Finite mathematics 8e lial

lial_22828_IRG_TTL_CPY.qxd 4/26/04 9:41 AM Page INSTRUCTOR’S RESOURCE GUIDE AND SOLUTIONS MANUAL to accompany FINITE MATHEMATICS Eighth Edition Lial • Greenwell • Ritchey lial_22828_IRG_TTL_CPY.qxd 4/26/04 9:41 AM Page This work is protected by United States copyright laws and is provided solely for the use of instructors in teaching their courses and assessing student learning Dissemination or sale of any part of this work (including on the World Wide Web) will destroy the integrity of the work and is not permitted The work and materials from it should never be made available to students except by instructors using the accompanying text in their classes All recipients of this work are expected to abide by these restrictions and to honor the intended pedagogical purposes and the needs of other instructors who rely on these materials Reproduced by Pearson Addison-Wesley from supplied electronic files Copyright © 2005 Pearson Education, Inc Publishing as Pearson Addison-Wesley, 75 Arlington Street, Boston, MA 02116 All rights reserved No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publisher Printed in the United States of America ISBN 0-321-22828-6 VHG 08 07 06 05 04 .4390398 KLQWV IRU WHDFKLQJ ILQLWH PDWKHPDWLFV = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = {lll SUHWHVWV = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = DQVZHUV WR SUHWHVWV = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = ILQDO H[DPLQDWLRQV = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = < DQVZHUV WR ILQDO H[DPLQDWLRQV = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 5: VROXWLRQV WR HYHQQXPEHUHG H[HUFLVHV = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 66 FKDSWHU U DOJHEUD UHIHUHQFH U14 Sro|qrpldov = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 66 U15 Idfwrulqj= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =67 U16 Udwlrqdo H{suhvvlrqv = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 67 U17 Htxdwlrqv = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 69 U18 Lqhtxdolwlhv = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =6; 719 H{srqhqwv= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =76 U1: Udglfdov = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =78 FKDSWHU OLQHDU IXQFWLRQV 633 Fkdswhu 44 Wkxv/ sod|hu D vkrxog fkrrvh urz zlwk sured0 : < = Sod|hu E elolw| 49 dqg urz zlwk suredelolw| 49 < vkrxog fkrrvh froxpq zlwk suredelolw| 49 dqg : froxpq zlwk suredelolw| 49 = Wkh ydoxh ri wkh jdph lv d44 d55  d45 d54 7+7,  8+6, @ d44  d54  d45 d55 7   +7, @ = 49  91 4 43  43  +4, 44 @ >  +4,  43 48 s5 @  s4 @ = 48 s4 @ Iru sod|hu E/ wkh rswlpxp vwudwhj| lv 43  ; @ >  +4,  43 48 : t5 @  t4 @ = 48 t4 @ Wkh ydoxh ri wkh jdph lv 95 9+43,  5+4, @ =  +4,  43 48 ;1 9 6  4 Wkh jdph lv vwulfwo| ghwhuplqhg vlqfh lw kdv d vdg0 goh srlqw1 Wkxv/ sxuh vwudwhjlhv fdq eh xvhg1 Wkh ydoxh ri wkh jdph lv 67 > zklfk lv wkh vdggoh srlqw dqg rffxuv dw wkh vwudwhj| +4> 5,= % &  45 451 :  67 ; Wkhuh duh qr vdggoh srlqwv1 Iru sod|hu D/ wkh rs0 wlpxp vwudwhj| lv s4 @ Wkhuh duh qr vdggoh srlqwv1 Iru sod|hu D/ wkh rs0 wlpxp vwudwhj| lv   431  Wkhuh duh qr vdggoh srlqwv1 Iru sod|hu D/ wkh rs0 wlpxp vwudwhj| lv 37 @ > 379.3 s5 @  s4 @ = s4 @ Iru sod|hu E/ wkh rswlpxp vwudwhj| lv 39 @ > 379.3 t5 @  t4 @ = t4 @ Wkh ydoxh ri wkh jdph lv 45 3+3,  9+7, @ = 379.3 JDPH WKHRU\  67  :;  46 6< ;   @ > @ 9:  9:  45  :;  56  67 57 5; = 9: Iru sod|hu E/ wkh rswlpxp vwudwhj| lv s5 @  s4 @ t4 @  67  56  4: 67 45   @ 9: @ 9: > :  57 5  ;  7 66 = 9: Wkh ydoxh ri wkh jdph lv      45  67  56 :;  57   @ 9: @ 9: = :  57 5  ;  7   ; 4; 471 7 t5 @  t4 @ Wkh jdph lv vwulfwo| ghwhuplqhg vlqfh wkhuh lv d vdggoh srlqw1 Wkh ydoxh ri wkh jdph lv ;/ zklfk lv wkh vdggoh srlqw dqg rffxuv dw wkh vwudwhj| +4> 4,=   4 491 5 Froxpq grplqdwhv froxpq 5/ vr uhpryh froxpq 51 Wklv jlyhv wkh sd|r pdwul{   4 = 5 Iru sod|hu D/ wkh rswlpxp vwudwhj| lv  +5, @ >  +5,  +4, s5 @  s4 @ = Iru sod|hu E/ wkh rswlpxp vwudwhj| lv s4 @ t4 @  +4, @ >  +5,  +4, t5 @ +froxpq zdv uhpryhg,/ t6 @  +t4 t5 , @ = Vhfwlrq 4415 Pl{hg Vwudwhjlhv Wkh ydoxh ri wkh jdph lv 634 Wkh ydoxh ri wkh jdph lv 6+3,  +4,+5, @ =  +5,  +4, 6 4 4;1 ; 5 Urz grplqdwhv urz 6/ vr uhpryh urz 61 Wklv jlyhv wkh pdwul{   4 = ; Iru sod|hu D/ wkh rswlpxp vwudwhj| lv 6; @ > 45 4  ;  : s5 @  s4 @ > 45 s4 @ s6 @ +urz zdv uhpryhg,1 < 4+3,  4+ zkhuh d55  d54 > d44  d54  d45 d55 d44  d45 s5 @ = d44  d54  d45 d55   t Wkh rswlpxp vwudwhj| iru sod|hu E lv > zkhuh t5 s4 @ d55  d45 > d44  d54  d45 d55 d44  d54 = t5 @ d44  d54  d45 d55 Iru sod|hu E/ wkh rswlpxp vwudwhj| lv 69 @ > 4  ;  6 t5 @  t4 @ = Wkh ydoxh ri wkh jdph lv t4 @ t4 @ 4: 4+6,  9+;, @ = 4  ;  6 ; 6 531 4 48 : < Uhpryh urz +zklfk lv grplqdwhg e| urz 6, dqg froxpq +zklfk lv grplqdwhg e| froxpq 6,1 Wklv jlyhv wkh pdwul{   4 = < Iru sod|hu D/ wkh rswlpxp vwudwhj| lv s4 @ +urz zdv uhpryhg,/ 3< < @ > 44 4  <  s6 @  +s4 s5 , @ = 44 Iru sod|hu E/ wkh rswlpxp vwudwhj| lv s5 @ t4 @ +froxpq zdv uhpryhg,/ 34 t5 @ @ > 44 4  <  43 t6 @  +t4 t5 , @ = 44 Wkh ydoxh ri wkh jdph lv DPE=     d44 d45  t4 DPE @ s4 s5 d54 d55 t5  @ s4 d44 s5 d54 s4 d45 s5 d55    t4 t5 DPE @ ^s4 d44 t4 s5 d54 t4 s4 d45 t5 s5 d55 t5 ` Vlqfh wklv lv d  pdwul{/ zh zloo gurs wkh pd0 wul{ qrwdwlrq1 d55  d54 d55  d45 DPE @  d44  d44 d54 d45 d55 d44 d54 d45 d55 d55  d54 d44  d45  d54  d44 d54 d45 d55 d44 d54 d45 d55 d44  d54 d55  d54 d45  d44 d54 d45 d55 d44 d54 d45 d55 d44  d45 d44  d54 d55  d44 d54 d45 d55 d44 d54 d45 d55 d44 +d55  d54 ,+d55  d45 , @ +d44  d54  d45 d55 ,5 d54 +d44  d45 ,+d55  d45 , +d44  d54  d45 d55 ,5 d45 +d55  d54 ,+d44  d54 , +d44  d54  d45 d55 ,5 d55 +d44  d45 ,+d44  d54 , +d44  d54  d45 d55 ,5 635 Fkdswhu 44 @ +d55  d45 ,^d44 +d55  d54 , d54 +d44  d45 ,` +d44  d54  d45 d55 ,5 +d44  d54 ,^d45 +d55  d54 , d55 +d44  d45 ,` +d44  d54  d45 d55 ,5 @ +d55  d45 ,+d44 d55  d54 d45 , +d44  d54  d45 d55 ,5 +d44  d54 ,+d45 d54 d55 d44 , +d44  d54  d45 d55 ,5 +d44 d55  d54 d45 ,+d55  d45 d44  d54 , +d44  d54  d45 d55 ,5 d44 d55  d54 d45 @ d44  d54  d45 d55 @ Wklv lv wkh h{suhvvlrq iru wkh ydoxh ri d qrq0vwulfwo|0 ghwhuplqhg jdph wkdw lv jlyhq lq wkh wh{w1 s4 d45 s5 d55 d55  d54 d44  d45  d45  d55 g g d45 d55  d45 d54 d44 d55  d45 d55 @ g g d45 d55  d45 d54 d44 d55  d45 d55 @ g d44 d55  d45 d54 @ g @ @ j/ wkhq DP @ ^j j`/ zkhuh j lv wkh ydoxh ri wkh jdph1   t4 eh wkh rswlpxp vwudwhj| iru +e, Ohw E @ t5 sod|hu E1 Wkxv/ t4 @ 591 Ohw  P@ d44 d54 d45 d55  eh wkh sd|r pdwul{ ri wkh jdph1 +d, Ohw D @ ^s4 s5 ` eh wkh rswlpxp vwudwhj| iru sod|hu D1 Wkxv/ s4 @ d55  d54 d44  d45 dqg s5 @ g g zkhuh g @ d44  d54  d45 d55 =  DP @ ^s4 s5 ` d44 d54 @ ^s4 d44 s5 d54 @ ^j j` d45 d55  s4 d45 s5 d55 ` Vlqfh s4 d44 s5 d54 @ @ @ @ d55  d54 d44  d45  d44  d54 g g d44 d55  d44 d54 d44 d54  d45 d54 g g d44 d55  d44 d54 d44 d54  d45 d54 g d44 d55  d45 d54 g @j dqg JDPH WKHRU\ d55  d45 d44  d54 dqg t5 @ g g zkhuh g @ d44  d54  d45 d55 =    t4 d44 d45 PE @ d54 d55 t5   d44 t4 d45 t5 @ d54 t4 d55 t5   j @ j Vlqfh d44 t4 d45 t5 d55  d45 d44  d54 d45  g g d44 d55  d44 d45 d44 d45  d45 d54 @ g g d44 d55  d44 d45 d44 d45  d45 d54 @ g d44 d55  d45 d54 @j @ g dqg @ d44  d54 t4 d55 t5 d55  d45 d44  d54 d55  g g d54 d55  d45 d54 d44 d55  d54 d55 @ g g d54 d55  d45 d54 d44 d55  d54 d55 @ g d44 d55  d45 d54 @ j> @ g   j / zkhuh j lv wkh ydoxh ri wkh wkhq PE @ j jdph1 @ d54  Vhfwlrq 4415  5;1 Pl{hg Vwudwhjlhv 4  +d, H4 @ 4s4 6+4  s4 , @ s4  6s4 @  7s4 H5 @ 8s4 4+4  s4 , @ 8s4  s4 @ 7s4 H6 @ 4s4 5+4  s4 , @ s4  5s4 @  s4 +e, 636 651 +d, Vwudlq   =9 =7 Phglflqh +e, Wkhuh lv qr vdggoh srlqw1 Wkh rswlpxp vwudw0 hj| iru suhvfulelqj phglflqh lv 43 43 @ @ @ > =9   =7 4=5 45 s5 @  s4 @ = s4 @ +t4 dqg t5 duh qrw ri lqwhuhvw khuh1, Wklv phdqv wkdw Gu1 Jrhghnhu vkrxog suhvfuleh phglflqh derxw ri wkh wlph dqg phglflqh derxw ri wkh wlph1 Wkh uhvxow +ydoxh, zloo eh =9 =9+4,  =7+3, @ @ @ > =9   =7 4=5 45 zklfk lqglfdwhv dq hhfwlyhqhvv ri 83(1   4 671 +d, 4 +e, Wkhuh lv qr vdggoh srlqw1 Iru sod|hu D/ wkh rswlpxp vwudwhj| lv +f, Iurp wkh judsk/ s4 @ 47 pd{lpl}hv wkh plql0 pxp h{shfwhg ydoxh wkh urz sod|hu uhfhlyhv1 631 Frpshwlwru3 v Vwudwhj| 7=< 7=:8   Erhlqj*v 7=< 7 Vwudwhj| 7=:8 Wkhuh lv qr vdggoh srlqw1 Wkh rswlpxp vwudwhj| iru sulflqj lv 35 5 @ @ > 7   ; s5 @  s4 @ = s4 @ +t4 dqg t5 duh qrw ri lqwhuhvw khuh1, Wklv phdqv wkdw Erhlqj*v sulfh vwudwhj| vkrxog eh wr dlp iru wkh '71< ploolrq surw 47 ri wkh wlph dqg wkh '71:8 ploolrq surw 67 ri wkh wlph1 Wkh ydoxh lv 7+3,  5+5, 7 @ @ > 7   ; zklfk phdqv wklv vwudwhj| zloo lqfuhdvh Erhlqj*v surw e| 45 ploolrq grooduv1 4  +4, @ @ >  +4,  +4, s5 @  s4 @ = Iru sod|hu E/ wkh rswlpxp vwudwhj| lv s4 @ 4  +4, @ @ >  +4,  +4, t5 @  t4 @ = Wkh ydoxh ri wkh jdph lv t4 @ 4+4,  +4,+4, @ @ 3>  +4,  +4, vr wklv lv d idlu jdph1 691 +d, Wkh sod|r pdwul{ iru wkh jdph lv Qxpehu ri I lqjhuv Qxpehu ri I lqjhuv   3 5 = 5 +e, Iru sod|hu D/ wkh rswlpxp vwudwhj| lv  +5, @ @ >  +5,  +5, ; s5 @  s4 @ = s4 @ 637 Fkdswhu 44 Iru sod|hu E/ wkh rswlpxp vwudwhj| lv  +5, @ @ >  +5,  +5, ; t5 @  t4 @ = t4 @ JDPH WKHRU\ Wklv olqhdu surjudpplqj sureohp fdq eh vroyhg e| wkh judsklfdo phwkrg1 Judsk wkh ihdvleoh uh0 jlrq1 Erwk sod|huv vkrxog fkrrvh qjhuv zlwk sured0 elolw| 67 dqg qjhuv zlwk suredelolw| 47 = Wkh ydoxh ri wkh jdph lv 3+7,  +5,+5, 7 @ @ =  +5,  +5, ; 4416  51 Jdph Wkhru| dqg Olqhdu Surjudpplqj   @   s5 @ j| @ @  H4 @ 8s4 dqg H5 @ 5s4 6s5 = H4 @ 8s4 j H5 @ 5s4 6s5  j=  s4 j s4 j Ohw sod|hu E fkrrvh froxpq zlwk suredelolw| t4 dqg froxpq zlwk suredelolw| t5 = Wkhq Ohw j uhsuhvhqw wkh pd{lpxp ri wkh h{shfwhg jdlqv/ vr wkdw H4 @ 8t4 5t5  j H5 @ 6t5  j=   4   s5 j  Zh fdq uhzulwh wkh sureohp dv wkh iroorzlqj olqhdu surjudpplqj sureohp= z @{.| 8{ 4 5{ 6|  {  3> |  3= t4 j     4= s5 s4 dqg | @ = j j Plqlpl}h vxemhfw wr= = Glylglqj e| j |lhogv Glylglqj e| j |lhogv  > H4 @ 8t4 5t5 6t5 = dqg H5 @ Ohw j uhsuhvhqw wkh plqlpxp ri wkh h{shfwhg jdlqv/ vr wkdw Ohw { @ s4 @ j{ @ Ohw sod|hu D fkrrvh urz zlwk suredelolw| s4 dqg urz zlwk suredelolw| s5 = Wkhq zlwk Wkh fruqhu srlqwv duh + 48 > 48 , dqg + 45 > 3,= Wkh plq0 lpxp ydoxh ri z @ { | lv 58 dw + 48 > 48 ,= Wkxv/ wkh ydoxh ri wkh jdph lv j @ z4 @ 85 = Wkh rswlpxp vwudwhj| iru D lv Ohw { @ t5 j t5 j  4   4= t4 t5 dqg | @ = j j Zh qrz kdyh wkh olqhdu surjudpplqj sureohp1 Pd{lpl}h vxemhfw wr= zlwk } @{.| 8{ 5|  6|  {  3> |  3= ... lial_ 22828_IRG_TTL_CPY.qxd 4/26/04 9:41 AM Page This work is protected by United States copyright