Chapter 12. Lesson Algebra 2 Worked-Out Solution Key. Prerequisite Skills (p. 792) 12.1 Guided Practice (pp )

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1 Chapter Copyright by McDougal Littell, a divisio of Houghto Miffli Compay. Prerequisite Skills (p. 9. The domai of f( is all real umbers ecept Þ 0.. The rage of g( is all real umbers.. The compositio f(g( is equal to y 0 y 0 y 0 y 0 ( y 0 y 9 The solutio is (, y 0 y 0 y 0 y 0 y 0 y The solutio is,. 0. y y 0..y 8. 8 y 8 0 The solutio is 0,.. f (g( ( Domai: all real umbers ecept 0. f( f( ( Domai: all real umbers. g(g( ( Domai: all real umbers ecept 0 y y Lesso.. Guided Practice (pp f ( ( a f ( ( f ( ( 8 f ( ( 8 a 9 f ( ( a 0 f ( (. a a a. The sequece, 8,,,... ca be writte as p, p, p, p,.... The et term is a p. A rule for the th term is (.. a There are 8 apples i the 9th layer.. a i i Lower limit Upper limit 0 Summatio otatio:. a i i i Lower limit Upper limit ifiity 0 i i i Summatio otatio: i i 8. a i i Lower limit Upper limit ifiity Summatio otatio: 9. i Lower limit Upper limit 8 Summatio otatio: ` ` i i 0 8 i ( i Algebra Worked-Out Solutio Key

2 Chapter, cotiued 0. 8i 8( 8( 8( 8( 8( i k (k ( ( ( ( ( i There are 8 apples i the stack.. Eercises (pp Skill Practice. Aother ame for summatio otatio is sigmotatio.. A sequece is a list of umbers ad a series is the sum of the terms i a sequece... a a a a a 8 a 0.. f ( a f ( 0 9 f ( 9 f ( a f ( a f ( a a 9 a a a 0 a 9. f ( 0. ( f ( a ( f ( ( f ( ( 9 f ( 0 a ( f ( a ( 8. f (. f ( a f ( f ( f ( a f ( a. (. f ( ( a ( f ( ( ( f ( ( ( f ( ( a ( 0 f ( ( 9 a ( f ( (. Give terms: p, p, p, p,... Net term: p Rule for th term:. Give terms:,,,,... Net term: Rule for th term:. Give terms: ( ( p, ( ( p, ( ( p, ( ( p,... Net term: ( ( p 0 Rule for th term: ( ( 8. Give terms:,,,,... Net term: Rule for th term: 9. Give terms: p, p, p, p,... Net term: p Rule for th term: 0. Give terms: p, p, p, p,... Net term: p 0 Rule for th term:. Give terms:,,,,,... Net term: Copyright by McDougal Littell, a divisio of Houghto Miffli Compay. Rule for th term: 8 Algebra Worked-Out Solutio Key

3 Chapter, cotiued. Give terms: p p 0, p p 0, p p 0, p p 0,... Net term: p p Rule for th term: 0.. Give terms:. 0.(,. 0.(,. 0.(,. 0.(,... Copyright by McDougal Littell, a divisio of Houghto Miffli Compay. Net term:. 0.(.9 Rule for th term:. 0.. Give terms:.8.(,.8.(,.8.(,.8.(,.8.(,... Net term:.8.(.8 Rule for th term:.8.. Give terms: 0., 0., 0., 0.,... Net term: 0.. Rule for th term: 0.. Give terms:..8(,..8(,..8(,..8(,.... D; Net term:..8( 0. Rule for th term:..8 ( ( a ( ( a i i Lower limit Upper limit Summatio otatio: 8. i Lower limit Upper limit Summatio otatio: 9. i Lower limit Upper limit ifiity Summatio otatio: i i ` i (i (i (i Algebra Worked-Out Solutio Key 9

4 Chapter, cotiued 0. ( i Lower limit Upper limit ifiity Summatio otatio:. i Lower limit Upper limit ifiity Summatio otatio:. i Lower limit Upper limit. Summatio otatio: i i Lower limit Upper limit ` i ` i i ( i (i i Summatio otatio: i i. i Lower limit Upper limit ifiity ` Summatio otatio: (i i. i p p p p p p i 8 0. i p p p p p i k p p p p k (k ( p ( p k ( p ( p ( ( ( ( ( ( i i i 8 0 k. k k 0. i..... i... i The first term is missig. The lower limit is 0, so the first term should be (0, or. Correct sum: B; 0 i i 0 9. true; ka i k ka k k... k i k( a... k a i i 0. true; (a i b i ( b (a b i ( b... ( b ( a... (b b b... b a i b i i i Copyright by McDougal Littell, a divisio of Houghto Miffli Compay. 0 Algebra Worked-Out Solutio Key

5 Chapter, cotiued Copyright by McDougal Littell, a divisio of Houghto Miffli Compay.. false; Sample aswer: Let a i i ad b i i. i i i i (i(i ( p ( p ( p ( p ( p ( p 8 ( ( i p p p (i p ( p ( p i i Because Þ 88,. false; (i ( 88 i a i b i Þ Sample aswer: Let a i i ad k. i i i i 9 i ( Because Þ, Problem Solvig i a i k Þ i 80( ( ( a ( a 08 80( a ø ( T p 80( a i k. a i i b i. Total measure of agles i skylight 80(, or Use Special Series Formula for sum of first positive ( itegers: i i 00(00 I 00 days, you will have 00 peies, or $0.0. To save $00, you eed 0,000 peies. ( 0,000 ( 00,000 00,000 0 ( Ï 00,000 0 Ï 00,000. You eed about days to save $00.. Give terms:,,,, Rule for th term: a a a 8 8 You eed moves to move rigs, moves to move rigs, ad moves to move 8 rigs.. a. d 0.( 0.. Mars is. astroomical uits from the su. 9,00,000 km b.. a.u. 9,0,000 km a.u. Mars is about 9,0,000 kilometers from the su. c. Mea distace from su (a. u. d Positio of plaet from su (. a. a There are balls i the fifth layer. b. a a ( ( ( ( ( 0 There are balls i the stack. c. The differece i each layer is ( (. Each layer of the square pyramid cotais eactly ( more balls tha the correspodig layer i the triagular pyramid. 8. The umber of balls i the top layers is (. i i Usig Eercise 9: i i ( Usig Eercise 0: i ( i i Usig the special formulas o page 9: i i ( ( ( Algebra Worked-Out Solutio Key

6 Chapter, cotiued Mied Review d Ï ( ( ( Ï Ï 0 9. d Ï ( ( ( Ï Ï 80. d Ï ( 0 ( Ï 9 Ï 8. d Ï ( ( (9 Ï 9 Ï 8. d Ï ( ( ( Ï Ï 8. d Ï ( ( (8 ( Ï 9 8. d Ï ( 9 ( Ï Ï 8. d Ï ( ( ( 8 Ï Ï 8. d Ï (9 ( 0 Ï 9 Ï 0 Graphig Calculator Activity. (p. 80. a., 9,,,,, 9,,, b. c. Sum 0. a. 9,,, 8,,,, 0,, b. c. Sum. a., 9,,, 0,,,, 8, b. c. Sum 8. a., 9,,,,, 9,,, b. c. Sum 0. a.,,, 9, 8, 9,,, 8, 0 b. c. Sum. a.,,, 8,,,, 8,, b. c. Sum 0 Copyright by McDougal Littell, a divisio of Houghto Miffli Compay. Algebra Worked-Out Solutio Key

7 Chapter, cotiued Copyright by McDougal Littell, a divisio of Houghto Miffli Compay. Lesso.. Guided Practice (pp a a 8 a 8 Each differece is, so the sequece is arithmetic.. ; d ( ( 0 a 0 0 (0 0. ( d 0( ( ( 0 a 0 0 (0 0. a ( d d ( d d ( ( ( 9 a 0 9 (0 9. ( 9 ( 8 S ( a 8 (8 S d d The house of cards cotais 08 cards.. Eercises (pp Skill Practice. The costat differece betwee cosecutive terms of a arithmetic sequece is called the commo differece.. A arithmetic sequece is a list of umbers that have the same commo differece betwee cosecutive terms. A arithmetic series is the sum of the terms.. a a ( 8 ( a (8 The sequece is arithmetic with commo differece.. a a The differeces are ot costat, so the sequece is ot arithmetic.. a 9 a 9 9 a 9 The sequece is arithmetic with commo differece 9.. a (0 a ( The differeces are ot costat, so the sequece is ot arithmetic.. a a a. 0. The sequece is arithmetic with commo differece a a 0 The differeces are ot costat, so the sequece is ot arithmetic. 9. a a The differeces are ot costat, so the sequece is ot arithmetic. 0. a a The differeces are ot costat, so the sequece is ot arithmetic.. a a ( a The sequece is arithmetic with commo differece.. ; d ( ( a 0 (0 8. ; d ( ( a 0 (0 9. 8; d 8 8 ( ( a 0 (0 Algebra Worked-Out Solutio Key

8 Chapter, cotiued. ; d ( ( ( a 0 (0. ; d ( ( 0 a 0 ( ; d ( ( a 0 ( ; d 0. a ( d (9 ( ( ( a 0 ( ; d (. ( d 9 ( 8 8 ( ( a 0 (0 0..; d.... ( (.. 0. a 0.( The geeral rule for a arithmetic sequece is ( d, ot d. So, ( ( 0.. The values of ad were substituted i the wrog place. It should be ( (, so 0.. ( d ( ( ( ( d ( ( ( (0 d 0 9 ( 8 Copyright by McDougal Littell, a divisio of Houghto Miffli Compay Algebra Worked-Out Solutio Key

9 Chapter, cotiued Copyright by McDougal Littell, a divisio of Houghto Miffli Compay. 8. ( d 9. C; 0 (... 0 (0 d 9( 9 9 ( ( 0. 0 (0 d 8 9d ( d d 8 9(9 d 9 d ( (9 9. ( d 9 d a ( d 9 d 9 ( 0 8d d ( ( 9. ( d 0 d ( d d 0 ( 8 d d 8 ( (. a 0 (0 d 8 9d a 8 (8 d 0 d 8 d d 8 9( 8 8 ( (. ( d d a 9 (9 d 89 8d (8 8 d 8 d ( (8 8. ( d 0d a ( d 0( d 8 9d d ( (. ( d 9 d a ( d d d d (. a 9 (9 d 8d a ( d d 8 9 ( d 9 d 8. ( d 0d 9. B; a ( d 0 d d d 0 ( ( d 8 d a ( d d d d 8 ( ( ( 0 0. ( 0 (0 S 0 0. ( a 8 (8 9 (9 S ( 8 8 (8 9 S (9 Algebra Worked-Out Solutio Key

10 Chapter, cotiued. 9 ( a 9 ( S 8. ( a 9 (9 8 8 S. a 9( 9 9( 9 S S 8 0. a 8 S a S 9.,,,,..., d ( ( 0.,,, 0,..., d ( (.,,, 9,..., d ( (. The graph of is a scatter plot whose poits lie o the lie that represets the graph of f(.. false; Sample aswer: Cosider the series 8, whose sum is 0. If d is doubled, the series becomes 0, whose sum is. Because Þ (0, the statemet is false.. true; Because is a arithmetic sequece, the first three terms a, b, ad c, are a,, ad. b 0 ( c 0 ( ( i ( i 8 8 8( ( ( ( 8( 0 8 or The umber of terms must be positive, so. (0 i 8 i 8 8( ( (0 0 ( ( or The umber of terms must be positive, so 8. (8 8i 0 i 0 0( (0 ( ( ( 0 or The umber of terms must be positive, so. ( i 0 i 0 0( (0 ( ( ( 0 or The umber of terms must be positive, so. Copyright by McDougal Littell, a divisio of Houghto Miffli Compay. 0 ( Algebra Worked-Out Solutio Key

11 Copyright by McDougal Littell, a divisio of Houghto Miffli Compay. Chapter, cotiued i ( i 0 ( 0 ( ( 0( ( [ ( ] 0 ( ( ( ( 0 or. The umber of terms must be positive, so. i ( i ( a ( (a ( ( [ ( ] 90 ( ( ( ( or The umber of terms must be positive, so (i 0 i 0 99,00. S a Because the commo differece is, the et term is, or 8. Problem Solvig. a., d ( ( b. Total umber of cells S 9 S 9 9 a 9 S 9 9 S 9 0 There are cells i the hoeycomb.., d S a ( ( S S There are bad members i each row. The bad has members.. a., d 8. a. ( (8 8 b. S S 9 S There are visible blocks. d( D( D(0 ( (0 D( D( ( ( 8 D( D( ( ( 80 D( D( ( ( b., d c. ( ( So, a rule is d(. d( Algebra Worked-Out Solutio Key

12 Chapter, cotiued. S 000,, d 0 8. a. ( d a ( (0 a 00 S a $00 should be give away o the first day. d (i. l (i. π π.0.0π.0.0π Sample calculatio: d 0.00((.008 b. l, l, l,... is a arithmetic sequece. l l ( (d l π ( (0.008π l [ ( (0.008]π c..008, d 0.008, ( d.008 ( ( The paper must be wrapped aroud the dowel times. l [ ( (0.008]π (from part b l.99π l π π.99π S ø 8. The legth of paper i the roll is about 8. iches. d. For a roll with a -ich diameter:.008 ( ( l π l [ ( (0.008]π.99π π.99π S ø 88.9 The legth of paper i the -ich diameter roll is about 88.9 iches. This is about. times the legth of paper i the -ich diameter roll, so you could epect to pay about.($.0 $. for the -ich diameter roll. Mied Review 0. /. / ( / ( / / /,80. / 9 / / ( / / /. / / 9 / ( / / / p / Ï Ï 9. ( 0 /. ( / [( 0 / ] [( / ] / / log log log log log log log log log log ø log 0 log 8 log 0 log log 8 log ø 0. ø ( ( Copyright by McDougal Littell, a divisio of Houghto Miffli Compay. 8 Algebra Worked-Out Solutio Key

13 Chapter, cotiued Copyright by McDougal Littell, a divisio of Houghto Miffli Compay. 8. Mea: Media: Mode: 8. Mea: 8 8 ø 0. Media: Mode: 8. Mea: Media: 8. Mode: 9 8. Mea: ( ( ( ( 9 ø.9 Media: Mode: oe 8. Mea: ø. Media:. Mode:.9 8. Mea: ø.9 Media:.8 Mode: oe 88. h umber of hats s umber of scarves h 8s 0 h 8s 0 h s h s s 8 s 9 h 9 h You sold hats. Lesso.. Guided Practice (pp a. a 8 a 9 a 9 a a The series is geometric with a commo ratio of. a. a a Because there is o commo ratio, the sequece is ot geometric. a. a 8 a 8 a a a The series is geometric with a commo ratio of.., r r ( a 8 ( 8 a 8,. a 9, r a r 9 ( (r ( a 8 ( 8 a 8 8. a, a r r r r r r r r r r Algebra Worked-Out Solutio Key 9

14 Chapter, cotiued If r : r a If r : r a S 8 r8 r S 8 (8 ( S (.09.0(.09 ø 8.90 The bo office reveue i 000 was about $8.9 billio.. Eercises (pp. 8 8 Skill Practice. The costat ratio of cosecutive terms i a geometric sequece is called the commo ratio.. If you divide each cosecutive term ad get the same ratio every time, the sequece is geometric. a. 8 a Because there is o commo ratio, the sequece is ot geometric. a. a a 0 The sequece is geometric with a commo ratio of. a. a a The sequece is geometric with a commo ratio of. a. a a 8 a a a 8 The sequece is geometric with a commo ratio of.. a a a a The sequece is ot geometric because there is o commo ratio. 8. a 8 a a 8 The sequece is ot geometric because there is o commo ratio. 9. a a 0 0. a. 0. a.. 0. a a The sequece is geometric with a commo ratio of 0.. a 0. a a The sequece is ot geometric because there is o commo ratio. Copyright by McDougal Littell, a divisio of Houghto Miffli Compay. 80 Algebra Worked-Out Solutio Key

15 Chapter, cotiued Copyright by McDougal Littell, a divisio of Houghto Miffli Compay. a. a a a 08 a a 08 The sequece is geometric with a commo ratio of. a. a a.8 0. a..8 a a.. The sequece is geometric with a commo ratio of. a. a 0 a 0 0 The sequece is ot geometric because there is o commo ratio. a. a. 0. a... The sequece is ot geometric because there is o commo ratio.., r., r 8 ( ( a ( 09 a (., r 8., r ( ( a ( a ( 8, 09, 9., r a , r a 9,., r a. 0., r ( a 0.( 9.., r (0. a ( , r. 0. (0. a ( , r.8 (.8 a ( , r B; 0(. a 0(..8, r 0 ( 8., r ( 0 Algebra Worked-Out Solutio Key 8

16 Chapter, cotiued 9., r ( 9, 00. a, r 8 a r a, r a r ( 00 ( a, r a r , r r Copyright by McDougal Littell, a divisio of Houghto Miffli Compay. 8 Algebra Worked-Out Solutio Key

17 Chapter, cotiued., r r ( ( a 8, r a r 8 ( ( I the first step, the epoet should be. r ( 8. I the first step, r should be raised to the ( th power, ot. r ( 9., r r r r ( or ( 0., a a r p r r ( or (., r r r r Copyright by McDougal Littell, a divisio of Houghto Miffli Compay , r r 00 ( ( ,00 (. 0, a 0 r 0 r 0 r a r 0 r 0 0 r r 0 0r r r r 0 9 ( 00 Algebra Worked-Out Solutio Key 8

18 Chapter, cotiued. a 0, 0 a r 0 r 0 r r 0 r 0 0 r r 0 0r r r 0 80 or a 0 80 r r a, a a r r r a r r r r r r r r (., a r r r a r r r r r r r r (., a r r r a r r r r r r r r or., a 8 r r r a r 8 r 8 r r 8 r 8 r Î 8 r Ï r Ï r Ï 8. S 0 r0 r S S 8 r8 r S 8 8,00 Copyright by McDougal Littell, a divisio of Houghto Miffli Compay. 8 Algebra Worked-Out Solutio Key

19 Copyright by McDougal Littell, a divisio of Houghto Miffli Compay. Chapter, cotiued 0. i 0 8 i i S 8 r8 r S 8 8. S r r S. S r r S 8 i, 0. ( i ( i i 0 i. C; S r r S ( ( 88,8 S 9 r9 r S 9 9 9,8. Sample aswer: S 00, choose r, fid. S r r ( 00 Series: a., r S r r S b., r S r r S ( S 8 Problem Solvig. a., r ( b. S r r S There are skydivers i four rigs. 8. a., r are the logical values because a. b. S r r S games are played i the touramet. 9. a. After the first pass, items remai, so ad r. b. Fid whe. log ( log log log 0 O the teth pass, oly oe term remais. 0. a., r 8 (8 8 S 8 r8 r S ,9,,9, squares are removed through stage 8. b. b 9 8 9, r 8 9 b b ø 0. The remaiig area of the origial square after the twelfth stage is about 0. square uit. Algebra Worked-Out Solutio Key 8

20 Chapter, cotiued. a. For Compay A: 0,000, d 000 ( d 0,000 ( (000 9, This sequece is arithmetic. For Compay B: b 0,000, r.0 b r b 0,000(.0 This sequece is geometric. b. Salary (dollars,000,000,000,000 b,000,000,000 0,000 9, Years of employmet c. For Compay A: S S 0 0 0,000 9,000 90,000 For Compay B: S r r S 0 0,000 ( ,. The sum of the wages durig the first 0 years of employmet is $90,000 for Compay A ad about $9, for Compay B. d. For Compay A S $0, $0, $, $, $, $, $, $8, $, $0, $, $, $, $, $, $88, $8, $, $9, $, $0, $, $, $0, $, $8, $, $, $, $0, $, $0, $, $, $, $, $8, $, $9, $90, For Compay B S $0, $0, $,.00 $8,99.8 $08,. $,9. $, $8,8. 9 $,.9 0 $0,. $9,.0 $00,. $,. $,88. $00,. $,90. $,90. 8 $, $,.9 0 $9,. The total amout eared by Compay B is greater tha the amout eared by Compay A after 9 years.. 000, r.0 S 0 r0 r S ø,8.0 You will have $,8.0 i the IRA after your last deposit. Mied Review ( 0( Copyright by McDougal Littell, a divisio of Houghto Miffli Compay Algebra Worked-Out Solutio Key

21 Copyright by McDougal Littell, a divisio of Houghto Miffli Compay. Chapter, cotiued ( ( ( ( ( 9 0 or ( ( ( 9 or ( 8( 8 8 i (i ( ( ( ( ( ( ( i ( ( ( ( ( i (8 (9 (0 ( ( i ( ( ( i ( ( ( 0 ( 9. (i (( ((8 i ((9... (( (( (i (( ((... i (( (( i ( ( ( ( (8 (9 i Quiz.. (p. 8. Give terms:,,,,... Rewritte terms: (, (, (, ( Net term: ( 9 Rule for th term:. Give terms:, 0,, 0,... Rewritte terms: ( ((, ( ((, ( ((, ( (( Net term: ( (( Rule for th term: ( (. Give terms: 0, 0, 0, 0,... Rewritte terms: ( (0, ( (0, ( (0, ( (0 Net term: ( (0 0 Rule for th term: 0(. Give terms:,,,,... Rewritte terms:,,, Net term: 0 Rule for th term:. Give terms:,,, 0,... Rewritte terms: (, (, (, ( Net term: ( 0 Rule for th term: (. Give terms: 9,, 8,,... Rewritte terms: 9(, 9(, 9(, 9( Net term: 9( Rule for th term: 9 (. i ( ( ( ( i (k ( ( ( k ( ( Algebra Worked-Out Solutio Key 8

22 Chapter, cotiued , d ( d ( ( ( 8 S S 8., d ( d ( ( S S., d ( d ( ( 8 8 ( S S ( 0., r 8 r ( (,80,9 S r r S,8,88., r r S r r,8,8,99 S ø.98., r r ( ( 8,0,,8 S r r S ( (,8,89,0. 0, r.0 r 0(.0 I 00, 8. a 8 0(.0 8 ø 09.9 I 00, the average tuitio at a public college was about $09.9. Mied Review of Problem Solvig (p. 88. a.,000, r.0 r,000(.0 b. a,000(.0 ø,8. Durig your th year of employmet, your salary will be $,8.. c. S r r S 0,000 (.00.0 ø,,00.8 After 0 years, you will have eared a total of $,, a. A πr π( π a π( π( π π( π( π π, d π π ( (π ( π b. (i π i Copyright by McDougal Littell, a divisio of Houghto Miffli Compay. 88 Algebra Worked-Out Solutio Key

23 Copyright by McDougal Littell, a divisio of Houghto Miffli Compay. Chapter, cotiued c. For : For : For : For 8: 8 i i i i (i π π (i π π π π (i π π π π π π (i π π π π π 9π π π π π Whe the umber of rigs is doubled, the total area is quadrupled.. Tables placed together by short edges: Sequece:, 0,, 8..., d ( ( Tables placed together by log edges: Sequece:, 8, 0,..., d ( ( Whe the tables are coected by the short edges, (, or, more people ca be seated.. Sample aswer: 8, S 0 S 0 8 a 8. a 8 Choose. ad a 8.. ( d a 8 (8 d.. d d d A series is 8 i or 0.. (i 0.. So,. ( (,., d ( d ( ( 0 S S There are 0 pieces of chalk i the pile.. 9, d ( d 9 ( ( The height at the top of the 0th stair is (0 iches. To fid the height of the bottom of the th stair, subtract the height of the th stair from. So,.. a. The sequece is geometric because each term is half of the previous term. b., r r c d. < < < ( log < log The poits lie o a epoetial decay curve. log < log <.0 Because represets the umber of two-hour itervals, there will be less tha oe gram of Platium-9 after about hours. Algebra Worked-Out Solutio Key 89

24 Chapter, cotiued 8. Sample aswer: Choose a geometric series with five terms: 8. Net, fid a arithmetic series with five terms ad a sum of. S a. a Choose. ad a 0.. ( d 0.. d 8 d d So,. ( ( 0. ad the series is Guided Practice (pp S 0.0 S 0. S 8 ø 0. S 8 ø 0. S 8 ø 0. As icreases, S appears to approach. 0. S Lesso. Ivestigatig Algebra Activity. (p. 89 Step. The et three areas are 8,, ad. These areas form a geometric sequece because there is a commo ratio of. Step. Number of pieces Combied area Number of pieces Combied area The total area appears to be approachig oe square uit.. A A r r A As gets large, A gets close to. For eample, whe 0, A , ad whe 00, A ø.., r S r. r Because, the series has o sum.., r S r. 0, r 0.8 d r The pedulum swigs a total distace of 0 iches (0. (0. (0.... r ( (0.0 (0.0 ( r ( Copyright by McDougal Littell, a divisio of Houghto Miffli Compay. 90 Algebra Worked-Out Solutio Key

25 Chapter, cotiued (0.0 (0.0 ( r ( Eercises (pp. 8 8 Skill Practice. The sum S of the first terms of a ifiite series is called a partial sum.. If r <, the series has a sum.. S 0. S ø 0. S 8 ø 0. S 8 ø 0. S 8 ø 0. S appears to be approachig. y. S S. S.8 S 08 ø 8.0 S 08 ø 9. S appears to be approachig 0. y. S 0. S. S. S 9 S 9. As icreases, S also icreases. The series has o sum. y Copyright by McDougal Littell, a divisio of Houghto Miffli Compay. 0.. S ø 0. S S ø. S. S ø.9 S appears to be approachig. 0. y 0. 8, r S r r Because, the series has o sum. 9. r Because, the series has o sum. 0., r 8 S r 8 88 Algebra Worked-Out Solutio Key 9

26 Chapter, cotiued., r S r., r S r., r 8 9 S r 8. r 0 9 Because 0, the series has o sum.. r Because, the series has o sum.., r S r., r S r 8. r 8 0 Because, the series has o sum. 9. For this series, r. Because, the series has o sum. 0. 8, r 8 S r 8 8., r 9 a S r., r 9 Because, the series has o sum.., r S r (0. (0. (0.... r ( (0. (0. (0.... r ( (0.0 (0.0 ( r ( (0.00 (0.00 ( r ( (0.0 ( r (0.00 0( r , Copyright by McDougal Littell, a divisio of Houghto Miffli Compay. 9 Algebra Worked-Out Solutio Key

27 Chapter, cotiued (0.0 9(0.0 9( r 9( (0. (0. ( r 0. ( , r If the series has a sum, < <. S r., r If the series has a sum, < <. S r Problem Solvig. d (0.8 ( Copyright by McDougal Littell, a divisio of Houghto Miffli Compay.. C; (0.0 8( (0. 9(0. 9(0.... r 9( Sample aswer: a. Choose r. S r ` The first series is i i. b. Choose r. S r The first series is ` i i. r The perso swigs a total distace of 0 feet. 8. S 0,000 0,000(0.88 0,000( D; r 0, ø,9,. The maimum amout of profit the compay ca make is $,9,.. S (0.8 ( r 0.8 ø 89.8 The compay will ship a total of about.9 billio cassettes. 0. Distace: 0, r 0 0 S d r 0 S d 0 Time:, r S t r S t 0. Because both series have fiite sums, Archilles catches up to the tortoise i secods after 0 feet is traveled. Algebra Worked-Out Solutio Key 9

28 Chapter, cotiued. a. The ball bouces, or feet betwee the first ad secod bouce, ad.., or 9 feet betwee the secod ad third bouce. b., r 0. r (0. Series: ` i (0. i c. S r 0. 8 The ball travels a total distace of 8 8 feet. d. (0.h, r 0. Total distace distace from each bouce origial distace r h (0.h 0. h h h h If the ball is dropped from a distace of h feet, it travels a total distace of h feet.. a., a, r r ` ` b. This aswer meas that evetually, oe square uit of area will be removed from the triagle, so o area will remai. Mied Review. P(A or B P(A P(B P(A ad B P(B P(B P(B %. P(A or B P(A P(B P(A ad B 0. P(A P(A. P(A or B P(A P(B P(A ad B P(A ad B P(A ad B 0.. d, ( d ( (. d 8, ( d ( ( d 8, a 8 ( d a 8 (8 d ( ( ( d, a 8 ( d a ( d 8 ( ( ( 0. d., a 9 ( d a ( d 9 (. ( ( d., a 9 ( d a 9 (9 d 8(. ( (.... r.,. r, a 8 r r r a r (. 8 ( (. (. r 0., a 0. r a r 0. (0. 0,8 0,8(0. Copyright by McDougal Littell, a divisio of Houghto Miffli Compay. 9 Algebra Worked-Out Solutio Key

29 Chapter, cotiued Copyright by McDougal Littell, a divisio of Houghto Miffli Compay.. r,. r 0., a 0 r r r a r ( 0 (0. 9 0,0 a 8 9 ( 0,0 8 (0.. r, 8 r r 8 ( ( Lesso. Ivestigatig Algebra Activity. (p. 8. Sequece: ; 8; 9; ; 90;,8; 0,9;,9, The sequece is geometric with a commo ratio of.., d. 8, r 8. Arithmetic sequeces: d Geometric sequeces: r. Guided Practice (pp a a 8 a 8. a 0 0.a 0 0.( 8 a 0. 0.( a 0.( ( a 0 a 0 a a. a ( a ( ( a ( 9., r r p. 9, d 9 d., d d 8., r 08 r p 9. a p a a p, a, ( ( 0. 0,000, The umber of members stabilizes at about, members.. f(, 0 f( 0 f( f( f( f( f( ( ( (. f(, 0 f( 0 f( f( f( f( f( ( ( (. Eercises (pp Skill Practice. The repeated compositio of a fuctio with itself is called iteratio.. A eplicit rule gives the value based o the positio of the term i the sequece. A recursive rule gives the value based o the previous term i the sequece. Algebra Worked-Out Solutio Key 9

30 Chapter, cotiued. a a 0 a 0. a 0 a 0 ( 8 a (8 a ( (. a a a. a 0 a 0 a a 9. a a a 8,0 8. a 0 (a a ( 0 0 (a 0 0 ( 0 0, 9. a ( 8 ( a ( 0 8 a ( 8 0. a 0 a a 0 a a. a a p p p a p 8 a p 8 p 08. A; a a p p p a p., d d., r r., r r., d 8 d., r r 8., a a 9 a, a, 9., d d 0., a a p p a, a, p., a 9 a a, a 9,. The previous terms must be defied first., a,. The rule does ot work for all the terms i the sequece., a,. f(, 0 f( 0 f( f( f( f( f(0 ( ( ( f(, 0 f( 0 f( f( f( f( f( ( ( ( Copyright by McDougal Littell, a divisio of Houghto Miffli Compay. 9 Algebra Worked-Out Solutio Key

31 Chapter, cotiued Copyright by McDougal Littell, a divisio of Houghto Miffli Compay.. g(, 0 g( 0 g( g( g( g( g( ( ( (. f(, 0 f( 0 f( f( f( f( f( ( ( ( 8. f(, 0 f( 0 f( f( f( f(9 f( ( (9 ( 9 9. h(, 0 h( 0 h( h( h( h( h( ( ( ( 0. f(, 0 f( 0 f( f( f( f( f(9 ( ( (9 9. f(, 0 f( 0 f( f( f( f( f(. g(, 0 g( 0 g( g( g(8 ( ( (8 ( g( g(08 (08 (08 0,08. C; f(, 0 f( 0 f( f( f( f( f( ( ( (., a 8 a 8 a a 0, a 8,., a ( a (a a (, a, (. a Ï Ï Ï a Ï Ï a Ï, Ï., a a a a 9, a, 8. 8, a a a a 8, a, 9., a (a ( a a (, a, ( 0. Sample aswer:, a,, First eight terms:,,,,, 0,, 8. Sample aswer: Because ad f( f(, the f( f(, f( f(, ad so o. So, if there is a fuctio f ad a iitial value 0 such that the first two iterates are equal, the all of the iterates must be equal. Algebra Worked-Out Solutio Key 9

32 Chapter, cotiued. a. a ( 8 a 8 9 (9 0 a 0 a a ( 8 a a 8 a 8 a a 9 a 8 0 a 9 b. Sample aswer: If, the first te terms are,,,,,,,,,. If, the first te terms are,,,,,,,,,. If 0, the first te terms are 0,, 8, 9, 0,, 8,,,. The terms of the sequece will always evetually repeat the umbers,,. Problem Solvig. a... Fish at start of year 0.8 p Fish at start of year New fish added , The first five terms are 000, 00, 00, 80,. At the begiig of the fifth year, there will be fish i the lake. b. Over time, the populatio approaches 00 fish. Amout of chlorie at start of week 0. p Amout of chlorie at start of week New chlorie added 0. p, 0. Over time, the amout of chlorie i the pool approaches 0 ouces. Curret balace Previous.0 p balace Paymet.0 p ,.0 00 It will take Gladys moths to pay off her credit card bill. Because., the balace at the begiig of the th moth is $.. So, she will be able to pay off the balace at the ed of the th moth.. f Ï Ï Ï Ï f Ï Ï Ï Ï f Ï Ï Ï Ï f Ï Ï Ï Ï. a. f Ï Ï Ï Ï Curret amout 0.0 p Previous amout New dose 0.0 p 0 0, b. The maiteace level of the drug is milligrams. c. The ew recursive rule would be (0, 0. (0, or 0, The ew maiteace level would be doubled as well, to milligrams. Curret Previous 8. a..08 p Amout balace balace withdraw.08 p 0,000 b..08 0,000 0, ,000 a.08 If a 0 0, the 9 0 0, , a 0 ø 9,. (use calculator. You should have at least $9,. i your accout whe you retire. Mied Review 9. a b c 0. a b c 9 Ï 8 Ï Ï Ï. a b c Ï 8 9. / Ï. ( / Ï ( 9. / / Ï 8 Copyright by McDougal Littell, a divisio of Houghto Miffli Compay. 98 Algebra Worked-Out Solutio Key

33 Chapter, cotiued Quiz.. (p. 8., r S S r 8. r., r S S r Because, the series has o sum (0. (0. (0.... r ( a a a a 0. a a a,., a p a Copyright by McDougal Littell, a divisio of Houghto Miffli Compay (0.0 9(0.0 9( r 9( (0.00 ( r ,000 a a 0 0 = a 8 8. a 0 (a 0 a ( (a = (, a p a 8 p, a, p. 8 a a a 8 8,. f(, 0 f( 0 f( f( f( f( f( ( ( (. g(, 0 f( 0 f( f( f( f(9 f( ( (9 ( 9 9. f(, 0 f( 0 f( f( f( f( f( ( ( (. f(, 0 f( 0 f( f( f( f( f( ( ( ( 9 Algebra Worked-Out Solutio Key 99

34 Chapter, cotiued. h(, 0 f( 0 f( f( f( f( f(9 ( ( ( f(, 0 0 f( 0 f( f( f(0 f( f( (0 ( ( 90 9., r 0.8. Graphig method: Y= : Mi u( 0.98u( 0 u( Mi,000 WINDOW : Mi Xmi 0 Ymi,000 Ma 00 Xma 00 Yma 0,000 PlotStart Xscl 0 Yscl 000 PlotStep 0 S r S 0.8 The pedulum swigs iches. Problem Solvig Workshop. (p. 8. Let L be the limit of the sequece. L 0.L 00 0.L 00 L 00 The sequece approaches 00.. Let L be the limit of the sequece. L 0.8L 0.L L ø 8.8 The sequece approaches Graphig method: Y= : Mi u( 0.9u( 00 u( Mi 0,000 WINDOW : Mi Xmi 0 Ymi 000 Ma Xma 00 Yma,000 PlotStart Xscl 0 Yscl 000 PlotStep =0 X=0 Y=000. Algebraic method: 0,000, L 0.9L L 00 L,000 With each method, the umber of members approaches,000. =0 X=0 Y=99.9 Algebraic method:,000, L 0.98L 0 0.0L 0 L,00 With each method, the umber of books i the library approaches,00.. Each year, % of the books are lost or discarded, so 98% of the books remai. The coefficiet of should be 0.98, ot 0.0.,000, Let L be the limit of the sequece. The: L 0.98L 0 0.0L 0 L,00. Sample aswer: Save a pey o day oe, ad icrease each day s savigs by a pey. 0.0, 0.0. Etesio (p. 8. Prove: i (i Basis Step: ( 0 Iductive Step:... (k k... (k ((k k ((k k k (k Therefore,. Prove: i i ( ( i ( (( Basis Step: 0 (i for all positive itegers. Copyright by McDougal Littell, a divisio of Houghto Miffli Compay. 00 Algebra Worked-Out Solutio Key

35 Chapter, cotiued Copyright by McDougal Littell, a divisio of Houghto Miffli Compay. Iductive Step: 9... k k(k (k 9... k (k k(k (k (k k(k (k (k (k [k(k (k ] (k (k k (k (k (k (k ((k ((k Therefore, i ( ( for all positive i itegers.. Prove: i i Basis Step: 0 Iductive Step:... k k... k (k k (k ( k k Therefore,. Prove: i i for all positive itegers. r i r i r Basis Step: r 0 r r Iductive Step: r r... r k r k r r r... r k (k r Therefore, itegers.. Prove: Basis Step: r i r i r i i(i ( 0 r k (k r r rk r rk rk r for all positive Iductive Step:... k(k k k... k(k (k (k k k (k (k k(k (k (k (k (k (k k k k (k Therefore i i(i for all positive itegers.. Prove: (i ( ( i Basis Step: ( p p ( ( p 0 Iductive Step:... (k k(k (k... (k ((k k(k (k ((k k(k (k (k k(k (k (k (k [k(k (k ] (k (k (k (k [(k ][(k ] Therefore, (i ( ( for all positive i itegers.. Series:,,, 8... A recursive formula for the th heagoal umber is H H. Prove: i ( i Basis Step: p 0 ( p Algebra Worked-Out Solutio Key 0

36 Chapter, cotiued Iductive Step: 9... k k(k 9... k (k k(k (k k k (k (k (k [(k ] Therefore, the th heagoal umber is give by ( for all positive itegers. 8. Prove: f f f... f f Basis Step: f 0 f 0 Iductive Step: f f... f k f k f f... f k f k f k f k f k f (k Therefore, the sum of the first Fiboacci umbers is f for all postive itegers. Mied Review of Problem Solvig (p. 88. a. ((0..8, r 0. r.8(0. Distace ` i.8(0. i b. S r.8 S 0. The ball travels a total of 8 feet.. a. Number of ew braches:,,, 8,, b. The sequece is geometric with a commo ratio of. c. Eplicit rule:, r r Recursive rule:, r r,. f( 8, 0 f( 0 f( f( f( f( f(8 8 ( Sample aswer: Let ad d. Eplicit rule: ( d ( ( Recursive rule: d,. Sample aswer: If < r <, the r gets closer ad closer to zero as icreases, ad the sum of the series approaches the value r. S r r ø 0 r r If r < or r >, the r gets further away from zero as icreases, ad the sum does ot approach a certai umber. If r, the sum does ot eist because the value is udefied. S r r 0 0 Udefied. The legth would be fiite because the commo ratio is 0.9, which is less tha., r 0.9 S r S The legth of the sprig would be 0 iches.. a.. ø 0. The mothly iterest rate is approimately 0.%. Amout owed ( iterest(curret balace Paymet Recursive rule: 0,000,.00 9 b. After moths, you will owe about $8.. c. 0,000;.00 If you pay $ a moth, the loa will be repaid i moths. d. Sample aswer: Yes, it is beeficial to pay the etra $0 each moth. This causes the balace to decrease faster, so you do t have to pay as much iterest. 8. Number of trees 0.9 p Previous umber of trees New trees , Over a eteded period of time, there will be about 000 trees o the farm. Copyright by McDougal Littell, a divisio of Houghto Miffli Compay. 0 Algebra Worked-Out Solutio Key

37 Chapter, cotiued Copyright by McDougal Littell, a divisio of Houghto Miffli Compay. 9. Sample aswer: S r Let S ad choose r. So, ` i sum of. i is a ifiite geometric series with a Chapter Review (pp The values i the rage of a sequece are called the terms of the sequece.. A sequece is arithmetic if the differece betwee cosecutive terms is costat.. A eplicit rule gives as a fuctio of the term s positio umber i the sequece.. I a geometric sequece, the ratio of ay term to the previous term is costat.. ( ( ( ( ( ( ( 8. (0 i (0 p (0 p i (0 p (0 p (0 p ( ( (0 ( 0 (. i (special formula i ( 8. k ( ( (special formula k ( ( p 9. 8, d 8 ( d 8 ( ( 0. a 8, d ( d a 8 (8 d ( ( (., 9 ( d 9 0d ( d d 9 0( 9 ( d 9 ( ( d d. (, ( S S S 8. (, a ( S S a ( S 0. (, a ( S S a S (, 0 8 8(0 S S S Algebra Worked-Out Solutio Key 0

38 Chapter, cotiued. 00, d ( d ( ( 00 After moths, Joe will have paid 00 dollars toward the computer.., r r 8. a 00, r r a r 00 ( 0 0( 9., r r r 9 r r or 0., r S r r S,8. 8, r S 9 r9 r S , r S r r S. 0, r S r r S , r 8 S r S 8 8., r S S r. r. Because r, the series has o sum.. 0., r 0. S r S (0. 8(0. 8(0.... r 8( (0.00 (0.00 ( r ( (0.0 8( r 00 8( Copyright by McDougal Littell, a divisio of Houghto Miffli Compay. 0 Algebra Worked-Out Solutio Key

39 Chapter, cotiued Copyright by McDougal Littell, a divisio of Houghto Miffli Compay (0.0 8( r ( a 9 9 a a a (8 0 a (0 00 ( a ( a ( a ( ( 8 a (8 0., r 8 r,. The th term is greater tha the previous term. d,., d d, 8. Curret populatio Previous.0 p populatio.0,000,.0 Chapter Test (p. 8. a 9 a 9 Arithmetic; the commo differece is. a. a a a Geometric; the commo ratio is. a. a 0 0 a 0 a 8 a Geometric; the commo ratio is.. Neither, there is o commo ratio or commo differece... p a a p p 89 0 p 8 a 9 a p 8 a 0 a p. a ( 0 a (0 00 (00 00 a (00 00 a a (00,00 8. a a a a a 9 9. Net term: 9, d ( d ( ( 0. Net term: 8, r r ( Algebra Worked-Out Solutio Key 0

40 Chapter, cotiued. Net term: 0 ( a 0 ( 8 (. Net term: 8..( a..(.8.(. 8 ( 8(8. i i 8. ( ( 8(8 ((8. ( 9, 0 (0 9 S S S (, 9 (9 S S S , r S r r S r r S , r S r r S r r S , r S S r , r 0 S S r (0. (0. (0.... r ( (0.0 (0.0 ( r ( (0.00 8(0.00 8( r 8( (0.0 ( r 00 ( a a 0 a a,, Copyright by McDougal Littell, a divisio of Houghto Miffli Compay. 0 Algebra Worked-Out Solutio Key

41 Chapter, cotiued. a a, 8. a 9 a These values are the same as the values i the table, so this rule defies the sequece represeted by the checkerboard quilts.. 0, r r 0 There are a 0 actors i the fifth roud of auditios. After this roud, three actors will remai, which is eactly the umber required for the three mai parts i the play. So, the rule makes sese for. Copyright by McDougal Littell, a divisio of Houghto Miffli Compay., 9. f(, 0 f( 0 f( f ( f( f( f (8 ( ( ( f( 8, 0 f( 0 f( f ( f( f( f ( 8 ( 8 ( 8 (. f(, 0 f( 0 f( f ( f( f( f ( (. a. represets the umber of rows ad colums. b. represets the umber of blue squares c. [ ( ] [ ( ] a [ ( ] [ ( ] [ ( ] 8 a [ ( ] a [ ( ] 8 a [ ( ] a 8 8 [ (8 ] Stadardized Test Preparatio (p. 8. B; (. A; S S S 0 Stadardized Test Practice (pp C; S r Basketball:., r 0.. S 0.. Baseball:, r 0.0 S 0. ø 8. Differece C; 0, r r 0, 0.. D; The sequece is,, 9,,,.... There is o commo differece or commo ratio, so this sequece is either geometric or arithmetic.. B; 0 a 8 Algebra Worked-Out Solutio Key 0

42 Chapter, cotiued. D;, r S r r S 8 r8 r S 8 8. B; p a p p 0 p (. D;, ( Sum i(i i 8., a 8 r r r a r 8 r 8 r r 8 r r r ( a 9. The sequece is geometric., r r ( a 8 ( ( i i (0.0 (0.0 ( r ( f(, 0 f( 0 f( f( f( f( f( ( ( ( a (9 8 a (0. a (8 ( a ( ( 9 a 8 ( 99. ( 9, ( 9 S S 9 9. d. First roud 0 people Secod roud 0 p 0 people Third roud 0 p 0 p 0 people th roud 0 people If 00 millio people already received the , the you are i the 8th roud because millio.. d (, d (( 0 d (( d (( d (( 9 d (( d 8 (8(8 0 There is o commo ratio or commo differece betwee cosecutive terms, so the sequece is either geometric or arithmetic. 8. This situatio should be represeted by a series. Sample aswer: The compay is pledgig a total amout, ot just a sequece of values. 9. a. The sequece is arithmetic because each radius is. meters greater tha the previous radius. b.., d. ( d. ( (..8. c. a 8.8.(8.0 The track meets the requiremet because the curve radius of lae 8 is.0 meters, which is less tha the 0-meter maimum. Copyright by McDougal Littell, a divisio of Houghto Miffli Compay. 08 Algebra Worked-Out Solutio Key

43 Chapter, cotiued 0. a. Curret balace Iterest p Previous balace Paymet 9. y p 0. y 8 y y.00 p 00,000, b. 8 ø,9. Mark will owe $,9. at the begiig of the 8th moth. c. Mark will pay the loa off i 9 moths, with the last paymet beig about $ d. If the paymets are $0, the Mark ca have the loa paid off i moths. He will pay less overall because the balace is less each moth ad the balace is paid off faster, so less iterest is paid.. y. y y y Cumulative Review, Chs. (pp y. y y y. (8 ( 8. (8 (( 8 Copyright by McDougal Littell, a divisio of Houghto Miffli Compay.. y 8. y y y. y ( (. y ( y. y Ï 8. y Ï y y y ( ( (0 0 (0. y a a a 8 y 8 Whe 8: y y a 9 a a y Whe 8: y y a 8 a y Whe 8: y (8 Algebra Worked-Out Solutio Key 09

44 Chapter, cotiued 0. y a a 9 a 8 8 y 8 8 Whe 8: y ( y y Vertices: (, 0, (, 0 Co-vertices: (0,, (0, y y 0 Trasverse ais: Vertical Asymptotes: y 0, y 0 Vertices: (0, 0, (0, 0. ( y ( (y 0 Verte: (, 0 9!. 9 P (9! 9!! 0!. P (!!!,0!. C! p!!. C! p 0!!! 8. P(A or B P(A P(B P(A ad B P(A ad B P(A ad B P(A ad B P(A p P(B A P(A ad B 0. p P(A ad B P(A p P(B p P(B 0.8 P(B y y y. Mea: ø.8 Media: Mode: Rage: 9 8 Std. Dev: (9.8 Î (.8... (.8 ø.0. Mea: ø.8 Media: 8 Mode: 8 Rage: Std. Dev: (.8 Î ( (.8 ø.8. Mea: ø. Media: Mode: Rage: 8 0 Std. Dev: (. Î (0.... (09. ø Mea: Media: Mode: Rage: ( Std. Dev: Î... ø.9. Mea: ø 88.8 Media: 9 Mode: oe Rage: 99 8 Std. Dev: ( Î ( ( ø 8.9 Copyright by McDougal Littell, a divisio of Houghto Miffli Compay. 0 Algebra Worked-Out Solutio Key

45 Copyright by McDougal Littell, a divisio of Houghto Miffli Compay. Chapter, cotiued. Mea: ø 8. Media: Mode: Rage: 88 8 Std. Dev: (8 8. Î ( ( 8. 9 ø.. i ( ( ( ( ( ( i 8. ( i ( (... i ( ( 0 9., r S r r S ø.9 0., r S r S.. Eplicit rule:, d ( ( d ( ( Recursive rule: d,. Eplicit rule:, d ( d ( ( Recursive rule: d,. Eplicit rule:, r r ( Recursive rule:. bags of cookies y y caledars 00 y Whe y 00: ( Bags of cookies If you sell 00 caledars, you will eed to sell 00 bags of cookies to meet your goal.. V π, h r V πr h π πr (r π πr (r r r 0 r r Possible zeros:,,, 9,,,,, Test r : 0 Caledars is ot a zero. Test r : is a zero. r r (r (r 9r Because r 9r does ot have ay real roots, r is the oly zero. The radius of the coe should be iches.. I, R. I Î P R Î P. 89 P. 88. P The hair dryer cosumes 88. watts of power. r, Algebra Worked-Out Solutio Key

46 Chapter, cotiued. 8,00(0.8,, r 0.8 r,( ,( ø 0.8 log 0.09 ø ( log 0.8 ø ø The car will be worth $8000 i about years. 8. Let the ceter of the circle be (0, 0 with radius 0, or r. y r y y, Whe y : y (0, (0, 0 (0, (0,,,000 0 Ï Distace betwee (0 Ï, ad (0 Ï, : d Ï (0 Ï (0 Ï ( Ï (0 Ï 0 Ï ø. The rope is about feet log. 9. Let evets A, B, C, ad D be choosig item, item, item, ad item, respectively, so that all items are differet. P(A p P(B p P(C p P(D 0 0 p 9 0 p 8 0 p 0,80 0,000 ø 0. The probability that all four people order a differet item is about 0., or.%. 0. a. Mea: 0,900 0,00 9,800 0,00 99,900,00 9,00 ø $0,9. Media: $0,900 Mode: oe Rage: $,00 Std. Dev.: (0,900 0,9. s Î... (9,00 0,9. ø $9.8 b. Commissios: $0,900(0.0 $0,09 $0,00(0.0 $0,0 $9,800(0.0 $990 $0,00(0.0 $0, $99,900(0.0 $999 $,00(0.0 $0, $9,00(0.0 $90 Mea: 0,09 0, , 999 0, 90 ø $0,.8 Media: $0,09 Mode: oe Rage: $0 Std. Dev.: (0,09 0,.8 s Ï... (90 0,.8 ø $.99 c. Each statistical measure i part (b is % of the correspodig statistical measure i part (a..,000, d 00 ( d,000 ( (00 9,00 00 a 9 9,00 00(9,800 I the ith year, the accoutat s salary will be $,800. Copyright by McDougal Littell, a divisio of Houghto Miffli Compay. Algebra Worked-Out Solutio Key