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Blank and Steven G. Kran tz" }{TEXT 258 55 "\n\nSection 6.5\nApplications of the Exponential Fu nction\n" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 4 "Load" }}{PARA 0 "" 0 "" {TEXT -1 93 "\nEach execution group of this section must be entered before continuing with the worksheet. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "with(plots): #Ignore any warnings!" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name chang ecoords has been redefined\n" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 32 " 1. A Basic Differential Equation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 4 "If " }{XPPEDIT 18 0 "A;" "6#%\"AG" } {TEXT -1 61 " is any constant (usually positive in applications) and \+ if " }{XPPEDIT 18 0 "lambda;" "6#%'lambdaG" }{TEXT -1 99 " is any po sitive constant, then the unique solution of the initial value problem \n\n " }{TEXT 260 2 " " }{TEXT 302 2 " " }{XPPEDIT 257 0 "Diff(y(t),t) = (-lambda*y(t))*` `,` `*y(0) = A;" "6$/-%%D iffG6$-%\"yG6#%\"tGF**&,$*&%'lambdaG\"\"\"-%\"yG6#%\"tGF/!\"\"F/%%~~~~ GF//*&F5F/-F(6#\"\"!F/%\"AG" }{TEXT -1 4 "\n is" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 19 " " } {TEXT 301 2 " " }{XPPEDIT 258 0 "y(t) = A*exp(-lambda*t);" "6#/-%\"yG 6#%\"tG*&%\"AG\"\"\"-%$expG6#,$*&%'lambdaGF*F'F*!\"\"F*" }{TEXT -1 1 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 56 "\nThe solution of this initial value prob lem is said to " }{TEXT 263 19 "decay exponentially" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 12 "2. Half-life" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "If a positive function " }{XPPEDIT 18 0 "f;" "6#%\"fG" } {TEXT -1 49 " is decreasing, then we may call the function " } {XPPEDIT 18 0 "proc (t) options operator, arrow; tau(t) end proc;" "6# f*6#%\"tG7\"6$%)operatorG%&arrowG6\"-%$tauG6#F%F*F*F*" }{TEXT -1 23 " \+ the halving time of " }{XPPEDIT 18 0 "f;" "6#%\"fG" }{TEXT -1 25 " \+ if\n\n " }{TEXT 287 2 " " }{XPPEDIT 256 0 "f(t+tau( t)) = f(t)/2;" "6#/-%\"fG6#,&%\"tG\"\"\"-%$tauG6#F(F)*&-F%6#F(F)\"\"#! \"\"" }{TEXT 288 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "In general the halving time is nonconstant. However:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 290 7 "Theorem" }{TEXT -1 4 " : " }{TEXT 291 15 "Suppose that " }{XPPEDIT 256 0 "lambda;" "6#%'lambdaG" }{TEXT 292 45 " is a positive constant. The halving time " }{XPPEDIT 257 0 "tau;" "6#%$tauG" }{TEXT 293 31 " of the exponential function " } {XPPEDIT 258 0 "proc (t) options operator, arrow; A*exp(-lambda*t) end proc;" "6#f*6#%\"tG7\"6$%)operatorG%&arrowG6\"*&%\"AG\"\"\"-%$expG6#, $*&%'lambdaGF-F%F-!\"\"F-F*F*F*" }{TEXT 294 42 " is constant. More pr ecisely, for every " }{XPPEDIT 259 0 "0 < t;" "6#2\"\"!%\"tG" }{TEXT 295 31 ", the halving time is given by" }}{PARA 0 "" 0 "" {TEXT 296 0 "" }}{PARA 0 "" 0 "" {TEXT 297 27 " " } {XPPEDIT 260 0 "tau = 1/lambda;" "6#/%$tauG*&\"\"\"F&%'lambdaG!\"\"" } {TEXT 298 2 " " }{XPPEDIT 261 0 "ln(2);" "6#-%#lnG6#\"\"#" }{TEXT 299 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 "\n" }{TEXT 300 6 "Proof:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "f \+ := t -> A*exp(-lambda*t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6 #%\"tG6\"6$%)operatorG%&arrowGF(*&%\"AG\"\"\"-%$expG6#,$*&%'lambdaGF.9 $F.!\"\"F.F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "tau = s olve( f(t+tau) = f(t)/2, tau); #Note tau is independent of t" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%$tauG*&-%#lnG6#\"\"#\"\"\"%'lambdaG!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 2 "\n\n" }}{PARA 0 "" 0 "" {TEXT -1 6 "Whe n " }{XPPEDIT 18 0 "f(t) = A*exp(-lambda*t);" "6#/-%\"fG6#%\"tG*&%\"A G\"\"\"-%$expG6#,$*&%'lambdaGF*F'F*!\"\"F*" }{TEXT -1 44 " is the mas s of an unstable isotope, then " }{XPPEDIT 18 0 "tau = ln(2)/lambda; " "6#/%$tauG*&-%#lnG6#\"\"#\"\"\"%'lambdaG!\"\"" }{TEXT -1 20 " is sa id to be the " }{TEXT 303 9 "half-life" }{TEXT -1 18 " of the substanc e." }}{PARA 0 "" 0 "" {TEXT -1 1 "\n" }}{PARA 0 "" 0 "" {TEXT 312 7 "T heorem" }{TEXT -1 4 " : " }{TEXT 313 15 "Suppose that " }{XPPEDIT 256 0 "lambda;" "6#%'lambdaG" }{TEXT 314 58 " is a positive constant . Then the exponential function " }{XPPEDIT 258 0 "proc (t) options o perator, arrow; A*exp(-lambda*t) end proc;" "6#f*6#%\"tG7\"6$%)operato rG%&arrowG6\"*&%\"AG\"\"\"-%$expG6#,$*&%'lambdaGF-F%F-!\"\"F-F*F*F*" } {TEXT 315 22 " can be written as " }{XPPEDIT 256 0 "proc (t) option s operator, arrow; A/(2^(t/tau)) end proc;" "6#f*6#%\"tG7\"6$%)operato rG%&arrowG6\"*&%\"AG\"\"\")\"\"#*&F%F-%$tauG!\"\"F2F*F*F*" }{TEXT 319 11 " where " }{XPPEDIT 256 0 "tau = 1/lambda;" "6#/%$tauG*&\"\"\"F &%'lambdaG!\"\"" }{TEXT 317 2 " " }{XPPEDIT 257 0 "ln(2);" "6#-%#lnG6 #\"\"#" }{TEXT 318 22 " is the half-life. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 "\n" }{TEXT 316 6 "Proof:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "formula1 := A*exp(-lambda*t) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)formula1G*&%\"AG\"\"\"-%$expG6 #,$*&%'lambdaGF'%\"tGF'!\"\"F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "formula2 := A/2^(t/tau);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% )formula2G*&%\"AG\"\"\")\"\"#*&%\"tGF'%$tauG!\"\"F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "formula2 := subs(tau = ln(2)/lambda, form ula2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)formula2G*&%\"AG\"\"\")\" \"#*(%\"tGF'-%#lnG6#F)!\"\"%'lambdaGF'F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "simplify( formula1 - formula2 );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{PARA 3 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 40 "3. Exercises \+ Involving Exponential Decay" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 320 8 "Exercise" } {TEXT -1 114 " \n(Exercise 10, page 472) The mass of a radioactive sa mple is 10g in 1950 and 7 g in 2000. What is its half-life?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 321 8 "Solution" } {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Let " }{XPPEDIT 18 0 "t = 0;" "6#/%\"tG\"\"!" }{TEXT -1 27 " correspond to 1950. Let " } {XPPEDIT 18 0 "m(t);" "6#-%\"mG6#%\"tG" }{TEXT -1 23 " be the mass at time " }{XPPEDIT 18 0 "t;" "6#%\"tG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "eqn1 := m (t) = 10*exp(-lambda*t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn1G/- %\"mG6#%\"tG,$*&\"#5\"\"\"-%$expG6#,$*&%'lambdaGF-F)F-!\"\"F-F-" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "eqn2 := 7 = subs(t=50, rhs(e qn1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn2G/\"\"(,$*&\"#5\"\"\" -%$expG6#,$*&\"#]F*%'lambdaGF*!\"\"F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "eqn3 := lambda = solve(eqn2, lambda);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn3G/%'lambdaG,$*&#\"\"\"\"#]F*-%#lnG6##\"\"( \"#5F*!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "eqn4 := tau \+ = ln(2)/lambda;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn4G/%$tauG*&-% #lnG6#\"\"#\"\"\"%'lambdaG!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "eqn5 := subs(eqn3, eqn4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %%eqn5G/%$tauG,$*(\"#]\"\"\"-%#lnG6#\"\"#F*-F,6##\"\"(\"#5!\"\"F4" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "eqn6 := map(evalf, eqn5);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn6G/%$tauG$\"+]5z;(*!\")" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 41 "4. Exercises Involving Radiocarbon Dating " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "Livin g matter contains two isotopes of carbon, " }{XPPEDIT 18 0 "` `^12*C; " "6#*&%\"~G\"#7%\"CG\"\"\"" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "` ` ^14*C;" "6#*&%\"~G\"#9%\"CG\"\"\"" }{TEXT -1 93 ", in a known fixed ratio. After death, carbon is no longer metabolized and the amount of " }{XPPEDIT 18 0 "` `^14*C" "6#*&%\"~G\"#9%\"CG\"\"\"" }{TEXT -1 59 " decreases due to radioactive decay. The mass of stable " } {XPPEDIT 18 0 "` `^12*C" "6#*&%\"~G\"#7%\"CG\"\"\"" }{TEXT -1 51 " \+ in a sample can be used to determine the mass " }{XPPEDIT 18 0 "m;" " 6#%\"mG" }{TEXT -1 7 " of " }{XPPEDIT 18 0 "` `^14*C" "6#*&%\"~G\"# 9%\"CG\"\"\"" }{TEXT -1 68 " that the sample had at the moment of dea th. Since the half-life of" }{XPPEDIT 18 0 "` `^14*C" "6#*&%\"~G\"#9% \"CG\"\"\"" }{TEXT -1 233 " is known to be around 5700 years, the tim e of death can be calculated from the law of exponential decay. Willar d Libby received the 1960 Nobel prize for physics for discovering th is method of dating.\n\nIn each exercise we will let" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 41 " \+ " }{TEXT 355 18 " " }{XPPEDIT 356 0 "y(t) = m/(2^(t/5700));" "6#/-%\"yG6#%\"tG*&%\"mG\"\"\")\"\"#*&F'F* \"%+d!\"\"F/" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 22 "denote the amount of " }{XPPEDIT 18 0 "` `^14*C" "6#*&%\"~G\"#9%\"CG\"\"\"" }{TEXT -1 16 " that exists " } {XPPEDIT 18 0 "t;" "6#%\"tG" }{TEXT -1 22 " years after death.\011" }}{PARA 0 "" 0 "" {TEXT -1 1 "\n" }}{PARA 0 "" 0 "" {TEXT 322 8 "Exerc ise" }{TEXT -1 24 "\n(Exercise 27, page 473)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 53 " \+ " }{BITMAP 220 182 182 1 "?TMWUo=:;JUJNJhKUk:>=> ::::^bp_lBOiRhbqgl?whO\\A?iP@cr_lAohM@cp_lC?iN`cuwlF_iR@d:`mLGjW`cuwlE OiOXctglCGiOHcs?mBGiSXcv_mCOiTXbn_lyngHPaeWkohMXboWl=gHIdWgvLoe`MTXmVKge_ADW 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_r>^^v_P^]`f[Tf^Ef]cv[U>_HN]b^[Rn^Dv\\_F[PV^A^\\Z^ZTn^BV]b>[VF_Cn]eV[Z f_G>^gf[hVaUf_tV]fFaTNaCG_D?es>dWwaHgexvcXwaJOf;gbRw`vvchFbNW`AGes^cVg a?odqnc[GbUogGWeiocLwf@wdfgcLofF_gwwf:xktGh;`g^ohW_flwdbGhQOfdGdWWfHWg i_e:Hjhoj=@hCPjjojxgg?`ieOkwdv^c[WbCgevFbMO`:Gdi^bO_`[Y>_Mn^iV]^v_Tv]ef\\i^a`f`aZn]f^\\\\v_RNdy_lLwiNdYD _OLES_WdX]fM@SbdaKWiLHduOlL_iMhcs?lGKbVlLoiO@dyglKoiQpCgfLmM\\HqAWHQtW cVMiE_fl?GhH`ac_jqffww`^wisffogK`bhGl=ggJPbfwk;WgHpabW kvnfCPaaGktvFucTSfJSu]ADV_FK]u^ATVaFJ[e^sCUWvHQU]oCTOvHMu\\msSMFG?E[ec SGfIWe]?tU[vHKU\\ecRAvGCU[osSKNo>bggk>OHMdWiFLee_GDWafISu\\kcSEfGCuZmc SEFHCuZaSR;vGEE[usTOFJWU]AdUUFKUe]C=CESRAKCbW>aCw^pihohmGfdOgPwdW?c<_cpFbIo`yNdpvbVoA ydImsLMkWdkPpmHNb]FaCO_kFb\\N_u^]hFb[>`>W^nFcbv`FW_d>bXV]jv[Sn_GN^n^\\ j>b\\^`]]N[[V_ON]]n[Vn^KV^jN]kFbdf aGo`uNclN_r>^u>ck3:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 349 "The Shroud of Turin is a cloth believed by some to b e the burial cloth of Jesus. In the 1980s, four strands of the cloth \+ were given to four different institutions for dating. The results of t he radiocarbon analysis were released on 13 October 1988. Each institu tion declared the Shroud of Turin to be a medieval forgery.\ni) What percentage of " }{XPPEDIT 18 0 "m;" "6#%\"mG" }{TEXT -1 35 " would the measured quantity of " }{XPPEDIT 18 0 "` `^14*C" "6#*&%\"~G\"#9% \"CG\"\"\"" }{TEXT -1 129 " have been in order to authenticate the S hroud?\nii) The Shroud was dated to around 1325 C.E. Approximately wha t percentage of " }{XPPEDIT 18 0 "m;" "6#%\"mG" }{TEXT -1 17 " was m easured?\n\011" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 323 8 "Solution" }{TEXT -1 1 ":" }}{PARA 0 "" 0 "" {TEXT -1 227 "The death date of Jesus is not precisely known. However, historical r eferences indicate that it did not take place later than \n36 C.E.. T he fibers of a burial cloth could not have come from living matter at \+ a later date.\n\n i)\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "s \+ := 1988-36; # time from death of Jesus to year of dating " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"sG\"%_>" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "100*y(s)/m*perCent = 100*evalf(1/(2^(s/5700)));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/,$**\"$+\"\"\"\"-%\"yG6#\"%_>F'%\"mG! \"\"%(perCentGF'F'$\"+qV(p)y!\")" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 77 "Any percentage substantially greater than 78.9 would have proved a forgery." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "ii)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "s := 1988-1325; # time from forgery to year of dating " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"sG \"$j'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "100*y(s)/m*perCent = 100*evalf(1/(2^(s/5700)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,$** \"$+\"\"\"\"-%\"yG6#\"$j'F'%\"mG!\"\"%(perCentGF'F'$\"+q`SD#*!\")" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "The measu red amount was about 92.25 percent of m. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT 324 8 "Exercise" }{TEXT -1 24 "\n(Exercise 2 8, page 473)" }}{PARA 0 "" 0 "" {TEXT -1 3 " " }}{PARA 0 "" 0 "" {TEXT -1 32 " " }{BITMAP 224 158 158 1 "?TMgia=:;JUJNJjC:w>=JFJ::::JhDpblojDOehO`tNcTn`@O]gcA?c?wa:Oanf`>oay^dHwc@?b;Oanf`Egb@O eNgdFobAgaq>aBOb=wcCOcBOb=_apv`QGdLG`e^_ov_jfaq>aFobAocBGcH?cC_c@wbp>` kf`jF`vv`r^by>bAOb=ocCOcXGeTGfVgeLgcHg`n>ao^`rVatnaAobDwcHOdDGcGoawFbq n`tVawnay>b>gb@Oc>WbAWasvaxNa:_cA?dRWdV_c:Od:o`=warvaD?cBWdN?dOGeMGeW? eRoeS?m]HmOyuSqtZAvDYtQYsLYuAAtSisQqu>ysMYu`QwYQwhiu`IwXYv`avaYwbQweQw cYwhYwgQweyveawbiwhQwgqwd=ZhyweQxgawkaweqwdIwgQw`iwdqvfYwaqwhAwiywciwj yvgAxaqwkAwiQxcywlIWnwkAwhIxbYxjAwnIxcaxjYwlqwcYxhYwoAxgaxiawjYwdYxkIx nywjYxgqwoawiqxgAxoywnIxjaxhIxoQv`YwUyu^itRauNytUIuVAv]iv`QwfiwgywjQwg IxfywmQwhYx\\IwlyubQxZIwpQu_YxKquharRQw^@qDaiI`ncoiMphE@nw_ldhhFPnpgk] PiJ`o?XmqxiQXpsGlg`iQPpyOmo@jW@q>HnvPkbXrDxnOevvbWgbO_ewfcNGaCwcxfcJ_cUoeCodRw`?_b uFcBOcM_e>gcPoa@gcp^a=Gb?OcFwcNwbDocCOcHocGodc@_cQgcP ?f>gcQOdWWgYOgrwbL?fdv`?GcP_fTwfo_fnGjj?jGpe^WdWOfMOeY?dN?eFwcM?cFwc@O cFobCWc?wcL?dHGdIocKwc?gb?WauVawNavnb=gbU?eTWdHgcHGcBgb:wa@?BUtKKTJoSI mcFesLCtKIDJAdKyCKIdIKtH]SIUsBYSH[cHQDKMdLAdKQTKGtJucImsFcSKyCJgDN]tK= dJaSEWsG]cFKdJAdK;TJscGisMMdLWDLMtImcH[cDQcFSSEISB?CFOsDbJsSIscGiCIgsG ;ePqdS;URwSHosEOsDqcGicMMdLOTKGDLADKWTLODMIDLmDPstN_dOuCI;DIqsIYdM_<;t IATMSDNMtLSdJAtKGtKQtLOdMATJItJySKIdJQDLADMKDIWDLwsLSTKSdMSDM]dNWtPyTP sDRqDM_TMYSI_sGATIGeV_ug?hk?XmeXfsGkAXnihj]Hoahq?YqKIu;YtSIrEItCysQAbu NWlxpaXqyxouxqiHpXqeHxXrmxsuxpEypmxqiXoqXoUxpCu:VwjyvfIxbawkAwgQxciwlI WZfIxbYX:Qiv`HPIcAvwgiwkawgYxgqwpiwjqxgAxkYwjIwbaw\\ivcAv^IwZaVHxmuwjQ Xhywk=XkIXm]xnmXqyXmaxpexpAYnmhYqviiu`AxYAwoyt\\AxEAubytfyyEAvoqo?YuS@ p@YnsPtV`pFIkZPqS@pDYnypuXPqSYlipsHXoDillHtBxn@ahMPpo_lmxgOxp>xnCYlgHt \\hpTqnrhuXpoMql`XsEhmgcwNa= ?byFcvfaAOavnbv>bE_a:Wc>OcO_a=Odwb?OcGWcF_cF?cB?cCobB_dKWdjggi?fUOe =oawn_c>_wv`rvdKOdHwbD?ak^`y>aunbyNbBGb@gbvnbwV`yf`ev`o^_pfapVaA?b=Wc? 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