ࡱ> M bjbj==  WW#l8L4 !:RL"$D" d$   ,&$F ՜nN T 0 !$$3.11 Mechanics of Materials F01 Exam #1 Solutions : Friday 10/05/01 (*show all of your work / calculations to get as much credit as possible) 1. Calculate the magnitude of the forces in members CD, DE, and DF, of the truss shown below indicating whether each member is in tension or compression (10 pts total)  1.A. METHOD OF JOINTS : 1. Draw a free-body diagram of the entire truss (*shown below) 2. Determine support reactions using the equations of static equilibrium SFx=0, SFy=0, SMxy=0 (CCW+, CW-) : By definition : REx=0 kips SFx=0= RAx+REx-2 RAx=2 kips SMA=0= -3(4.5)-5(4.5)(2)-2(4.5)(3)+2(3)+REy(4.5)(4) SMA=0= -13.5-45-27+6+ REy18 SMA=0= -79.5+ REy18 REy=79.5/18 REy=4.417 kips SFy=0=-3-5-2+REy+RAy= -10+4.417+RAy RAy=5.583 kips (3 pts for determining reaction forces) 3. Identify a joint where you know the maximum amount of forces (e.g. a support with two members). Start at Joint E. 4. Draw a free-body diagram of the joint and determine whether forces are compressive or tensile. We know that there has to be a force in the downwards y-direction counteracting the upwards reaction force at the roller. Hence, FDE must be in compression, which also indictaes that FFE must be in tension to counteract FDEx. 5. Write and solve equations of static equilibrium* for diagram drawn in step 4. From geometry : FDEx=FDEcosa=0.832FDE FDEy=FDEsina=0.5546FDE where : a=33.69 SFy=0=4.417-FDEy=4.417-0.5546FDE FDE=7.963 kips (COMPRESSION) Ans. (3 pts for determining FDE) 6. Move to an adjacent joint and repeat steps 4-5 until entire truss is solved Move to Joint D and choose directions arbitrarily. From geometry : FDEx=FDEcosa=0.832FDE=6.625 kips FDEy=FDEsina=0.5546FDE=4.4162 kips FFDx=FFDcosf=0.4472FFD FFDy=FFDsinf= 0.8943FFD FCDx=FCDcosa= 0.832FCD FCDy=FCDsina=0.5546FCD SFx=0=-2-FDEx+FCDx+FFDx=-2-6.625+0.832FCD+0.4472FFD SFx=0=-8.625+0.832FCD+0.4472FFD (1) SFy=0=-2-FCDy+FFDy+FDEy=-2-0.5546FCD+0.8943FFD+4.4162 SFy=0=2.4162-0.5546FCD+0.8943FFD (2) Hence, we have two equations and two unknowns : Solve eq. (1) for FCD=8.625-0.4472FFD/0.832=10.366-0.5375FFD Plug into eq. (1) into eq. (2) and solve for FCD: 0=2.4162-0.5546[10.366-0.5375FFD]+0.8943FFD 0=2.4162-5.749+0.298FFD+0.8943FFD 0=-3.3328+1.1923FFD FFD=2.79 kips (COMPRESSION) Ans. Plug into eq. (1) : 0=-8.625+0.832FCD+0.4472(2.79) FCD=8.866 kips (COMPRESSION) Ans. (4 pts for determining FFD and FCD) 2. The stress state for a linear elastic isotropic material is shown on the following page (E=45kPa and n=0.35). a) What is the magnitude and direction of the largest normal strain? b) What is the magnitude of the largest shear strain? What plane does it take place in? (10 pts)  2.A. The stress pseudovector is :  EMBED Equation.DSMT4  Hooke's Law for a linear elastic, isotropic material under multiaxial loading in matrix form is :  EMBED Equation.DSMT4  which corresponds to the following set of six equations :  EMBED Equation.DSMT4  Substituting in the given numerical values yields :  EMBED Equation.DSMT4  a) The largest normal strain is ex=-0.23 (compressive). Ans. b) The largest shear strain is gxz=-0.24 Ans. 3. Calculate the force needed to compress the steel bar shown below to Lo/2 if the bar is simultaneously heated up by 50oC (E=200GPa, aL=12(10-6 /oC, Lo=0.5m, Ao=0.05m2). (10 pts) 3A. s=F/Ao=Ees( es=F/EAo eTOTAL=eTHERMAL+es=+aLDT-F/EAo eTOTAL=DL/Lo=-0.5Lo/Lo=-0.5 -0.5=+aLDT- F/EAo solve for : F=[0.5+aLDT]EAo substitute in numerical values : F=[0.5+(12(10-6/oC*50 oC)]200GPa*0.05m2 F=5.006GN Ans. 4. Define the following words in one to two sentences : (2pts each=10 pts) a) statically indeterminate : a structure which has more variables than number of equations of static equilibrium; hence, additional equations are needed to solve the entire structure (i.e. geometric compatibility, constitutive eqs., etc.) b) membrane stresses : tensile stresses in the wall of a thin walled spherical pressure vessel tangential to the curved surface of the vessel, sf=sq c) thermal shock : failure or fracture of a materials or structure due to stresses induced by thermal expansion d) hydrostatic stress : a stress state with three normal, equal stresses and no shear stresses, s=sx=sy=sz, txy=txz=tyz=0 e) moment lever arm : the perpendicular distance from the point of interest to the line of action of the applied force PAGE  PAGE 1  D(6789;@ASTVWYZ $Dֵּzrif`fif`f[if`fEHaJ EHH*aJ aJCJOJQJaJOJQJaJo(6OJQJ]aJ5OJQJ\aJjOJQJUaJ 5<>*OJPJQJ\^JaJ5>*OJQJ\aJ CJOJQJ jU OJQJaJ5B*CJOJQJ\ph5CJOJQJ\6CJOJQJ]5CJ,OJQJ\5>*CJ,OJQJ\# D78:;TVFJL&^`^$a$hDFLjnr$&(*,TX^`bd|: õH*OJQJaJjOJQJUaJ5B*CJOJQJ\ph5OJQJ\aJH*aJ EHH*aJ CJOJQJaJ 5H*\aJ 5\aJaJ OJQJaJ<OJQJaJ@;<  : |   n p^p``^      ( : < > @ T Z v z | ~   . 4 8 < B F R ͽ CJOJQJ <OJQJjOJQJU5B*CJH*OJQJ\ph5B*CJOJQJ\ph5OJQJ\aJ56\]aJ 5H*\aJ 5\aJ EHH*aJ CJOJQJaJ OJQJaJH*aJaJ OJQJaJH*OJQJaJ2R V d l n p v z ~ "(*:>@BHLPVXhlnprtv 5H*\aJ 5\aJ EHH*aJ CJOJQJaJ5CJOJQJ\aJ OJQJaJ OJQJaJH*aJMn @np Z$CefP ^` "$&4:>DHNdhx|'*02@BWYhkux #$35BCDF`def׽5CJOJQJ\56\]aJ 5H*\aJ 5\aJ OJQJaJ EHH*aJ aJCJOJQJaJH*aJ 6]aJFf~NP`dfrVWnopqs·¯£wmj5EHU\jg? CJUVaJj5EHU\j g? CJUVaJj5EHU\jL g? CJUVaJj5U\jCJOJQJU5\5CJOJQJ\5CJOJQJ\ OJQJaJ5B*CJH*OJQJ\ph5B*CJOJQJ\ph'PbdhUVrs:6t08xz~,. zsl CJOJQJ CJOJQJ5B*CJOJQJ\ph j5CJOJQJ\5CJOJQJ\5CJH*OJQJ\5CJH*OJQJ\5CJOJQJ\j5CJOJQJU\56\]aJH*OJQJ5\j5U\j4 5EHU\j_oh? CJUVaJ) "&(2468BDFTVXZ^`bdprtv^`dhjltx)5B*CJOJQJ\ph56\]aJ5CJOJQJ\CJH*OJQJ jCJOJQJ5\ jCJOJQJCJH*OJQJ CJOJQJ CJOJQJCJH*OJQJ?)hqrs~&`#$)D. 0ghiopqstz{|}~0JmHnHu0J j0JUCJH*CJH*OJQJ CJOJQJ CJOJQJ5CJOJQJ\0 1h/ =!"#$%`!*F bd߄ԕZ]@[@MLx t|ܹ3ZA K  ZB&(QЄ5hAX%J|T$ e7]a<7;wٙ%c &f?5bBeq cϥ5kװh ,`M5dtւejJzjj&w|߮[ۦ ڹк4t2t?p™C!dWG"qo^YwF称Z~<,D6r G<#M}!h84=kϴSDO!4Mca,^y?}}Q묆I}߱Blt" s,hA]=kCƏc{P`=cV{eL'>@Xǵ_k:D*/^qyq=+1nЋ 68@'ɇWd8t銟'YP$N~߂TW)()F1(6-|[w$99 '?AS$O?ypKNR_Mp;.ZIBf },&7Ww$)/|_ 'ɃS|HIUp+d#Foߣpc%DT^Lo$ N2~I?N2~I/T28I9|_l߬$UU? '9JUL4PRpm N(~4dO<I/Uj8I|7IvPpjj ' ԿP_+pv N)NR_G$s*Y8  N~IQA81cN"lԿԿ]Lo$ Ne^V =mbWɸR9߰|רbtX n UW'|_ 0O1S{Wpߧ!OK'| [F K~qpg" WJ~1p ߯C5[o{>&|8K|yVG~=\"|? 0'!7l pko@τw!G>'|[la7"|yK6;"I~pCߏB> nPLRBJ#_?poJ#?1?3p-QyVRȯB(\?apc OF(|6B/!GτRw'G~sB kC(#\&?~pCO(PyBJ#{P;p cD5#GN(>Rȳr +$:]{v=ݽN; džf|kcFwF'aboCqyx1&MGhwFa;xZc}M)|S29Jsc]W'u>h@?ЏM 5hNuu=ڿϯׯu֠bP ;RK(f{DLC:KcN.C(' >{D x;'"#=H}Y'y~6՝y~Q\HR\oAhAq6ݕvwv>NvAOEG{/ŮaMP-#B#"Uυ{GJ‹"oD}`7al4v!b߫⟲un$=e/E/h/OB<=3yB~3+mDU/SBv8=r z>$}}:lj?]~{Cy>0*Vݳ"jF1cONۏVDv֭[#|N?ƷB33\lc}N?c ٟ"xe{pPMJ+z|EbsC[(習+av@кLp9w#G\n/b{Q+Gdc!3X%kď2=A.#9 2֛=S}nFg͎h]^n>\W7Sj#$n]xG,?kI:)臺^O ~~Qs_P=ObyќKǽS4lX4#|(W{g{ms(xlFƠ#6*Y˧*s]͓x;iМil~a%z]>dc|7b}:cu?`qP5ԍ\OYИ}"N dX1nkbXc2bc[OГ|Pcf,Xiz74f͟]\33vv fkn^smkε=sl&?91|q0֙W1(3_2syyOn:/_@:Ondn3 UAssЍGЅW& fԋ 'vt~綾jIBb+u Ť}Xf4/=Uя`A\xYIoLG4iIF[q=8-mEPVT5 i0 isUJi]]nE.x<t)|v2fR>%AQ])\TSGQt8Eل2EQVdi%NVU-")5:vSxǎݤTF>m2(N[3UVɖRIDŁPJ87qlfN;08#.b=oWdGQrSȯc.TW1!XC>;\)01R<_QK}5W9l\y8r.J3^!u!r(踆lmh0P+;K3>ʢ9S񛠯SN8ߜ8hl0y+\_ukt0k)fT3jȑ#FIIɑ;wlܸqɓ7֮]{r-QKF4_/P*J]g25鐕<> >GIg"g̷q\51kϤǟ8~ƗVg/82dn'>O,1ۣ߯?^QjsE2pEpERnqWy[Ju \QƵrwkJI+י<%<:(~ZUU VIu^gBu禎y^8 NMxB %ԚBJ3ږ/֌A,`YSV\QRII5A}l)w|tZweq!RzX pF>kkWx#4xZ+Ox:]|lO;?wmh>Iω E|cf >#-Q>xwFgsۤi~Qwzj]}\<{Wұy:L kҬg!N'-6 $&L/C꙾_w _o^T_7'![Ih N 4-"2ʼnN{V9 nz&9w;k|_NerĠv۽r!cdEGް!QTKzP?'vMsq&NNr*W0I) +p,Z|R7Iph.M4s Cy⊤bZһgx /Ig<%ycmRWJ.rʟ ?EU;tpL73 ntp~>.T.:gSە-uDyl;noGv&GaRwmlU\ҟ9:.Z ӝOi1]7*W{|;ߜaB+eGhQue]uLZ)2]'GG(\6O&A\fq ѯ¿Rxϫ^ 6%A\q`;%j?gxۼoŏK9hg?E36oے uZx/WIڏfbm^=!@A!1G?Jxx6o=$ R,b}'z7;$j?gyYY7 RLï+ P @Crsr.F/_* P :z2~$@G3 b+rNI:-WSKJԠR)h!V}1?SAYzre3y/"J9hyXOE$@G3 16/=1r$A\cq{# P z}JܿSϢ:jUH~45l\.rnI7s4#^Iڏf⠣|-[m;rfס_# P $@G3 16o8&Hܿ3b%j?gy={"[͜iSᷕhgwT}{ꦦAN.Iu&Z֬[ĿANjϫ^> L,W]jW,INCKKuU5|Iu&Z+b3ӵϙl#INCKKupی L,XE_ZNZoW^\$:-˵bgJҩRR]smƢGn$:-`փгJҩRR]ClzI$:-"u;m$ihx4ۼF%w$:-\5kWS⥺γ{y%ԙhY.!Gđ:h: INCKKuvێ L,67XoD߀^Nz6ܫ L,RYG?WN MD?dIu&ZkpЇS⥺/%ԙhYۣIҩRR]yMЛ7%w$:-|ֵk'INCKKu]p_߻:_V[If%Uwmˬ_A__@|}/WaLM'?Py{qc ƒKX`0m^2g 執yD ׃|zyﰷe5~+rȷ ͫϺ^C26@MѫW%W|5_y{ׅn|9y}o8tpwz;h執{ފvNvflazukɭ%Pyw BX_D|}E/eN!̇y0ml2Ρg.lƱ>"'7 0w >A7?od z!ׇ|ߛ9NJQ;ç:ǻ*!v- mMC׻Zd}z u-[iA.z(uZ~E֕+#PBx!m@4I:-_H"a^H_>3?'YC?Xz5*u-潂 br(uZ,3gODh!>6o 'ȍD-YD_B ۼN$Jiіu$0Z/k!F(uZ^ߢJ"a^Hi&DuAQ}XDh!^"mEߊvI:-_bzuk%0Z/o+D/_JHCחXc=}lI: mƣǟ@n$Jz`Au-҇Bߛ\I:-_i: =$BF yס_ߘ\I:-_֢ů)PBx!m݋D.YC-ECq I: ŕۉ r_JyhMD$BF l^E(uZ^lfB$BF mdS=)R,F~}80I: ݍ~7=Iyhz_,֙%0Z/hw=5D|ڒu-/Sȝ#R5OA߫ $0Z/U}侑D]oI"a^H65_'RCWX䱎/H"a^HϷ{ )iKyhTGX?u-ͻ>D{-: u-oͻZIܿR֗_$BF l*WƯB$J9,~v?9QI: 8A߃>I3_mWByW!V͜Uϰ^?_By3g"7[͜~tQI: #m&7H͜YwBfI: ]lڠoK$J9-`]r$0Z/6:zujJܿSBa}u,Dh!;~ I#_o1$BF 8G_FI7sYyDh!m\͗D3gE. HI: A?0I7s[te}]$0Zgy1(}|ߓrǩ=Rk#508%;\SLPwS%3wKcݛ"er/7tމ!ߏcy ̓jwX>~FM]S\E}me(+p||[9 ?{g8 Lpt,;-=zŎ/l}$o̱T5IC8kݤ9מHjth&8lۡ?}qR(ʲ %)%)E)KR %,A+P/NY,Ik7zDp}O"gR_gGbzV|mӷySL>}cI1^^o|}jz3ϒ3yE/d$zx| '7ɶy~j?z$9}~ >{7M}}6u']șt0Ǥ)4a:|הmPA/L3DdI^|$l1HULb.u^(B_VW~#Ac>1YY/Cd5ۼ穟GEr&/<1d*1Y+YWҷնy/Q2+L^%04MN g捦8r&7Q1Yߏ~7yݨ߉9;F1i XDߤ%Vm^=z3O_=2ǤTe]2IUFu 29IMsL0Cч %??`ׇz_əcYᛤoO_yM7ſISߘ9&Cmֵ7E6}m^2u2#gӗLsLO\~X~D=X(6q)Z^Cb/D߅9]}I1yb M|Mߢmfz9 șo9˘c"!|0} =I$TșLI0!xMF6u?{#gr/}过9&FY&Yo75nr&-kN3\ | _D_bw9+3D90Gb@=Y),E+|7ߠCd7}5sLރwY7|wȿK{oz%*LVѷיcYGMӷEyөId}e&>~m^3@_g-p37t$3}6u+mș-cr9\RKM.%}_v#q|9/>N"j?7%URK2D(戦lt`?)h SkRiצ 4J6Xk2HRTG_luzj[3ev.sS{%OC7X; pFm=mgwзm6xȬ# n7lRCZbbr+)|6o9d34͛{Xz$rOo"`< hCLv=Cál^o#Z}K/}} =ᥒI%FO:!5=5a)ׄpԆZhK\Sp>KZ2k_'K? >BVP;?GE3 {(׾d ]w%_Nx 67ȾI&z߂l2=-gWзe*CZo(<_wK𥴍wvW;}o68݋0mxzi{3"2wf-o8oea/[kD;;ޗR'l"d|uK~/򴨜2_TI'̕TI6[TJ%7[?~7QG{/$~z/>}G^nG{]2}>[ 5%ZпV3m!^>| YD3,͛鋈Yg&7lf?f@hޣd%k7#l@z r/0:C'[n&s3[Doge-vȵ#ߎpA22COz@ټ} sq6 12&ȏq~:qSk.uwYWU9}O U.'Uݩ\r_vr^!YJUIiܢR]!+N(79F|r+|n^{ NSu5Xt u~]g}+1v`ہ9sḻc;0v`ہ9sḻc;0v`ہ9sḻc;0v`ہ9sḻc;0v`ہ9sḻc;0v`ہ9sḻc;0v`ہ9sḻc;0v`ہ9sḻc;0v`ہ9sḻc;0v`ہswaoݡN煝6'Vlͫtбc/`.)*b9`!juU05[c4*"* 6?xZ]h\E>sgv%i" C oa[QSA(Z H0Jl ZBb R$TMV.ۇS[!zf޻wf{aܝs̙X @vXWxn( ]`ۦH,,ea]^^8tp` S5tPxR(8~: X>q2hzRʊ {'M7xܹf1|c$x+_OBUMßIz}Uoz4>F!~/޲3<У h#?h*zFu\ չE˨υ^ϘPɎb4 SXC7[E)ka?RkT:}n_1NW&''󋋋+++|>>>>f/u4,2{} OR".3_#^z e5BIпBI7K*8GeB=xb-DZhD Se;&oC~aCⰽO4V GĔ ZhVD1)/vZf]4H}iШ)F|MD[ +߽33i6 3˷sw=sg ňBYNTKIq%©8L׈CtQ=cvZCŧ6G05sO&Hh矡):wEP=Cm¤u@]ׄo30u oMr_VhxYK$܊8.T爢C-0]1$_\~U_Z_Y0))? &4 >Yπ {>0x÷ 57zx LhYmIj|g<,$ʂg=<&R)ZoEіPy_` 3ʬQpbb'Cnʅŗ"'[AL{2!˄y$ePwq7 ;gMóFxԀg+3yo2fւ/kT)uh ǻ?7sX8鉆co yT/ӯ\]EwmmJ<;A.rdGێ_ =t/6ꥇ+Jp.@ F\8^f'q6ˇ4+͆1Ά)>[k6d0%<~/W[V;~ ;O ~C_͗9gS4'ʔqUD/;k:b;Z6kJGDʥ׸ԊGH6^|eC~6N8OΈ\gCx՚O=xU> hY|ܞ}_J/ȕK+D:9E'u Jnĸl˾f^VcUcyoCk^_"ڝћvt9>bdeId^vΘǵj_iXOM~*۷zC|'M,.9\]3\{lN3gK<;6&Qأ`R.ONry7n)'a'F[Ƅ\{lN3gK<;6&Qأ`R.ONry7$$HTWf~f~ 13~UiV{/峨WͥjԾի}Z3TٵկVPkfPkfPkfPkfPkfPk,)gjs_ȡC=Q$y`!_cQ-cxxٝf6л(T^CФ@B^ew,t&XJ,H!sR&{>z_s'<2lB,.YxȢҖY,beUb֓ Y$t -,Iv-nyɲR/b)NGh#-R;mtR\ mîW1V[tXbR8ɒ{ps8nU~"+!SJ7[ͷʿ:R tuT/ץ(;MBT> rٌٴN?ʪK7 m5,{eqM f/j}ljqxm'N?Vڶm[ӧT&MD/h$˥ǮWK7ya%'袌k2JT[ puՙ.̙?d ύ=,ύ͕.GutuV!䘸vr^^JZxmNhPƟ\_'نW:u+Gm\Qӹ*qE5ElҤI~E\ƹV\׾sv%NnCu'tAX^qm]~nZt{Hz7IL:t\@gh5׶Kn*':˽=7r3`7HM׏~Uq륻zuX$?n|3K|MO{^:{En*-Z;Z!45QFWWgx9r2:['ɖ߉Տ?XyF:~OSޖ=~o~ TFwRO+KuO-&f[}NWlӢLB>b{ۓ sl3=Sяg_n\Y3w3WKrpWo*㿼!_+Ye%4/W*m@w;ɂu&Z$udJGctHy1Goп?@[A:-K`3&~j+m1[5r ԙhYR\9g >4/WgEL,iA:}P!?͋畆=~_rL,AkV-[>4/Wjc!~#rL,A9eK >4/W*o&#PgeI.RKӤc?͋NJC{r?ԙhYAmo"Sмx^iiߛo%ND˒0X@|y?͋?Ә PgYQ2 >4/Wi7}@r ԙhYRvo#Sмx^iߝwEnA:-Kà}E ~j+dAǒSL,I18eϔ~Ӥ?͋N [D˒~4# >4/WĴm! @%eeE ~j+b7}s3ѲO?8HڇJLFGL,!N;}h^<ٴ^{#DD˒1f7 >4/Wb_\qA:-K38g˔4OCy}? C5o[Dy.Ӿ_On @=,} +?͋畖"Lrjy`,~j+3{ 1ܿA7]; >4/WJ0k 95o45N_ *~ڇJ5LJ/E @=% .X34/Wh-wQrj,^=w >4/W̴o&ܿٳ`% ~j+2훃>.y5o6H?}<8ڇJL gAw$DڇJɦ}[&F͞kį!Sмx^i_YȕA=5M(Sмxk}Oޖ_~\žIrI{w :-U.ngKsmy[w :-/A`+w0rG)h!=w$Q'%%{M6oDn :-U`~92$Dyils$Sʳ &ЏG?VD O4>?\ :-UbDMD w7kւdTA-5 ڦ}e%WNL*6GEkw :-/A2;~Wr')h1WD ۊr)h  :-/AGdTyS'O$Q'%%Oeg'7BL*3HOAO.HNDKK{Eo?vdTm-HNDKKo7^"JdTކnŏ$Q'%%ȒioQ$]>%HNAKO|E%Iԉh x QE}-<_|뵟lYV{Urϱ:sNj0ӏC?8f3;/ K#N6tfӭn47^;$A"}^72&0l}Nw xȴ&̶ۚ*h ]jAMxT'[ֺ.V,rd1SrXX<Dź/FyvEb2ߕ:A#3ʣ @g;2׉ۧ{&M5%ߔ7*U -7W&WUL'Sl fJ0[šrpv^9k;3ԧN)O+?;L0s?dSv.>vZe>z mdf#ٱ.(aRW[BUB3ʋ3alrfn6386o4d03q}F?PCl$Bu7]u!ߕnPzҌ9oI%̵d{CMѪU%Sl5f3[zR/C 2̕a-Q"ֈrzQ`Y\+sysh甂#G9N83Ǚ=Ύcpg}nran/+6䶒VX-͠*+[N~s+aix';lsQ0z"ƓD(xѧ0 r3`~{!;Z^"DItg6ٴ}k6mɵ%ߖ̷VVC]~Mr5b6SҗU _ ̗gO9( %苣Ï\T1SE~>\ |oKZLBH-.VKh9jygzV:Wg! ΂}aA:-G`NOwBA4 OSwG_FA:-G}`RWBA4 OSD?lA:-G}`X1! MBn L>f@ +~A:iji_s-ȵrf5諣Wï*Q4r VӇЃ DD45lڷ}Bral%)O??Yixi#=.Sg z@OOh2k=0u6ZzA]oÿUixZϴz%UrԊz ݎoh:MNكROyr +/! >Fr;al#59aA:ik}Bn Llय़,3uMTiߓO"7ZFQ0xzo^uMԾ};w&EFQ;4o^ DD4i-W!WUFQo6wNA:i۴-<.Ssg ?-A:i};w"[͞O ޤu! kM/_Fn L=觡}uMLӾcǑ/Ssfh4 ?\ L=] 1~#A:i]}ѫ WS͞jz=-Q4JmW w|cxlQ[\}|ԑMjW(sJW=EJ+SokURDmc;J6XFkg7OԏOCj?DPJїD/S|)J3_.ډ.9wGXѸ#qiq#q2ۏ;@}[te߰Gg l߄_gma~ikԯ9י{5љsיC~.so3caN>_'|:sØ0K~r:i[iE5smok׈zc;4f ٣SїE/S|93_MFW0f>=:'!Pj.U7Z{}w?@ߊmmgCӾkENgsk=: aKu^"/2ˤDŸNN'L #臣g '??n7z y`N{hG?z[|ɷc=L n{]}E :Wb&+RǢqtTbGtءcŦ ~7~&`߿o7sb'|!G?fwCyw=:bL~1s0Ĵ9П= 茂'@_ O27Ѧ}?(9G ;wHsbw}Mi}*p un& sUjWzqt3Wx xg=2s=~ar:٣ v_S;nӾ7o$̿eU%:_2旛ͤ> 69YdE范cG!?q̏7L=}Pr:CcIn]ѻt%ߍMS7GoߒN 3ߌ=:5:}5:Wg5MJQB/_NiJ1_=:! P{fUy0 R(Qcc(Gأ߃_g}>Ӿԛѷo%ob*XI}9 +[żf7z. ?0ҙOcN2tOBOI"ߝdSLZSFoߖNZ3ߊ=:u6}-:f.^9̕c,{tEoAĤ ,䣘2o3;3~"r0otNƤq"_s/c mކCr:ۙY ! $bn5aӾ /"̿)4Su"4sSƴo懳G'OEOI%߃2kG=~r:kѩ ߆~+mog.L*QWB 29|E8@u%;MNO/ϸihܿC_}S r:01Gg gu0LS/F 9W[TgџyTL~}hr:{'أ _7>=|_ӾNԝ;w!әNwdNCh@_N} k|#(׈N# )b6]'Q |Fv33K>PoNn;mc 9[MF'L>Rn ?%=hIѳl%.wA]YFLGEpqsU:;"%-tV!wYB}9zֲq֪\:q)w)>A S;.Kd8{8\pz8ƥ:ť@2t$D8 o&5o9p0`ߎOw$N>e~|khkr]k̾o}Kz9^_RK{% śJf*Y/3>f0 .1~,Xrcɏen ah!ӇeKqtߍ\7ݘ| ZCdmL#fpi_ոqTǯN:Uc*TpEȸF3f6rj_GzC/Ñ ;d~Kv{%^0v:X&d &3k]o}+W[I~%s+_yxmt2:3?̿Nz6U9CX 1 [\wz~;j_bARG9mާ[*\oe&;x9>Soz@J~YnN\yUw)w#ƕG\qG&_\/_h~ۭG?F{WgiU{yS)w}]#OHs]OI\OIϻ:z?G{cxc%yucxk4:\ïXp,r 9 q,OnqIﺜ-~݋?Wx X.^#u~LrEG{Jh\׵q~Scqx,swvQM|sN"7Gors߭I܍nj&e$b\DyF;<ϓu#Wud#| D-0~?|>_/ѿZxdKӾ'=#$So"Ͽi߯Q2Ge(s~)'O&P[<'=aڗG Α9Gs?Ϟ\)/Gjgr(';zA{mm%V2{vA!G "Øд3o7dw3= }D1ϴ;-2`߲ |'SGGA{ߛ? ~! _C9 GyQcja5;KN"s)N1=g O9~?W9>屁Ux%_ X\vA";50f%`J c {B3A~KA{M:wxȴ#ێv̷gO(S:w"Psci_*} $u'ӝlw3̞H)={D1ôA0?d3ן $Psc!Ӿa鐆7PC|{a O@?Bli80o4dG371 yx ja̎7Og! g{g3<Ňǟ&Pscg" |yд^$3΋̽yfOG?tff0|Aek&530{5K32|^Gg7߉'ٝbv71F o;lgnGg+lߌ3lav+|`;xyλ̽y ѯE_Zf1me!\ A^C53AfCڴo W,%*sK=:eE].WLfSƛCf.Y9f~{tf‹/yٙ0˴o4L:1gSѧO%;/>M7d:茇qc%;q̎ }#G$8Y̍`~8{t2`}:z:~:tf1MQ{̣duanG!x~ @d2 æ}{&ӇNoz1ߓ=:= >=?l 3}]u%ӍNW0ߙ=:#}v`#i_Kxȴ&ӊ̷`NshF)~SMilshawxwΝ|c4 7 ۀ6Ʀ}k!s+Y:f{tjB 񫓭L fkB-{_9^\gMxn?kZQp͒~Gm;jGmq^ 7O:D@Qt:F~(g຿r0'9O v>+RJ}ĵT{}-!X";ܶ8ӣ[])ˡ}Pkhcqc\Mp5˺m:,ny!\xD/uG٣݃ӷ[W5R tuT) .UB?5?8NWJU\-E믽d(Koߌw7n܏gϞvĉ?cm۶5>}J*mҤI$e-Z,u c/9GemMKI1s[s}8 1{X+9\菱\OZ,8""䘸vr?oku+h=W`w^r>Rw; RkǟMړ/߻R&|lk]Ww]/+ tVs0m ,B\wMDG}JQO/0>&`ׯ斾y8GEE~ixʩO ˀ{{<;<|+OpYҏBfOeҞφ,}|[vG9OيvjoYqD N=`FXZY:YjZZ-yB~b{ۓv>#dxL)X7xα9֕sOF?f{e~)ryg܍~_/]W>3B& d}l4^3Ѳ(_΀4?͋9 }3ѲG]ڇJLV3ѲL_^y,Ӿ)S! @%=m01G}h^<4Ĵ'zO^z ԙhYRЛ7}h^<´.z]z ԙhYK/.Sмx^i݁$' ԙhYdpԑ)qÎi!ڇJAoL,k7o}h^<Ŵo Ƚ.PgeI9sƟ%Sмx^ii݋#3ѲiC>4/Wf}3Ѳ>[o!Sмx^i_C53Ѳˢ/-Sмx^i_4z4\ @% NHүR@:.M >4/W:); PgeI OCy}o:A:-Kz`K >4/WZhGLLcR'@g]VFGɠ&Sмx^ii=(PgeI ߏOCy}w߉L,J+}h^4/W:m/O"oA:-Kc?.Sмx^Ӿܻu&Zb/}h^3ۄ 3-5ol4XI}92ڇJLKn @= &O@?Ny$Ӿ44trjjН> =?AyRi_+VɵAmZ5k>4/WcW,~9rj)cEoADM. >4/o\v#=y[]\žIrI{w :-U.ngKsmy[w :-/A`+w0rG)h!=w$Q'%%{M6oDn :-U`~92$Dyils$Sʳ &ЏG?VD O4>?\ :-UbDMD w7kւdTA-5 ڦ}e%WNL*6GEkw :-/A2;~Wr')h1WD ۊr)h  :-/AGdTyS'O$Q'%%Oeg'7BL*3HOAO.HNDKK{Eo?vdTm-HNDKKo7^"JdTކnŏ$Q'%%ȒioQ$]>%HNAKO|E%Iԉh x QE}-<_|뵟lYV{Urϱ:sNj0ӏC?8f3;/ K#N6tfӭn47^;$A"}^72&0l}Nw xȴ&̶ۚ*h ]jAMxT'[ֺ.V,rd1SrXX<Dź/FyvE-EX,77=>>+uN)징GgGgϡt #^2wdO)MkJ)sMobUB[oƯL2+N̔`;C1ȵ*rvN9c; gOR ^oV~F&wa3Ȗ];?!8K}˴}7Ff7c]P2/!-gg,flgg&q0m h2Ɏaf,L>(27!0vQIn]B+sݠY}shߒ\K-k| 4PUëJ*jTg9$Z)(_\e+|i[/DrzAɳ)V>))G#sqf3{(`/v^ ^oW6m-[m%oZ33Z DD4ui`C r 軡w"Q4r VӇЃ DD45lڷ}Bral%)O??Yixi#=.Sg z@OOh2k=0u6ZzA]oÿUixZϴz%UrԊz ݎoh:MNكROyr +/! >Fr;al#59aA:ik}Bn Llय़,3uMTiߓO"7ZFQ0xzo^uMԾ};w&EFQ;4o^ DD4i-W!WUFQo6wNA:i۴-<.Ssg ?-A:i};w"[͞O ޤu! kM/_Fn L=觡}uMLӾcǑ/Ssfh4 ?\ L=] 1~#A:i]}ѫ WS͞jz=-Q4J)07N7JiݤvR >>tYk2%jF_{q1F[[%uJύ>Zjxed!Z/~^|~0{tA4C}I:%ɗb4e u']wӑs2`α!qCGG↡e +w|aL }#&99'sr{^K^ YfV#Ԯ&bJbըjo-j?ol\'jߣGP ]PAr6y3{swknr]z>g;z;\wH(##O??_q?. Qr7d}n|]vRX¾"tbK-%7y fA~:=!0}AB>׋&׋B0{MkJ:su5 jNAQ" /J⦾ٹ@ }q>vY%]"Ѣ\}7op&>/n)8~qt 'ɝ"Էy/|In/˹/C,ǿJ ̳>3rJ`İ.?0M}}p/>A%`oν9Zoe b^ 6>A-rAk#( %GK$R9BύO\N#x>WYu5L jH(fw&YHe?4 Trgɟ30pߏ~`?DU'[M~/9<őK`d}8`Fm gOA.|=І5VېkK.s]#W|ze {i!W|9S_^'G./<Bv w+\b\VL}wwx.{EX ۫"\`?< .+!я&wAzaZkk#S_̽)z3|7GP+s.ߏ\=sftTKh_/km;Śv{Phzo:%1ϯSQӧ#ȧOYIOn>$%8b'SOEGI}2 =\sI]xkVL+xUDx*9nA;n7!p;v )ɜ! g!oVّ¾}?K!B"▀J dVMՐ7WsıCoyυ90Fp6 0Ɍ ;FAZ_V/pa@knZid@[H뫣;Cz꒯ VJ7Ȕ&[j-/˞=|K>,`f+;^ ?GVH]qง~=uWr8Y>">m>m$E|Zski)q0AnCU:|:(NcDkm Jy e<^!s7pMK;uTND?$N?` $qی &$[ﻠuR@_o䖓_Kar6Ē!/Z eA?P0Ah]tەLn|Ҝz |- ל|3h ϭ ?U W\3/C1OnhNW'v! ?~/j>3>CZDZjmo}sM_9Vb0 #; d'o,X Htu $30SL}sgh;l3Jt5<g8Srem %x[B~)=`DWWp_ 1y]aȾ֣ó:rȯgl&ѓ$30L}wh;ӳvItu7/H fadwh{9%zWG%30GL}gS i<&wzRDWq?~Ab0 #{-5*xJz];%30M}o/h7{ snIt6;J fado?h<y%Sgg'>#+duX\Yfupj$(gHS0'o4hC鉁a]}H3pSD 0mqxǑG~<=`DWcOB,1y5}> f3 r3Ϥg|*?C#1ym[̾O.Bzb.}`Fvo-X'o"DYk%z guɰ m+xJ~=ɰ];Sb0 #52Xc+?yHf)(bZrgnprza|Jm˟ɷ{{ zo7Gi%73o{3 Vw߭ioa\(x5^g/zø+sŚ`[N^og}+'B|HX,Tg6\V?Pݕj75[WE5\d\$5\'N9WpI'cGE?(cd S_ s ~< K!^}={!%ٽԗ̜oG'qOBO›Df ٭Էy Z< ֒[C~5=DXŞ==oUdao"x,&Bz}>'l<,4b)xE~&=0}iN#3 i<m"X&O`|>Xxǒ8oaAg8^0r1#!샹Fw0!da/9-O$^ApaA_>ۇL_ OOB@;tc7L7S_;vht+hO@kV[ۊLkm1sc&x kL=Ѐ>'ӀlChd dD'&@5T}OdC SZ4ubO:Hgoǽz;ȴ':17Ck^Asr7G Msooc2MCSS_mhu+C6Z&{ @@dj ZUS =rU#*~BL}eˠS,2K#(%}%Sl)( s'~ݿ?5cSj&U٠^o+jm|EmY>6 ggojXw,懶^o/y/6ſgpHo | wǏSׯtkKWKq|?f/cW_W̏ pW|=9ݿ8_x[80ޞokzfNGy9 7`~f:hC g m;ϻG9kj;y?i,Yi@~sG_K͛[ۘ?b&`! 벢dN =$' xLU=ACL[(ĊD F6W;D(ծlvvfԄԊilM67 +ںuރsiy y3|?>{~;wya1ߤ6_iP5e^Ke;ۖ‹n l"~9鬁&ޯtg­b>4"LҩB$vα)&{q蓣W_$yR]&Q?]L9]s{aCGㅇJ]-U -}Hb}a|wOĵ\@|tN=)N%J0  සiIpwE d`Z'O8!0={|;f^U^`Z>L+%ƩEf:8/_BK#iXgī/JZ"(ڤ_[>b /v;~%?wËjW?hI~Uۥe_b3#zv#G"ZGwn;fƮ(+="c/;vw3Ѻcz8~9/;sݟKRB㷆o o_ w;_ Waor&?7(]6dx&L0dx7oݼU^"$ð%^I`[F2 vLtlX ٫hEgiGxM69{2 ^/DTV!':B})C6]2p&P8|?"m&-Y,aQ;. F !'2Hc2<c Rx | zLkU ^ƀqV uw8`ZLi_/eJ~Rcnm_^Ƣӭ Ycӽ!e >3SNw7`ئGq+#U|˂L2*rd9"~$w 7Ւ _{!l%KJ3K&~JSW-d9"~$w 7Ւ _{!l%K HiQ$B"85~T쿙{xTYOD,mVӒBbibYw'U=0y^}sjW,gGoԣW n0zMC<`ǒc]߷ۯ%T/?h_Pۺ 4[]hYp:'ߊ"ۜ!3 Ã`Z>L{>˞ vrg6orG/qy`Zu`Z.0VV? |? |S=1_q* 2qJ(.^ፋ=@Q\1秏P"?Sˤ9tS9jl`$Ka%bU$7dx#F$[`& 2 OG)F φ VH2v)\a/[E2|#`[I7n;@2v iᝰ;>&.}qE(WV؉1^X{Ň.?gߧr'?mRO' 秹q\|C_~[=!V.ByΣj}$o؞/F{u`aC0{<]|SMsɫD# 85I(>8#3VeeE^(^d"Oi*[B(z˙f S 9uD*Sa((.Qa ?p5oAJ!8hxPyb7&wxF6- +SX8܂rDd$p0  # A20cܑJe q&`!i0cܑJe Ö . pGH7xڭWMlE~3];c 8D*$.=T(r]$vRYp W8P U%H[Ԑ5T?13]ڱ+{fޛ,.a}\W*f3#&uc0$/,vJϋ~ sdA{6zV^Lfw8a(] 3(."R!W?pn<3B*^.3(h;"f43tu;0ufኢ:.r Iߨs>iV 9 9k;ZP Yx/5׽O$a[–Z'gnq -aU6(9ٌ <Ė*1B]W1[$_C0&vAnYL.GgfD禇3Y̹EMk %Ʌ[ZЪȞ?%1x=hqd?w͈A|vJ},n5r:7~ge4jc_yG ׵-hQ,J`iEp.JYhrLgyxo݋݋'{_mə>I,,sog+SY%y2xݥye B59J4o(<1*S>C?dXJ5r$u5/)*$"T> cLY0LB|q*Q#Ӹ/@U- ti Ձ: rUQ W3JR!E '] G)$Ӡ Wаk޶7::>zt8Tڄdn &SP?\]A:UP7IqAUT {A dpgB\^Yn?ur*L9Dd H0  # A2 E1*1$~ן? `! E1*1$~ן @ WxڵVMLA~3[8hD 4BF$g(h+gjp$3p%ں!uqWA~Dv5톟Q7R] +jmva.x:Ǩy÷ki-X g~W]zyXEұé;R02#.ރv8I|B`/gf#S~ mKa_hwX}P0Z60?žh1zp __x$/j!(mꚘ L=|A3E[ ?"N7NaI C@/hF.p5H8[>@)U:r=ܯp%w,SgL+Vr@ڪS姈 ԫ5]PW){_`*"|DdHJ  C A? "2xtkix `!xtkiҠ@.@ `GxڵXoE3ڋ#)AB4Z\[vv׻^,N =$"8j%AUPHAS \KSRRAV.HXNؙ{}100~$0B Ngy>aD{&=íB^aNЄ9: ^Uj[ZC 3<̤g2&Nt>G|[Z]oNQpzqi&+̌7a = i 1 TԃN&22[۷eeq EwMfM͙)+\ffԬTTs} _\gyH =n& 9=b.sdFϒSR]t-,MS,  !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxz{|}~Root Entry FnNData yWordDocument ObjectPoolsnNnN_1063717708FsnNsnNOle ObjInfoEquation Native   !"#$%&'()*+.123456789:;<=>?@ABCDEFGIJKLMNOQRSTV DSMT4WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT Extra!/ED/APG_APAPAE%B_AC_AE*_HA@AHA*_D_E_E_A  s x s y s z t yz t xz t xy {}=="-825"-2"-41{}۬ DSMT4WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT Extra!/ED/APG_APAPAE%B_AC_AE*_HA@AHA*_D_E_E_A  e x _1063718122 FsnNsnNOle  ObjInfo Equation Native  e y e z g yz g xz g xy {}==1/E1"-n"-n000.1"-n000..1000...2(1++n)00....2(1++n)0.....2(1++n)[]s x s y s z t yz t xz t xy {}0 DSMT4WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT Extra!/ED/AP_1063718600FsnNsnNOle ObjInfo Equation Native LG_APAPAE%B_AC_AE*_HA@AHA*_D_E_E_A  e x ==s x /E"-ns y /E"-ns z /Ee y =="-ns x /E++s y /E"-ns z /Ee z =="-ns x /E"-ns y /E++s z /Eg yz ==2(1++n)t yz /Eg xz ==2(1++n)t xz /Eg xy ==2(1++n)t xy /E FMathType 4.0 Equation MathType EFEquation.DSMT49q_1063808863FsnN nNOle ,CompObj-iObjInfo/Equation Native 01Table$SummaryInformation(HDocumentSummaryInformation8P,@6M6GxDSMT4WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT Extra!/ED/APG_APAPAE%B_AC_AE*_HA@AHA*_D_E_E_A  e x =="-8/45"-0.35*2/45"-0.35*5/45=="-0.177"-0.0155"-0.038=="-0.2305e y =="-0.35*"-8/45++2/45"-0.35*5/45=="-0.06++0.044"-0.0388==0.0548e z =="-0.35*"-8/45"-0.35*2/45++5/45==0.0622"-0.015++0.111==0.1583g yz ==2(1++0.35)*"-2/45=="-0.12g xz ==2(1++0.35)*"-4/45=="-0.24g xy ==2(1++0.35)*1/45==0.06+pÔ(QUfZ95M+zII1tu,!|C9Uc2C1O/:b}nc 꺧fu`d(ѣ[dbAfź|&BT#BnVQ+M]h}I>CHRv+l^y^}遼/}/|V*1ȯbks< e  Oh+'0 , H T ` lxTHINGS TO DO WORK :HINChristine OrtizhrihriNormalnChristine Ortiz40iMicrosoft Word 9.0@]@CM@&M@rPONz! ՜.+,0 hp|  MITY THINGS TO DO WORK : Title  FMicrosoft Word Document MSWordDocWord.Document.89q i8@8 NormalCJ_HaJmH sH tH <A@< Default Paragraph Font, @, Footer  !&)@& Page Number R            RF D78:;TV#$5@Au1Xo01efgNr<n % G H l m % } * + G H / ] ^ _  - L h z (;+2#.9:S0000000000000000000000000000000000000000000000000000000000000000000000000000000000000@00@0@0@0@0@0@0@00@00@000@0@00000000000000000@0@0@0 0 D R f)!n P " + C E R:::: !!4ef2$F bd߄ԕ2"F2$qg}ANc!.:$TU2$x&a K.Y#2$kl'/uߒ(2$juU05[c4*y2$9|N͵ě<lmn(2$Gן`3 -(2$f{ģp̧;2$$`QIș]2$|VӒԙ@K~2$Adz"axr 2$cK>v:< 2$QH.65@#;<S33333333333333333333333333Christine Ortiz<C:\Christine\Teaching\3.11F01\Exams\Exam1\Exam1Solutions.docChristine OrtizRC:\WINDOWS\Application Data\Microsoft\Word\AutoRecovery save of Exam1Solutions.asdChristine Ortiz<C:\Christine\Teaching\3.11F01\Exams\Exam1\Exam1Solutions.docChristine Ortiz<C:\Christine\Teaching\3.11F01\Exams\Exam1\Exam1Solutions.docChristine OrtizRC:\WINDOWS\Application Data\Microsoft\Word\AutoRecovery save of Exam1Solutions.asdChristine Ortiz<C:\Christine\Teaching\3.11F01\Exams\Exam1\Exam1Solutions.docChristine OrtizRC:\WINDOWS\Application Data\Microsoft\Word\AutoRecovery save of Exam1Solutions.asdChristine Ortiz<C:\Christine\Teaching\3.11F01\Exams\Exam1\Exam1Solutions.docChristine OrtizRC:\WINDOWS\Application Data\Microsoft\Word\AutoRecovery save of Exam1Solutions.asdChristine OrtizRC:\WINDOWS\Application Data\Microsoft\Word\AutoRecovery save of Exam1Solutions.asd@   R`@``@` ` `` @``(@```4@``<@UnknownG:Times New Roman5Symbol3& :Arial9GaramondI& : ?Arial Unicode MS"qh`$Zg4Zo,Z(z!!20dY2QTHINGS TO DO WORK :Christine OrtizChristine OrtizCompObjUj