Solutions Manual Elementary Linear Algebra 4th edition by Stephen Andrilli & David Hecker

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STUDENT SOLUTIONS MANUAL KEVIN BODDEN

RANDY GALLAHER

LEWIS AND CLARK COMMUNITY COLLEGE

��e6ra &

8 �ri8onometry

MICHAEL SULLIVAN PEARSON

----­

Prentice Hall

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Upper Saddle River, NJ

07458

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Review

Real Numbers ............................................................................................................................... 1 R.l ... . . . .. . . .. .. . . .. . .. . . 3 R.2 Algebra Essentials . ....

R.3 Geometry Essentials ..................................................................................................................... 6 R.4 Polynomials .................................................................................................................................. 8 .. . .. . .. . . . . . . . II R.5 Factoring Polynomials .

R.6 Synthetic Division ...................................................................................................................... 14 R.7 Rational expressions .....................: ............................................................................................. 15 R.8 nth Roots; Rational Exponents ................................................................................................... 20 Chapter Review ..................................................................................................................................... 24 Chapter Test .......................................................................................................................................... 28 . .

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Chapter 1 Equations and Inequalities Linear Equations ......................................................................................................................... 30 1.1 Quadratic Equations.................................................................................................................... 39 1.2 Complex Numbers; Quadratic Equations in the Complex Number System ............................... 48 1.3 Radical Equations; Equations Quadratic in Form; Factorable Equations ................................... 50 1.4 Solving Inequalities .................................................................................................................... 59 1.5 Equations and Inequalities Involving Absolute Value ................................................................ 65 1.6 Problem Solving: Interest, Mixture, Uniform Motion, and Constant Rate Job Applications ..... 69 1.7 Chapter Review ..................................................................................................................................... 73 Chapter Test .......................................................................................................................................... 80 Chapter 2

Graphs

The Distance and Midpoint Formulas......................................................................................... 82 2.1 Graphs of Equations in Two Variables; Intercepts; Symmetry .................................................. 87 2.2 Lines 93 2.3 2.4 Circles ....................................................................................................................................... 101 Variation ................................................................................................................................... 106 2.5 Chapter Review ................................................................................................................................... 108 Chapter Test ........................................................................................................................................ 114 Cumulative Review ............................................................................................................................. 116 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Chapter 3 Functions and Their Graphs 3.1 Functions................................................................................................................................... 118 The Graph of a Function ........................................................................................................... 125 3.2 Properties of Functions ............................................................................................................. 129 3.3 Library of Functions; Piecewise-defined Functions ................................................................. 136 3.4 3.5 Graphing Techniques: Transformations ................................................................................... 142 Mathematical Models: Building Functions ............................................................................... 150 3.6 Chapter Review................................................................................................................................... 153 Chapter Test ........................................................................................................................................ 159 Cumulative Review ............................................................................................................................. 163

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Linear and Quadratic Functions

Linear Functions and Their Properties . . . 165 4.1 Building Linear Functions from Data ....................................................................................... 170 4.2 Linear and Quadratic Functions . . . . . . 172 4.3 4.4 Properties of Quadratic Functions ............................................................................................ 182 Inequalities Involving Quadratic Functions . . 185 4.5 Chapter Review . .. . . . . .

195 Chapter Test 201 Cumulative Review ................................................................................................ :............................ 203 ...... ............................. . .. ......................................... ....

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Chapter 5

Polynomial and Rational Functions

Polynomial Functions and Models . . :.................................................................. 205 5.1 Properties of Rational Functions . . . . . . . . ..

215 5.2 The Graph of a Rational Function . . . . . .. .

219 5.3 Polynomial and Rational Inequalities . . . . . .

242 5.4 The Real Zeros of a Polynomial Function . . . . .

249 5.5 Complex Zeros; Fundamental Theorem of Algebra . . . . 270 5.6 Chapter Review . . . . : ................................................................... 273 Chapter Test . . . .. . . . . . .

289 Cumulative Review . . . . . . . .

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Exponential and Logarithmic Functions

Composite Functions ................................................................................................................ 296 One-to-One Functions; Inverse Functions . . . .. . . . . 303 Exponential Functions . . . . . . . .

313 Logarithmic Functions .............................................................................................................. 321 Properties of Logarithms .......................................................................................................... 329 Logarithmic and Exponential Equations . ... . .

333 Compound Interest . . . . . . . . .

341 Exponential Growth and Decay Models; Newton's Law; Logistic Growth and Decay Models . . . . . . 345 Building Exponential, Logarithmic, and Logistic Models from Data....................................... 348 6.9 Chapter Review . . . . ..

351 Chapter Test . . . .

361 Cumulative Review . . . . . . .. . . .

364 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8

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Trigonometric Functions

Angles and Their Measure . . . . . . .

367 7.1 Right Triangle Trigonometry . . . . .

371 7.2 Computing the Values of Trigonometric Functions of Acute Angles . . . 378 7.3 Trigonometric Functions of General Angles ............................................................................ 384 7.4 Unit Circle Approach: Properties of the Trigonometric Functions . . .

391 7.5 7.6 Graphs of the Sine and Cosine Functions ... . . .. . . 394 Graphs of the Tangent, Cotangent, Cosecant, and Secant Functions.. . .. . 404 7.7 Phase Shift; Sinusoidal Curve Fitting . . . . .

407 7.8 Chapter Review . . . .

413 Chapter Test . . . . . . . . . . . 421 Cumulative Review . . . . .. . . . .. . . . . . . .

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Analytic Trigonometry

The Inverse Sine, Cosine, and Tangent Functions .

8.1 The Inverse Trigonometric Functions (continued) 8.2 Trigonometric Identities 8.3 Sum and Difference Formulas 8.4 Double-Angle and Half-Angle Formulas 8.5 Product-to-Sum and Sum-to-Product Formulas 8.6 Trigonometric Equations I 8.7 Trigonometric Equations 11.

8.8 Chapter Review Chapter Test Cumulative Review

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Chapter 9 Applications of Trigonometric Functions Applications Involving Right Triangles .

9.1 The Law of Sines . . . . . . .

9.2 The Law of Cosines . . .. . .

9.3 9.4 Area of a Triangle .. . .. .

Simple Harmonic Motion; Damped Motion; Combining Waves 9.5 Chapter Review . . . . . . ..

Chapter Test .. .. . . .

Cumulative Review . . .

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Chapter 10 Polar Coordinates; Vectors 10.1 Polar Coordinates .

10.2 Polar Equations and Graphs 10.3 The Complex Plane; DeMoivre's Theorem 10.4 Vectors 10.5 The Dot Product.

Chapter Review . .

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Cumulative Review

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11.4 The Hyperbola 11.5 Rotation of Axes; General Form of a Conic . . . . . . .

11.6 Polar Equations of Conics .. . . .

11.7 Plane Curves and Parametric Equations Chapter Review . .. . . ..

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12.1 12.2 12.3 12.4 12.5 12.6 12.7 12.8

Systems of Equations and Inequalities ... . ...... . .

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Systems of Linear Equations: Determinants . . . . . .

Matrix Algebra. . . . . .

Partial Fraction Decomposition . . .. . .. . .

Systems of Linear Equations: Matrices

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Systems of Nonlinear Equations Systems of Inequalities Linear Programming

Chapter Review Chapter Test

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Sequences; Induction; the Binomial Theorem

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Mathematical Induction The Binomial Theorem

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Chapter 14 Counting and Probability .. . .

14.1 Sets and Counting . .

14.2 Permutations and Combinations 14.3 Probability . . . . . .. . . . .

Chapter Review . . . . . . . .. . . .

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Graphing Utilities

Section I The Viewing Rectangle

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Section 3 Using a Graphing Utility to Locate Intercepts and Check for Symmetry Section 5 Square Screens Section

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Preface This solution manual accompanies Algebra & Trigonometry, 8e by Michael Sullivan.

The Instructor Solutions Manual (ISM) contains detailed solutions to all exercises in the text and the chapter projects (both in the text and those posted on the internet). The Student Solutions Manual (SSM) contains detailed solutions to all odd exercises in the text and all solutions to chapter tests. In both manuals, some TI-84 Plus graphing calculator screenshots have been included to demonstrate how technology can be used to solve problems and check solutions. A concerted effort has been made to make this manual as user-friendly and error free as possible. Please feel free to send us any suggestions or corrections.

We would like to extend our thanks to Dawn Murrin, Christine Whitlock and Bob Walters from Prentice Hall for all their help with manuscript pages and logistics. Thanks for everything!

We would also like to thank our wives (Angie and Karen) and our children (Annie, Ben, Ethan,Logan, Payton, and Shawn) for their patient support and for enduring many late evemngs.

Kevin Bodden and Randy Gallaher Department of Mathematics Lewis and Clark Community College 5800 Godfrey Road Godfrey,IL 62035 kbodden@lc.edu

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Chapter R Review

Section R.t

23. a.

I. rational

b.

3. Distributive

c.

5. True

II. 13.

15.

1 7.

1 9.

{O,I}

{0,1'21 '3I '4I }

d. None

7. False; 6 is the Greatest Common Factor of 1 2 and 1 8 . The Least Common Multiple i s the smallest value that both numbers will divide evenly. The LCM for 1 2 and 1 8 is 36.

9.

{I}

e.

2 5 . a.

AuB = {I, 3, 4,5, 9} u{ 2, 4,6,7,8} = {I, 2,3,4, 5, 6, 7,8, 9} AnB = { l, 3, 4, 5 ;9} n{2, 4,6, 7,8} = {4}

None

b.

None

c.

None

d.

(AuB) nC = ({I, 3, 4,5, 9} u{2,4,6, 7,8})n{I,3,4,6} = {l,2,3,4,5,6, 7,8,9} n{1,3,4,6} = {I, 3,4,6}

{0,1'21 '3I '4I }

e.

{�,Jr,�+1,Jr + �} {�,Jr,�+1,Jr+�}

27. a.

1 8 .953

b.

1 8.952

28 .653

b.

28.653

A = { 0, 2, 6, 7, 8}

29. a.

3I. a.

0.063

b.

0.062

AnB = {I, 3, 4, 5, 9} Il {2, 4, 6, 7, 8} = {4} = {O, 1, 2, 3, 5, 6, 7, 8, 9}

33. a.

9.999

b.

9.998

35. a.

0.429

b.

0.428

37. a.

34.733

b.

34.733

Au B = { 0, 2, 6, 7, 8} u { 0, 1, 3, 5, 9}

39.

3+2 = 5

{2,5}

4I.

x+2 = 3·4

b. {-6,2,5}

43.

3y = 1+ 2

45.

x-2 = 6

47.

�=6 2

49.

9 - 4+2 = 5+2 = 7

5I.

-6+ 4 . 3 = -6+ 1 2 = 6

53.

4+5 - 8 = 9 - 8 = 1

= {O, 1, 2, 3, 5, 6, 7, 8, 9}

2I. a.

c.

{-6,� ,- 1 .333 . .. = - 1 .3,2,5 }

d. {Jr} . e.

{-6,� ,-1 .333 ... = -1 .3,Jr,2,5 } 1

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Test x-axis symmetry: Let y = -y (_y) 2 = x + 4 y2 = x + 4 same Test v-axis symmetry: Let x = -x / = -x + 4 different Test origin symmetry: Let x = -x and y = -y . (_y) 2 = -x + 4 / = -x + 4 different Therefore, the graph will have x-axis symmetry.

57.

59.

61.

9x2 + 4y2 = 36 y-intercepts: x-intercepts: 9x2 + 4(0) 2 = 36 9(0) 2 + 4/ = 36 4/ = 36 9x2 = 36 y2 = 9 x2 = 4 y = ±3 x = ±2 The intercepts are (-2, 0) , (2, 0), ( 0, -3), and (0,3) Test x-axis symmetry: Let y = -y 9x2 + 4(_y) 2 = 36 9x2 + 4/ = 36 same Test v-axis symmetry: Let x = -x 9(_X) 2 + 4/ = 36 9x2 + 4y2 = 36 same Test origin symmetry: Let x = -x and y = -y 9(_x) 2 + 4 ( _y) 2 = 36 9x2 + 4/ = 36 same Therefore, the graph will have x-axis, y-axis, and origin symmetry. .

y = V; y-intercepts: x-intercepts: y=W=O O = V; O=x The only intercept is (0, 0) .

Test x-axis symmetry: Let y = -y -y = V; different Test v-axis symmetry: Let x = -x y = h = -V; different Test origin symmetry: Let x = -x and y = -y -y = h = -V; y = V; same Therefore, the graph will have origin symmetry.

63.

x2 + y - 9 = 0 y-intercepts: x-intercepts: x2 -9 = 0 02 + y - 9 = 0 y=9 x2 = 9 x = ±3 The intercepts are (-3, 0) , (3, 0) , and (0,9) .

Test x-axis symmetry: Let y = -y x2 - Y -9 = 0 different Test y-axis symmetry: Let x = -x (-X) 2 + y - 9 = 0 x2 + Y 9 = 0 same Test origin symmetry: Let x = -x and y = -y (-X) 2 _ y - 9 = 0 x2 Y - 9 = 0 different Therefore, the graph will have y-axis symmetry.

-

y = x3 - 27 y-intercepts: x-intercepts: 0 = x3 - 27 y = 03 - 27 y = -27 x3 = 27 x=3 The intercepts are (3, 0) and (0, -27) .

Test x-axis symmetry: Let y = -y -y = x3 - 27 different Test v-axis symmetry: Let x = -x y = (_X)3 - 27 Y = _x 3 - 27 different Test origin symmetry: Let x = -x and y = -y _y = (_x)3 _ 27 Y = x 3 + 27 different Therefore, the graph has none of the indicated symmetries.

-

90

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c.

d.

x-intercepts: _x2 -4x = 0 -x(x + 4) = 0 x = O; x = -4 The vertex is at x = -2ab = -(-4) 2(-1) = � -2 = -2 .

Since p( -2) = 4 , the vertex is (-2, 4).

f(x) = g (x) _X2 + 1 = 4x + 1 0 = X2 + 4x 0 = x(x + 4) x = 0; x - 4 Solution set: {-4, O} . f(x) > 0 We graph the function f(x) = _x2 + I . y-intercept: f(O) = 1 _x2 + 1 = 0 x-intercepts: x2 - I = 0 (x + I)(x - I) = O x = -I;x = 1 -(0) = O . Since The vertex is at x = -2ab = 2(-1) f(O) = I , the vertex is (0, 1).

10

10

-10 The graph ofp is above the x-axis when -4 < x < O . Since the inequality is strict, the solution set is { x 1 - 4 < x < O} or, using interval notation, (-4, 0) . f(x) ;::: 1 g. _x2 + 1 ;::: 1 _x2 ;::: 0 We graph the function p(x) = _x2 . The -(0) = O . Since vertex is at x = -2ab = 2(-1) p(O) = 0 , the vertex is (0, 0). Since a = -1 < 0 , the parabola opens downward.

10

10

10 -10

c.

The graph is above the x-axis when -I < x < I . Since the inequality is strict, the solution set is { xl - I < x < I } or, using interval notation, (-I, I) . g(x) :o; O 4x + I :O; 0 4x :O; -I x :o; --41

10

-1 0

{ l -±} or, using interval notation, ( - ±] .

The solution set is x x :O;

29.

-00 ,

f.

f(x) > g(x) _x2 + 1 > 4x + 1 -x2 - 4x > 0 We graph the function p(x) = _x2 - 4x .

The intercepts of p are y-intercept: p(O) = 0

The graph of p is never above the x-axis, but it does touch the x-axis at x = O. Since the inequality is not strict, the solution set is {O}. f(X) = X2 _ 4; g(x) = -x2 + 4 a. f(x) = 0 x2 - 4 = 0 (x - 2)(x + 2) = 0 x = 2;x = -2 Solution set: {-2, 2} .

1 90

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exist. No portion of thi s material may be reproduced, in any form or by any means, without permission in wri tin g from the publi sher.

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e.

f.

g.

g( x) = ( x -3 )( 2x +l )( x + 5 ) Tolastfindtwo thefactors remaiequal ningtozeros0 andofsolve. g, we set the 2x + 1 = 0 x + 5 = 0 2x=-1 x= -5 x= --21 Therefore, the zeros are -5, --21 , and 3.

Notice thesearatil rationalonalzeros the list how of potenti zeros.were all in x-intercept of a graphInarethetheprevisameous as theThezeros of thesfunction. part, we found the zeros to be -5 , --21 , and 3. Therefore, the x-intercepts are -5 , -.!.2 , and 3.

To fmd the y-intercept, we simply find g( O) . g( O) = 2 ( 0)3 + 5 ( 0)2 -28 ( 0) -15 = -15 So, the y-intercept is -15 .

Whether theisgraph crosses orthetouches atlicity. an x-iEachntercept determi n ed by mul t i p ynomizeroal occurs once,odd somultiplici thefactor multy,tipofthelicithetgraph y polof each is 1.

For wil cross thethex-axix­ s ataxithes atzero.

Thus, the graph crosses each of the three x-intercepts.

The power functi o n that the graph of g resembles for large values of I xl is given by the term with the highest power ofx. In this case, the power function is y = 2x3 . So, the graph of will resemble the graph of y = 2x3 for large values of I xl .

h.

The three intercepts are -5 , -.!.,2 and 3.

Near -5 : g ( x) = ( x -3 ) ( 2x + 1 ) ( x + 5 ) -8 ( -9)( x+5 ) = 72 ( x +5 ) (a line with slope 72) Near --21 : ( x) = ( x -3 )( 2x + 1 )( x + 5 ) ( - �} 2X+l) (%) 63 63 = --634 ( 2x+l) = --x-2 4 63 ( a line with slope - 2 or -31. 5 ) Near( 3: g x) = ( x -3 )( 2x + 1 )( x + 5 ) = ( x-3)( 7 )( 8) = 56 ( x-3) (a line with slope 56) We couldvalues first forevaluate thep determi functionneatthe several x to hel scale.ng all this information together, we Putti obtain the following graph: �

g

�

i.

y

60

:1

-50

g

3.

x

(1. -30)

x3 -4x2 +25 x-100= 0 x2 ( x -4) + 25 ( x - 4) = 0 ( x - 4) (X2 + 25 ) = 0 x - 4 = 0 or x2 + 25 0 x= 4 x2 = -25 x = ±..J-25 x =±5i The solution set is {4, -5 i,5 i} . =

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105.

csc 0 = -2, tan 0 > 0 => 0 in quadrant III Since 0 is in quadrant III, cos 0 < 0, sec 0< 0, sin 0< 0 and csc 0 < 0, while tan 0 > 0 and cot O > O. If a is the reference angle for 0 , then csc a = 2 .

Now draw the appropriate triangle and use the Pythagorean Theorem to find the values of the other trigonometric functions of a .

113.

y

115. x

J3 . = -1 sma cos a = 2 2 J3 = -J3 = J3 seca = 2 ._ 2J3 tana = -1 · 3 J3 J3 J3 J3 3 cot a = J31 = J3 Finally, assign the appropriate signs to the values of the other trigonometric functions of 0 . sinO = .!..2 cos O = _ J32 J3 2J3 tan B = secO = 3 3 cot B = J3 sin 40° +sinI300+sin 2200+sin310° = sin 40° +sin( 40° +90°) + sin( 40°+ 1 80°) + sin( 40° +270°) = sin 40° + sin 40° - sin 40° -sin 40° =0 Since f (B) = sin B = 0.2 is positive, 0 must lie either in quadrant I or II. Therefore, B + tr must lie either in quadrant III or IV. Thus, f ( B + 7t) = sin( 0 + 7t) = -0.2

117.

1 = -1 = 5 G·Iven sm. B = -51 , then csc B = -sin O 1 5 Since csc 0 > 0 , B must lie in quadrant I or II. This means that csc ( 0 + tr) must lie in quadrant III or IV with the same reference angle as B .

Since cosecant is negative in quadrants III and IV, we have csc(B + tr) = -5 . sin 1° + sin 2° + sin 3° +... + sin 357° + sin 358° + sin 359° = sin 1° +sin 2° + sin3° + . . . + sin(360° _3°) + sin(360° - 2°) + sin(360° _1°) = sin 1° +sin 2° +sin3° + . . . +sin( _3°) + sin( -2°) + sin( -10) . = sin 1° +sin 2° +sin3° + . . -sin 3° -sin 2° -sin 1°

a.

�

b.

--

109.

111.

.

�

-

107.

322.J2 [sm( 2( 60°)) - cos( 2( 600)) -1 = --:n] 32.J2 (0.866 - (-0.5) - 1) 16.6 ft .

Let 1'; = 322.J2 32 [sm(2x) - cos {2x) - I] R

c.

Since F ( 0) = tan 0 = 3 is positive, B must lie either in quadrant I or III. Therefore, B + 7t must also lie either in quadrant I or III. Thus, F (O+ 7t) = tan (B + tr) = 3 .

119.

--

Using the MAXIMUM feature, we find: 20

Answers will vary.

390

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91.

(

)

cos sm S3 - cos "21 et a = sm -53 and 13 = cos 21 a . m. quadrant I; 13 is in quadrant I . Then sin a = �5 , 1 0 � 13 � -n o � a � -n , and cos 13 = -, 2 2 2 cos a = .Jl - sin 2 a = = 1 - :5 = =

L

.

_I

_I

.

-I

-I

-

.

tan a - tan 13 1 + tan a tan 13 J3 3 3 4

IS

.

FG) ) J¥s � sin 13 = �1 - cos2 13 = Jl - (�J = )I - ± = l = � (

-4J3 - 9 12 1 - 3J3 12 -9 - 4J3 · 12+3J3 12 - 3J3 12 +3J3 -144 - 75 J3 1 17 -48 - 25J3 39 48 + 25J3 39

�)

cos sin- I �- cos-J = cos ( a - 13) = cos a cos 13 + sina sin 13 J3 = -45 . -21 + _35 . 2 4 3J3 4 + 3J3 =+-= 10 10 10 93.

[ ( �) �] ( )

---

tan sin-J - - tan-J Let a = sm. -J -"21 and 13 = tan -J 4"3 . a m quadrant IV; 13 is in quadrant I. Then, . = --21 ' 0 � a � -2 ' and tan 13 = -43 ' sma 0 < 13 < -n2 · cos a = .J'1- si-n:-2-a

( J

Let a = cos-J -� a is in quadrant II. Then cos a = - -35 , -2 � a � n . sin a = .Jl - cos 2 = 1 - :5 = = 1-

IS

7r

� (-�J � � � [ ( �)] = = (�)( -�) = �: a

n

J ( �J

=

sin 2 cos- J

) l=�

= 1 - - = 1 -± = J3 tana = --1 = -J3 3

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sin 2a

-

2sin a cos a

=

2

-

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23.

Vertices: (0, -6), (0, 6); asymptote: y 2 x ; Center: (0, 0); Transverse axis is the y-axis; a = 6 . Find the value of b using the slope of the a = b6 = 2 => 2b = 6 => b = 3 asymptote: b =

27.

2

Write the equation:

2

25

-

the center of the hyperbola is at (0, 0). a = 5, b = 3 . The vertices are (5, 0) and

( -5, 0) .

Find the value of c: 2 c 2 = a + b 2 = 25 + 9 34 => c = J34 The foci are ( $4, 0) and ( -$4, 0) .

Find the value of c: c 2 = a 2 + b 2 = 3 6 + 9 = 45

c = 3..[5

x l - 9 =1

=

2

The transverse axis is the x-axis. The asymptotes

�6 - x9 = 1 .

F2 1

3

3

=

(0, 3� )

V =

0'

are y = S' x; y = - S' x . y

2x

10

x

-10

25.

Foci: (-4, 0), (4, 0); asymptote: y = - x ; Center: (0, 0); Transverse axis is the x-axis; c = 4 . Using the slope of the asymptote: - � = - 1 => - b = -a => b = a

a

.

Find the value of b: b 2 = c 2 a 2 => a 2 + b 2 = c 2 b 2 + b 2 1 6 => 2b 2 = 1 6 => b 2 = _

=

b = J8 = 2-fi

a

= J8 = 2-fi

Write the equation:

29.

g

( = 4)

4x 2 - y 2 = 1 6 Divide both sides by 1 6 to put in standard form: y 2 16 x 2 y 2 4x 2 - = - => -- = 1 16 16 16 4 16 The center of the hyperbola is at (0, 0). a = 2, b 4 .

The vertices are (2, 0) and (-2, 0).

Find the value of c: c 2 = a 2 + b 2 = 4 + 16 = 20 =

C

c = J2Q = 2..[5 The foci are (2..[5,0) and (- 2..[5, 0) .

( a = b) 2 2

Xg � = 1 .

-

The transverse axis is the x-axis. The asymptotes are y = 2x; y = - 2x .

�90

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(0, 1), (1, 0), (0, 6), (6, 0). = 2x + y (0, 1) = 2(0)+ 1 = 1 (0, 6) = 2(0)+6 = 6 (1, 0) = 2(1)+0 = 2 (6 0) = 2(6 )+0 = 12 12 (6, 0). = 2x + 5Y x 0, y 0, x + y 2, x 5, y 3.

3x + 2y = 12, 2X+3Y == 1212 {3x+2y

Vertex

Value of z z

z

z

11.

Minimize

z

�

�

�

subject to �

constraints.

=

�

Graph the

....... ....... ... ...... .... . - -. . - .

: - : - : . :a: .

-

: : - : ti

. '

-

:-:-:-:-:-:-: :-:.

.

.

y: y = 6 -�x2

2X+3 ( 6-% X) 12 2x+18-�x2 = 12 5 --x=-6 2 12 X = -5 y = 6-�2 (.!25 ) = 6-�5 = .!25 (2.4, 2.4) .

(0, (2, 0), (0, 4), (4, 0), (2.4, 2.4 ).

3x + 5y (0, 2) = 3(0)+5(2) = 10 (0, 4) = 3(0) 5(4) = 20 (2, 0) = 3(2)+ 5(0) = 6 (4, 0) = 3(4)+ 5(0) = 12 (2.4 2.4) = 3(2.4) + 5(2.4) = 19.2 20 (0, = 5x 4y x 0, y 0, x + y 2, 2x 3y 12, 3x + y

Substitute and solve:

at

12 and

solve the system:

Solve the second equation for

z

The maximum value is

2x + 3y =

To find the intersection of

The comer points are Evaluate the objective function:

.

:.: : : :-: :.:-: :

The point of intersection is

The comer points are 2), Evaluate the objective function: Vertex

(0, 2), (2, 0), (0, 3), (5, 0), (5, 3). = 2x + 5y (0, 2) = 2(0)+ 5(2) = 10 (0, 3) = 2(0)+ 5(3) = 15 (2, 0) = 2(2)+ 5(0) = 4 (5, 0) = 2(5)+ 5(0) = 10 (5 3 1 2(5) + 5(43 ) =(2,250). = 3x + 5y x 0, y 0, x + y 2, 2x + 3y 12, 3x + 2y 12.

z

The comer points are Evaluate the objective ftmction: Vertex

z

z

z

z

z

The maximum value is

z

z

1 5.

Maximize

z

�

�

�

�

+

z

�

at

4).

subject to

+

�

�

�

12. Graph

the constraints.

at

subject to

Minimize �

z=

13.

+

z

Value of z

The minimum value is

Value of z =

�

Graph

the constraints.

3x y = 12, 3y = 12 {2X+ 3x+ y = 12

To find the intersection of +

2x + 3y = 12

and

solve the system:

690

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