 12 xy   − = − − =     . (b) 1 x y x y − = − is true for all values of x and y , providing that x y ≠ to prevent division by zero. 58. (a) x y x y + = + is true for values where both x and y have the same sign or either are zero: x y x y + = + , when 0 and 0 x y ≥ ≥ or when 0 and 0 x y ≤ ≤ Example: 6 3 6 3 9 and 6 3 6 3 9 Also, 6 ( 3) 9 9 6 3 6 3 9 + = + = + = + = − + − = − = − + − = + = x y x y + = + is not true however, when x and y have opposite signs x y x y + ≠ + , when x 0 and 0 ; or 0 and 0 y x y > < < > .

  15. Page 15
    Read page 15 as text

    Section 1.2 Fundamental Operations of Algebra 11 Example: 21 6 15 15, 21 6 21 6 27 15 − + = − = − + = + = ≠ 4 ( 5) 1 1, 4 5 4 5 9 1 + − = − = + − = + = ≠ (b) In order for x y x y − = + it is necessary that they have opposite signs or either to be zero. Symbolically, x y x y − = + when 0 and 0 x y ≥ ≤ ; or when 0 and 0 x y ≤ ≥ . Example: 6 ( 3) 6 3 9 and 6 3 6 3 9 − − = + = + − = + = Example: 11 7 18 18 11 7 11 7 18 − − = − = − + − = + = x y x y − = + is not true, however, when x and y have the same signs. x y x y − ≠ + , when x 0 and 0; or 0 and 0 y x y > > < < . Example: 21 6 15 15, 21 6 27 15 − = = + = ≠ 59. The total change in the price of the stock is 0.68 0.42 0.06 ( 0.11) 0.02 0.29 − + + + − + = − . 60. The difference in altitude is 86 ( 1396) 1396 86 1310 m − − − = − = 61. The change in the meter energy reading E would be: ( ) 2.1 kW h 1.5 kW 3.0 h 2.1 kW h 4.5 kW h 2.4 kW h change used generated change change change E E E E E E = − = ⋅ − = ⋅ − ⋅ = − ⋅ 62. Assuming that this batting average is for the current season only which is just starting, the number of hits is zero and the total number of at-bats is also zero giving us a number of hits 0 batting average at bats 0 = = − which is indeterminate, not 0.000. 63. The average temperature for the week is: 7 ( 3) 2 3 1 ( 4) ( 6) C 7 7 3 2 3 1 4 6 C 7 14 C 2.0 C 7 avg avg avg T T T − + − + + + + − + − = ° − − + + + − − = ° − = ° = − °

  16. Page 16
    Read page 16 as text

    12 Chapter 1 Basic Algebraic Operations 64. The vertical distance from the flare gun is ( )( ) ( )( ) ( ) 70 5 16 25 350 400 350 400 50 m d d d d = + − = + − = − = − The flare is 50 m below the flare gun. 65. The sum of the voltages is ( ) ( ) 6V 2V 8V 5V 3V 6V 2V 8V 5V 3V 10V sum sum sum V V V = + − + + − + = − + − + = 66. (a) The change in the current for the first interval is the second reading – the first reading 2 2 2 1 2 lb/in 7 lb/in 9 lb/in Change = − − = − . (b) The change in the current for the middle intervals is the third reading – the second reading ( ) 2 2 2 2 2 2 9 lb/in 2 lb/in 9 lb/in 2 lb/in 7 lb/in Change = − − − = − + = − . (c) The change in the current for the last interval is the last reading – the third reading ( ) 2 2 2 2 2 3 6 lb/in 9 lb/in 6 lb/in 9 lb/in 3 lb/in Change = − − − = − + = . 67. The oil drilled by the first well is 100 m 200 m 300 m + = which equals the depth drilled by the second well 200 m 100 m 300 m + = . 100 m 200 m 200 m 100 m + = + demonstrates the commutative law of addition. 68. The first tank leaks ( ) L 12 7 h 84 L h = .The second tank leaks ( ) L 7 12h h 84L. = 12 7 7 12 × = × 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 time spent browsing the second site on each day minutes minutes 7 days 25 7 days 15 day day 175 min 105 min 280 min OR minutes 7 days (25 15) day minutes 7 days 40 day 280 min t t t t t t = × + × = + = = × + = × = which illustrates the distributive law.

3371 more pages in the full document

Unlock it once and it stays in your library, ready to chat with or turn into flashcards.

Unlock the full document

More Mathematics study materials

See all

Study this document with CramX

Once it is in your library, every tool can work from it.