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Use an initial interval of (0,pi/2).Use an intial inte rval epsilon=0.2." }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 222 6 "Inputs" }} {EXCHG {PARA 3 "" 0 "" {TEXT 217 123 "The following simulation approxi mates the optimal angle (theta) to maximize the area of a gutter given with a edge length 2" }}{PARA 3 "" 0 "" {TEXT 217 98 " a) Cross- sectional area of the gutter, A b) lower limit of the range , a " }}{PARA 3 "" 0 "" {TEXT 217 86 " c) Upper limit of the range , b d) Interval range, epsilon" }} {PARA 3 "" 0 "" {TEXT 217 4 " " }{TEXT 217 0 "" }{TEXT 217 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {XPPEDIT 0 1 "Typesetting:-mrow(Typesetting:- mi(\"restart\", bold = \"false\", italic = \"true\", mathvariant = \"i talic\"), Typesetting:-mo(\";\", bold = \"false\", mathvariant = \"nor mal\", fence = \"false\", separator = \"true\", stretchy = \"false\", \+ symmetric = \"false\", largeop = \"false\", movablelimits = \"false\", accent = \"false\", lspace = \"0.0em\", rspace = \"0.2777778em\"), Ty 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centGF1/%'lspaceGQ&0.0emF'/%'rspaceGQ,0.2777778emF'-F96.Q\"~F'F/F/F AF1FBFDFFFHFJFL/FPFN-I'mspaceGF$6&/%'heightGQ&0.0exF'/%&widthGFN/%&dep thGFfn/%*linebreakGQ(newlineF'FW-F,6&Q\"AF'F/F2F5-F96.Q*≔F'F/F FUFBFDFFFHFJ/FMFQFO-I(mfencedGF$6%-F#6%-F,6&Q(θF'F//F3F1F<-F9 6.Q\",F'F/FF@FBFDFFFHFJFL/FPQ,0.3333333emF'-F,6&Q\"lF'F/F2F5F/F<-F9 6.Q(→F'F/FFUFBFDFFFHFJFLFV-Ffo6%-F#6,-I#mnGF$6%Q\"6F'F/F<-F96 .Q(−F'F/FFUFBFDFFFHFJ/FMQ,0.2222222emF'/FPFeq-F^q6%Q\"2F'F/F< -F96.Q'⋅F'F/FFUFBFDFFFHFJFLFVFcp-F96.Q\"+F'F/FFUFBFDFFFHFJF dqFfqFcpFjq-F,6&Q$cosF'F/F]pF<-Ffo6%-F#6#-F,6&Q&thetaF'F/F]pF " 0 "" {XPPEDIT 0 1 "`assign`(range1, `+`(`*`(100, ` *`(epsilon)))):" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF '6=-I#miGF$6&Q'range1F'/%%boldGQ&falseF'/%'italicGQ%trueF'/%,mathvaria ntGQ'italicF'-I#moGF$6.Q*≔F'F//F6Q'normalF'/%&fenceGF1/%*separ atorGF1/%)stretchyGF1/%*symmetricGF1/%(largeopGF1/%.movablelimitsGF1/% 'accentGF1/%'lspaceGQ,0.2777778emF'/%'rspaceGFN-I#mnGF$6%Q$100F'F/F<-F 96.Q'⋅F'F/FF@FBFDFFFHFJ/FMQ&0.0emF'/FPFY-F,6&Q-ϵF'F //F3F1F<-F96.Q\":F'F/FF@FBFDFFFHFJFLFO-F96.Q\"~F'F/FF@FBFDFFFHFJ FXFZF\\oF\\o-I'mspaceGF$6&/%'heightGQ&0.0exF'/%&widthGFY/%&depthGFdo/% *linebreakGQ(newlineF'F\\o-F,6&Q*iterationF'F/F2F5F8-FR6%Q\"0F'F/F/FAF4FBFDFFFHFJFXFOF_oF \\o-F,6&Q#xuF'F/F2F5F8-I%msubGF$6%-F,6&Q'intialF'F/F2F5-F#6#-F,6&Q\"lF 'F/F2F5/%/subscriptshiftGFapFep" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"! " }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 217 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 217 84 "The loop calcul ates the and prints the interval with each iteration as program runs." }}{PARA 0 "" 0 "" {TEXT 217 69 "This loop contains the algorithm for \+ the equal interval search method" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "while `>`(range1, epsilon) do `assign`(Area[l], A(initi al[theta], l)); `assign`(iteration, `+`(iteration, 1)); `assign`(d, ev alf(`*`(`*`(`+`(sqrt(5), `-`(1)), `/`(1, 2)), `*`(`+`(xu, `-`(xl)))))) ; `assign`(x1, `+`(xl, d)); `assign`(x2, `+`(xu, `-`(d))); `assign`(va l1, subs(l = x1, Area[l])); `assign`(val2, subs(l = x2, Area[l])); `as sign`(interval, [xl, xu]); `assign`(range1, `+`(xu, `-`(xl))); `assign `(optimization[point], `*`(`+`(xu, xl), `/`(1, 2))); if `>`(val1, val2 ) then `assign`(xl, x2); `assign`(x2, x1); `assign`(xu, xu); `assign`( d, evalf(`*`(`*`(`+`(sqrt(5), `-`(1)), `/`(1, 2)), `*`(`+`(xu, `-`(xl) ))))); `assign`(x1, `+`(xl, d)) else `assign`(xl, xl); `assign`(xu, x1 ); `assign`(x1, x2); `assign`(d, evalf(`*`(`*`(`+`(sqrt(5), `-`(1)), ` /`(1, 2)), `*`(`+`(xl, `-`(xu)))))); `assign`(x2, `+`(xu, `-`(d))) end if end do; -1; `assign`(optimization[length], optimization[point]); - 1; printf(\"Optimization point for lenght=%g\", optimization[length]); " "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6ay-I#moGF$60Q \"~F'/%'familyGQ,Courier~NewF'/%+foregroundGQ*[255,0,0]F'/%0font_style _nameGQ,Maple~InputF'/%,mathvariantGQ'normalF'/%&fenceGQ&falseF'/%*sep aratorGF=/%)stretchyGF=/%*symmetricGF=/%(largeopGF=/%.movablelimitsGF= /%'accentGF=/%'lspaceGQ&0.0emF'/%'rspaceGFL-I'mspaceGF$6&/%'heightGQ&0 .0exF'/%&widthGFL/%&depthGFT/%*linebreakGQ(newlineF'FOF+-F,62Q&whileF' F//%%boldGQ%trueF'F2F5/F9Q%boldF'/%+fontweightGF]oF;F>F@FBFDFFFHFJFMF+ -I(mfencedGF$6)-F#6%-I#miGF$6(Q'range1F'F//%'italicGF[oF2F5/F9Q'italic F'-F,60Q\">F'F/F2F5F8F;F>F@FBFDFFFH/FKQ,0.2777778emF'/FNFap-Ffo6(Q-&va repsilon;F'F//FjoF=F2F5F8F/FinF2F5F\\oF^oF+-F,62Q#doF'F/FinF2F5F\\oF^o F;F>F@FBFDFFFHFJFMF+FOF+-I%msubGF$6%-Ffo6(Q%AreaF'F/FioF2F5F[p-F#6#-Ff o6(Q\"lF'F/FioF2F5F[p/%/subscriptshiftGQ\"0F'-F,60Q*≔F'F/F2F5F 8F;F>F@FBFDFFFHF`pFbp-Ffo6(Q\"AF'F/FioF2F5F[p-Fao6)-F#6%-F[q6%-Ffo6(Q( initialF'F/FioF2F5F[p-F#6#-Ffo6(Q&thetaF'F/FfpF2F5F8Feq-F,60Q\",F'F/F2 F5F8F;/F?F[oF@FBFDFFFHFJ/FNQ,0.3333333emF'FbqF/FinF2F5F\\oF^o-F,60Q\"; F'F/F2F5F8F;F_sF@FBFDFFFHFJFbpFOFOF+-Ffo6(Q*iterationF'F/FioF2F5F[pFhq Fes-F,60Q\"+F'F/F2F5F8F;F>F@FBFDFFFH/FKQ,0.2222222emF'/FNF\\t-I#mnGF$6 'Q\"1F'F/F2F5F8FbsFOF+-Ffo6(Q\"dF'F/FioF2F5F[pFhq-Ffo6(Q&evalfF'F/FioF 2F5F[p-Fao6)-F#6%-I&mfracGF$6)-F#6&-Ffo6(Q%sqrtF'F/FfpF2F5F8-Fao6)-F#6 #-F_t6'Q\"5F'F/F2F5F8F/FinF2F5F\\oF^o-F,60Q(−F'F/F2F5F8F;F>F@FBF DFFFHF[tF]tF^t-F#6#-F_t6'Q\"2F'F/F2F5F8/%.linethicknessGFat/%+denomali gnGQ'centerF'/%)numalignGFgv/%)bevelledGF=F2-F,60Q'⋅F'F/F2F5F8F;F >F@FBFDFFFHFJFM-Fao6)-F#6%-Ffo6(Q#xuF'F/FioF2F5F[pF[v-Ffo6(Q#xlF'F/Fio F2F5F[pF/FinF2F5F\\oF^oF/FinF2F5F\\oF^oFbsFOF+-Ffo6(Q#x1F'F/FioF2F5F[p FhqFfwFhsFbtFbsFOF+-Ffo6(Q#x2F'F/FioF2F5F[pFhqFcwF[vFbtFbsFOF+-Ffo6(Q% val1F'F/FioF2F5F[pFhq-Ffo6(Q%subsF'F/FioF2F5F[p-Fao6'-F#6'Fbq-F,60Q\"= F'F/F2F5F8F;F>F@FBFDFFFHF`pFbpFiwF\\sFjpF/F2F5F8FbsFOF+-Ffo6(Q%val2F'F /FioF2F5F[pFhqFbx-Fao6'-F#6'FbqFixF\\xF\\sFjpF/F2F5F8FbsFbsFOF+-Ffo6(Q )intervalF'F/FioF2F5F[pFhq-Fao6+-F#6&FfwF\\sF+FcwF/FinF2F5F\\oF^o/%%op enGQ\"[F'/%&closeGQ\"]F'FbsFOF+FeoFhqFcwF[vFfwFbsFOF+-F[q6%-Ffo6(Q-opt imizationF'F/FioF2F5F[p-F#6#-Ffo6(Q&pointF'F/FioF2F5F[pFeqFhq-F]u6)-F# 6%FcwFhsFfwF^vFcvFevFhvFjvF2FbsFOF+FO-FP6&FRFUFW/FZQ%autoF'FOF^[lF^[lF ^[lF+-F,62Q#ifF'F/FinF2F5F\\oF^oF;F>F@FBFDFFFHFJFMF+-Fao6)-F#6%F_xF]pF \\yF/FinF2F5F\\oF^oF+-F,62Q%thenF'F/FinF2F5F\\oF^oF;F>F@FBFDFFFHFJFMFO F+F+F+F+FfwFhqF\\xFbsFOF+F+F+F+F\\xFhqFiwFbsFOF+F+F+F+FcwFhqFcwFbsFOF+ F+F+F+FbtFhqFetFhtFbsFOF+F+F+F+FiwFhqFfwFhsFbtFbsFOF+-F,62Q%elseF'F/Fi nF2F5F\\oF^oF;F>F@FBFDFFFHFJFMFOF+F+F+F+FfwFhqFfwFbsFOF+F+F+F+FcwFhqFi wFbsFOF+F+F+F+FiwFhqF\\xFbsFOF+F+F+F+FbtFhqFet-Fao6)-F#6%F\\uF\\w-Fao6 )-F#6%FfwF[vFcwF/FinF2F5F\\oF^oF/FinF2F5F\\oF^oFbsFOF+F+F+F+F\\xFhqFcw -F,60Q*&uminus0;F'F/F2F5F8F;F>F@FBFDFFFHF[tF]tFbtFbsFOF+-F,62Q$endF'F/ FinF2F5F\\oF^oF;F>F@FBFDFFFHFJFMF+Fb[l-F,60Q\":F'F/F2F5F8F;F>F@FBFDFFF HF`pFbpF+F+F+FOF+Fj\\lF+FgpF]]lFOF+-F[q6%Fbz-F#6#-Ffo6(Q'lengthF'F/Fio F2F5F[pFeqFhqF`zF]]lFO-Ffo6(Q'printfF'F/FioF2F5F[p-Fao6'-F#6&-Ffo6#Q!F '-I#msGF$6#QAOptimization~point~for~lenght=%gF'F\\sF`]lF/F2F5F8F^^lFOF ^[lFOF^[lFOFOF+" }}{PARA 6 "" 1 "" {TEXT 223 36 "Optimization point fo r lenght=2.6459" }}}{EXCHG {PARA 0 "> " 0 "" {TEXT 224 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "`?`;" }}{PARA 11 "" 1 "" {XPPMATH 20 "I\"?G6\"" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 222 47 "Golden Search Meth od along the Length dimesnion" }}{PARA 0 "" 0 "" {TEXT 217 0 "" }} {EXCHG {PARA 0 "" 0 "" {TEXT 217 172 "creating a varibale which calcul ates the range of the interval, intially just start the while loop we \+ take this value to be greater than epsilon, here we choose 100*epsilon " }}{PARA 0 "" 0 "" {TEXT 217 0 "" }}{PARA 0 "" 0 "" {TEXT 217 66 "ite ration is a variable name which counts the number of iterations" }}} {EXCHG {PARA 0 "> " 0 "" {XPPEDIT 0 1 "`assign`(range1, `+`(`*`(100, ` *`(epsilon)))):" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF '6>-I#miGF$6&Q'range1F'/%%boldGQ&falseF'/%'italicGQ%trueF'/%,mathvaria ntGQ'italicF'-I#moGF$6.Q*≔F'F//F6Q'normalF'/%&fenceGF1/%*separ atorGF1/%)stretchyGF1/%*symmetricGF1/%(largeopGF1/%.movablelimitsGF1/% 'accentGF1/%'lspaceGQ,0.2777778emF'/%'rspaceGFN-I#mnGF$6%Q$100F'F/F<-F 96.Q'⋅F'F/FF@FBFDFFFHFJ/FMQ&0.0emF'/FPFY-F,6&Q-ϵF'F //F3F1F<-F96.Q\":F'F/FF@FBFDFFFHFJFLFO-F96.Q\"~F'F/FF@FBFDFFFHFJ FXFZF\\oF\\o-I'mspaceGF$6&/%'heightGQ&0.0exF'/%&widthGFY/%&depthGFdo/% *linebreakGQ(newlineF'F\\o-F,6&Q*iterationF'F/F2F5F8-FR6%Q\"0F'F/F/FAF4FBFDFFFHFJFXFOF_oF \\o-F,6&Q#xuF'F/F2F5F8-F,6&Q&evalfF'F/F2F5-I(mfencedGF$6%-F#6$-I&mfrac GF$6(-F#6#-F,6&Q#PiF'F/FhnF<-F#6#-FR6%Q$2.0F'F/F " 0 "" {XPPEDIT 19 1 "while `>`(range1, \+ epsilon) do `assign`(Area[theta], A(theta, optimization[length])); `as sign`(iteration, `+`(iteration, 1)); `assign`(d, evalf(`*`(`*`(`+`(sqr t(5), `-`(1)), `/`(1, 2)), `*`(`+`(xu, `-`(xl)))))); `assign`(x1, `+`( xl, d)); `assign`(x2, `+`(xu, `-`(d))); `assign`(val1, subs(theta = x1 , Area[theta])); `assign`(val2, subs(theta = x2, Area[theta])); `assig n`(interval, [xl, xu]); `assign`(range1, `+`(xu, `-`(xl))); `assign`(o ptimization[point], `*`(`+`(xu, xl), `/`(1, 2))); if `>`(val1, val2) t hen `assign`(xl, x2); `assign`(x2, x1); `assign`(xu, xu); `assign`(d, \+ evalf(`*`(`*`(`+`(sqrt(5), `-`(1)), `/`(1, 2)), `*`(`+`(xu, `-`(xl)))) )); `assign`(x1, `+`(xl, d)) else `assign`(xl, xl); `assign`(xu, x1); \+ `assign`(x1, x2); `assign`(d, evalf(`*`(`*`(`+`(sqrt(5), `-`(1)), `/`( 1, 2)), `*`(`+`(xl, `-`(xu)))))); `assign`(x2, `+`(xu, `-`(d))) end if end do; -1; `assign`(optimization[theta], optimization[point]); -1; p rintf(\"Optimization point for theta=%g\", optimization[theta]);" "-I% mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6`y-I#moGF$60Q\"~F'/ %'familyGQ,Courier~NewF'/%+foregroundGQ*[255,0,0]F'/%0font_style_nameG Q,Maple~InputF'/%,mathvariantGQ'normalF'/%&fenceGQ&falseF'/%*separator GF=/%)stretchyGF=/%*symmetricGF=/%(largeopGF=/%.movablelimitsGF=/%'acc entGF=/%'lspaceGQ&0.0emF'/%'rspaceGFL-I'mspaceGF$6&/%'heightGQ&0.0exF' /%&widthGFL/%&depthGFT/%*linebreakGQ(newlineF'FOF+-F,62Q&whileF'F//%%b oldGQ%trueF'F2F5/F9Q%boldF'/%+fontweightGF]oF;F>F@FBFDFFFHFJFMF+-I(mfe ncedGF$6)-F#6%-I#miGF$6(Q'range1F'F//%'italicGF[oF2F5/F9Q'italicF'-F,6 0Q\">F'F/F2F5F8F;F>F@FBFDFFFH/FKQ,0.2777778emF'/FNFap-Ffo6(Q-&varepsil on;F'F//FjoF=F2F5F8F/FinF2F5F\\oF^oF+-F,62Q#doF'F/FinF2F5F\\oF^oF;F>F@ FBFDFFFHFJFMF+FOF+-I%msubGF$6%-Ffo6(Q%AreaF'F/FioF2F5F[p-F#6#-Ffo6(Q&t hetaF'F/FfpF2F5F8/%/subscriptshiftGQ\"0F'-F,60Q*≔F'F/F2F5F8F;F >F@FBFDFFFHF`pFbp-Ffo6(Q\"AF'F/FioF2F5F[p-Fao6'-F#6%Fbq-F,60Q\",F'F/F2 F5F8F;/F?F[oF@FBFDFFFHFJ/FNQ,0.3333333emF'-F[q6%-Ffo6(Q-optimizationF' F/FioF2F5F[p-F#6#-Ffo6(Q'lengthF'F/FioF2F5F[pFeqF/F2F5F8-F,60Q\";F'F/F 2F5F8F;FerF@FBFDFFFHFJFbpFOFOF+-Ffo6(Q*iterationF'F/FioF2F5F[pFhqFes-F ,60Q\"+F'F/F2F5F8F;F>F@FBFDFFFH/FKQ,0.2222222emF'/FNF\\t-I#mnGF$6'Q\"1 F'F/F2F5F8FbsFOF+-Ffo6(Q\"dF'F/FioF2F5F[pFhq-Ffo6(Q&evalfF'F/FioF2F5F[ p-Fao6)-F#6%-I&mfracGF$6)-F#6&-Ffo6(Q%sqrtF'F/FfpF2F5F8-Fao6)-F#6#-F_t 6'Q\"5F'F/F2F5F8F/FinF2F5F\\oF^o-F,60Q(−F'F/F2F5F8F;F>F@FBFDFFFH F[tF]tF^t-F#6#-F_t6'Q\"2F'F/F2F5F8/%.linethicknessGFat/%+denomalignGQ' centerF'/%)numalignGFgv/%)bevelledGF=F2-F,60Q'⋅F'F/F2F5F8F;F>F@FB FDFFFHFJFM-Fao6)-F#6%-Ffo6(Q#xuF'F/FioF2F5F[pF[v-Ffo6(Q#xlF'F/FioF2F5F [pF/FinF2F5F\\oF^oF/FinF2F5F\\oF^oFbsFOF+-Ffo6(Q#x1F'F/FioF2F5F[pFhqFf wFhsFbtFbsFOF+-Ffo6(Q#x2F'F/FioF2F5F[pFhqFcwF[vFbtFbsFOF+-Ffo6(Q%val1F 'F/FioF2F5F[pFhq-Ffo6(Q%subsF'F/FioF2F5F[p-Fao6'-F#6'Fbq-F,60Q\"=F'F/F 2F5F8F;F>F@FBFDFFFHF`pFbpFiwFbrFjpF/F2F5F8FbsFOF+-Ffo6(Q%val2F'F/FioF2 F5F[pFhqFbx-Fao6'-F#6'FbqFixF\\xFbrFjpF/F2F5F8FbsFOF+-Ffo6(Q)intervalF 'F/FioF2F5F[pFhq-Fao6+-F#6&FfwFbrF+FcwF/FinF2F5F\\oF^o/%%openGQ\"[F'/% &closeGQ\"]F'FbsFOF+FeoFhqFcwF[vFfwFbsFOF+-F[q6%Fjr-F#6#-Ffo6(Q&pointF 'F/FioF2F5F[pFeqFhq-F]u6)-F#6%FcwFhsFfwF^vFcvFevFhvFjvF2FbsFOF+FO-FP6& FRFUFW/FZQ%autoF'FOF[[lF[[lF[[lF+-F,62Q#ifF'F/FinF2F5F\\oF^oF;F>F@FBFD FFFHFJFMF+-Fao6)-F#6%F_xF]pF\\yF/FinF2F5F\\oF^oF+-F,62Q%thenF'F/FinF2F 5F\\oF^oF;F>F@FBFDFFFHFJFMFOF+F+F+F+FfwFhqF\\xFbsFOF+F+F+F+F\\xFhqFiwF bsFOF+F+F+F+FcwFhqFcwFbsFOF+F+F+F+FbtFhqFetFhtFbsFOF+F+F+F+FiwFhqFfwFh sFbtFbsFOF+-F,62Q%elseF'F/FinF2F5F\\oF^oF;F>F@FBFDFFFHFJFMFOF+F+F+F+Ff wFhqFfwFbsFOF+F+F+F+FcwFhqFiwFbsFOF+F+F+F+FiwFhqF\\xFbsFOF+F+F+F+FbtFh qFet-Fao6)-F#6%F\\uF\\w-Fao6)-F#6%FfwF[vFcwF/FinF2F5F\\oF^oF/FinF2F5F \\oF^oFbsFOF+F+F+F+F\\xFhqFcw-F,60Q*&uminus0;F'F/F2F5F8F;F>F@FBFDFFFHF [tF]tFbtFbsFOF+-F,62Q$endF'F/FinF2F5F\\oF^oF;F>F@FBFDFFFHFJFMF+F_[l-F, 60Q\":F'F/F2F5F8F;F>F@FBFDFFFHF`pFbpF+F+F+FOF+Fg\\lF+FgpFj\\lFOF+-F[q6 %FjrF`qFeqFhqF`zFj\\lFO-Ffo6(Q'printfF'F/FioF2F5F[p-Fao6'-F#6&-Ffo6#Q! F'-I#msGF$6#Q@Optimization~point~for~theta=%gF'FbrF]]lF/F2F5F8Ff]lFOF[ [lFOF[[lFOFOF+" }}{PARA 6 "" 1 "" {TEXT 223 36 "Optimization point for theta=0.86655" }}}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "" "%#%?G" }}}}} {MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }