{VERSION 6 0 "IBM INTEL NT" "6.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 2 0 1 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "Blue Emphasis" -1 256 "Times" 0 0 0 0 255 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Green Emphasis" -1 257 "Times" 1 12 0 128 0 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Maroon Emphasis" -1 258 "Times" 1 12 128 0 128 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Dark Red Emphasis" -1 259 "Times" 1 12 128 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Purple Emphasis" -1 260 "Times " 1 12 102 0 230 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Red Emphasis" -1 261 "Times" 1 12 255 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 265 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 261 266 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 261 267 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 } {CSTYLE "" -1 270 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 271 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 272 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 258 273 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 259 274 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 } {CSTYLE "" 258 275 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 259 276 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 277 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 278 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 279 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 } {CSTYLE "" -1 280 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 281 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 282 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 283 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 284 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 } {CSTYLE "" -1 285 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 286 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 287 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 288 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 289 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 290 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 291 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 292 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 293 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 294 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 295 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 296 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 297 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 298 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 299 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 300 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 301 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 302 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 303 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 304 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 305 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 306 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 307 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 308 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 1 " -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 128 1 2 1 2 2 2 2 1 1 1 1 } 1 1 0 0 8 4 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 128 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 12 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Plot" -1 13 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Normal" -1 257 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 61 "The curve formed by a uniform han ging chain .. the catenary " }}{PARA 0 "" 0 "" {TEXT -1 37 "by Peter \+ Stone, Nanaimo, B.C., Canada" }}{PARA 0 "" 0 "" {TEXT -1 19 "Version: \+ 26.3.2007" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 79 "The derivation of the eq uation of the curve formed by a hanging chain or cable " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 86 "S uppose that a uniform cable or chain hangs under its own weight betwee n two supports." }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 555 115 115 {PLOTDATA 2 "6(-%'CURVESG6$7[al7$$!\"\"\"\"!$\"3y>b(=Ip--\"!#< 7$$!3?ccccOj)y*!#=$\"3b!>a\"eaT>5F-7$$!3?%RRRHzmp*F1$\"3@.3&[3+(=5F-7$ $!35JJJJ\\s/'*F1$\"3I$o,bO<\"=5F-7$$!3mKKK#y=Ib*F1$\"3)e=UD9Cz,\"F-7$$ !34LLLLEJ,&*F1$\"3LG#R!z)F1$\"3!fS_%*H=b,\"F-7 $$!3U*******Rj1u)F1$\"3qM>-u0`:5F-7$$!3S.///Y$4p)F1$\"3nPmZ]D\\:5F-7$$ !3T2333e?T')F1$\"3%f`o!H7P:5F-7$$!3Q6777qZ\"f)F1$\"3J!G[9.a^,\"F-7$$!3 IFGGGyh)[)F1$\"3j?g.G7Z95F-7$$!3IWWWW'edQ)F1$\"3wX;Lt#oP,\"F-7$$!3E)yy y)R\\M$)F1$\"3plMkum_85F-7$$!35JJJJ$HKG)F1$\"3`!\\q.a)Q85F-7$$!3&RZZZn k>B)F1$\"3*3$H$zPXL,\"F-7$$!3#z\"===+q!=)F1$\"3M&>z8bkL,\"F-7$$!3pnnn< !oz7)F1$\"3PrH85F-7$$!3!fhhh,s(pzF1$\"3+*=xyv'485F-7$$!3]mmmm#zo(yF1 $\"3$41!)H_DD,\"F-7$$!33<<<j45F-7$$ !3+!)zzz^7gnF1$\"3MR\\'*RJb45F-7$$!3[\\\\\\\\4cnmF1$\"3q#=be!G;45F-7$$ !3)*=>>>n*\\d'F1$\"3OE:eL*p&35F-7$$!3Kmmm;'f6_'F1$\"3?%*R;j5C35F-7$$!3 o8999DKnkF1$\"3v$[YYZ\"*z+\"F-7$$!3!oyyG'RSSkF1$\"3QoOJPs!z+\"F-7$$!3/ hhh6a[8kF1$\"3TOL%>5_y+\"F-7$$!3GNNNgoc'Q'F1$\"3!\\wm;/Dy+\"F-7$$!3S34 44$['fjF1$\"3`YFdGI#y+\"F-7$$!3<,,,^L*>J'F1$\"3mbD*[>ly+\"F-7$$!3%GHHH RQVE'F1$\"3%yB;3gLz+\"F-7$$!3^%[[[V$o;iF1$\"3'*pUhAi)z+\"F-7$$!3Ixwww% G!phF1$\"3'[^6yv%)z+\"F-7$$!3y#HHHReW1'F1$\"3YIB(>#4p25F-7$$!3G3444$)) )ffF1$\"3#)fE*)of225F-7$$!3]fgggGJgeF1$\"3sIIK,C_15F-7$$!3s5777utgdF1$ \"3rR\"Qb(eH15F-7$$!3ywxxx3%ol&F1$\"3ShjO.*)R15F-7$$!3%GMMMMWHb&F1$\"3 1i@BgS^15F-7$$!3G\">>>*[`daF1$\"3:\\AD%)3L15F-7$$!3sRSSSa7i`F1$\"3Wn;^ 45%e+\"F-7$$!3kddddM@f_F1$\"3xQ.X$*eC05F-7$$!3Yuuuu9Ic^F1$\"3X$Q3_/P\\ +\"F-7$$!3l666hJ&G5&F1$\"3Gk8`#*e$\\+\"F-7$$!3sZZZZ[S\\]F1$\"3$Q?:2+8] +\"F-7$$!3y$QQQ`cf*\\F1$\"3\"=!31Ql605F-7$$!3&)>???#3D%\\F1$\"3+t6>@w= 05F-7$$!3511118)f*[F1$\"3\"fCA@)R=05F-7$$!3#=>>>Ra%\\[F1$\"39!eA^9*405 F-7$$!3^xxxxu#H![F1$\"3]xQIO=$\\+\"F-7$$!3wjjjj0ScZF1$\"35NQ5+&)p/5F-7 $$!3[zzzz)**el%F1$\"3I%*G([[CT+\"F-7$$!3?&fff>*RbXF1$\"3\\!G*>/Kv.5F-7 $$!3vuvvvc[.XF1$\"3Rm..U:r.5F-7$$!3'[bbb:s:X%F1$\"3Pe/fBDw.5F-7$$!3'\\ ```je'*R%F1$\"3VP[L#=oQ+\"F-7$$!3_9:::^uZVF1$\"3$z53$Hk(R+\"F-7$$!35jj jjy&pH%F1$\"3uWt__V./5F-7$$!3s67771% F1$\"3nU%G%[6)Q+\"F-7$$!3#*3444hfWTF1$\"3o%RDA4lO+\"F-7$$!3SaaaavJYSF1 $\"3amS\"[()GJ+\"F-7$$!3')*********Q![RF1$\"3!Hz\"G]ys-5F-7$$!3eggg5#y M*QF1$\"3>1fG>cl-5F-7$$!3w?@@@u\"*QQF1$\"3&))G#[rUp-5F-7$$!3#4===jcVy$ F1$\"3CVEed!3G+\"F-7$$!3nTUUUezHPF1$\"3;\"\\,!e(RH+\"F-7$$!3iXYYY0x!o$ F1$\"3RU1s??-.5F-7$$!3:]]]]_uJOF1$\"3'GED9%>..5F-7$$!3maaaa*>Fe$F1$\"3 'Gl#[x-&H+\"F-7$$!3jeeeeYpLNF1$\"3i&za\")ezF+\"F-7$$!3'GKKKs/!HMF1$\"3 'op^)=oC-5F-7$$!34(yyyy9VK$F1$\"3B(37p5C=+\"F-7$$!3s;<+\"F-7$$!35******\\%HF3$F1$\"3Qd c%eLB@+\"F-7$$!3E%\\\\\\Rt3.$F1$\"363S2!e&>-5F-7$$!37UUUn`%\\+$F1$\"3! )f%fB;)>-5F-7$$!3U*)*)*)Rt,zHF1$\"3X\"[w+$[<-5F-7$$!3uOPP7$*3`HF1$\"3X 64Nn]7-5F-7$$!3/%[[[Ghr#HF1$\"3PY]?V,0-5F-7$$!3IrrrridHGF1$\"3Uf&f')H5 ;+\"F-7$$!3ceeee7*>t#F1$\"3q;\\$\\tm6+\"F-7$$!3ZWWWW,\\1FF1$\"3B8?L474 ,5F-7$$!3QIIII!*)4o#F1$\"3#fXR\\9S5+\"F-7$$!3G;;;;z[bEF1$\"3t=wZm],,5F -7$$!3j,---o)*HEF1$\"3AU3xMe,,5F-7$$!3)GPPPd%)*yDF1$\"3d$Q%['>'3,5F-7$ $!3qWXXXB)z_#F1$\"3A'G[-z>7+\"F-7$$!3C%\\\\\\M=\"yCF1$\"3ggc$)4iO,5F-7 $$!3xVWWWVDGCF1$\"3UQa+!3![,5F-7$$!3I$RRRM!RyBF1$\"3#RGL([<_,5F-7$$!3% GMMMME&GBF1$\"3u/j))*Qp9+\"F-7$$!3Doooo.7CAF1$\"3_)fp$)[)3,5F-7$$!3l$R RRR9(>@F1$\"3-**o$f]81+\"F-7$$!31GGG`bd%4#F1$\"3l<3)*G:`+5F-7$$!3Aiii7 nVp?F1$\"3=%yU!f:Z+5F-7$$!3P'pp>(yHW?F1$\"3mQpe\\gV+5F-7$$!3zIJJJ!f\"> ?F1$\"3_SzHKfU+5F-7$$!3?lll!>?S*>F1$\"3t(>6&e0W+5F-7$$!3O******\\8))o> F1$\"311-**RxZ+5F-7$$!3^LMM4DuV>F1$\"3.nv[')Q`+5F-7$$!3%z'oooOg=>F1$\" 3-_?WJUg+5F-7$$!3%pxxxZ(=n=F1$\"3q[F$z0o2+\"F-7$$!3%foooGrd\"=F1$\"3)[ fk0m94+\"F-7$$!3%\\fff4bVw\"F1$\"3&f\"=nLj*4+\"F-7$$!3&R]]]!*QHr\"F1$ \"34;ZJNC)4+\"F-7$$!3'esss-e4h\"F1$\"3iS;$\\Wv1+\"F-7$$!3xZ\\\\\\r(*3: F1$\"3RR+'>hB-+\"F-7$$!3M!===BA@Y\"F1$\"3q.[xWN2+5F-7$$!3$HTTTJn_T\"F1 $\"33I&[6n+++\"F-7$$!3AHIIb)R=R\"F1$\"3K$[EY-q*****F17$$!3_XYY'R7%o8F1 $\"3\"=2Oh)[,+5F-7$$!3\"=EEw$\\)\\M\"F1$\"3C>+VQE0+5F-7$$!35yyyyub@8F1 $\"3t'*>5-v5+5F-7$$!3?JKKKa:97F1$\"3wAX`5(f/+\"F-7$$!3H%eeeQ`n5\"F1$\" 3]6]UbRk+5F-7$$!3z8:::$\"3@h SVdt0+5F-7$$!3()\\ggg:>M')F_[n$\"3/:!)yezk)***F17$$!3#omnnn#3A\")F_[n$ \"3!=ns3Hiu***F17$$!3y$GHHzt*4wF_[n$\"3M.PPYnF(***F17$$!3u+444\\'y4(F_ [n$\"3%\\!\\S(e(4)***F17$$!3*o!===yb(3K%F_[n$\"3bYgfsqS+5F-7$$!35r'oo oX([SF_[n$\"3yCR:')yM+5F-7$$!37]mmmJ\\/NF_[n$\"3G5n(zbm,+\"F-7$$!3;HYY Y1CgHF_[n$\"3%*GLrAQX****F17$$!3!*Raaaz&*)[#F_[n$\"39sO^rZp(***F17$$!3 j]iii_nUZ'***F17$$!3QhqqqDRY:F_[n$\"3_!Hlw,\\g***F17$$ !37syyy)4^2\"F_[n$\"3'=\">cZX]'***F17$$!3mbfsttt*G&!#A$\"3Y1)3<%)y**** *F17$$\"3A?////`k5F_[n$\"3@6](44_.+\"F-7$$\"3+xjjjG/_:F_[n$\"3ra:MZZS+ 5F-7$$\"3!QLKKKb&R?F_[n$\"3\"yOP9<))****F17$$\"3=]SSS!)3))pF_[n$\"3O=vB_oN+5F-7$$\"3o_TT Tm4]sF_[n$\"3dt'[RQM/+\"F-7$$\"3:bUUU_57vF_[n$\"3yV\\Z;(*[+5F-7$$\"3kd VVVQ6uxF_[n$\"3aBC.,'>0+\"F-7$$\"37gWWWC7O!)F_[n$\"3&GIwXaA0+\"F-7$$\" 33lYYY'R,c)F_[n$\"3!4:0)**3X+5F-7$$\"3/q[[[o:%3*F_[n$\"3.o\\Qx')H+5F-7 $$\"3#phfffJ.f*F_[n$\"3ri6>N!=,+\"F-7$$\"3QOMMM1l45F1$\"3uZ]())\\\\&** **F17$$\"3ttrrr$f\\.\"F1$\"3/G4$)=$[*)***F17$$\"316444\"o-1\"F1$\"3axd D+Kb)***F17$$\"3U[YYYod&3\"F1$\"3a`LZugR)***F17$$\"3w&QQQe&)36\"F1$\"3 5G`2tQ\\)***F17$$\"3?+**)*)Ra?@\"F1$\"3l:,-[87+5F-7$$\"3j9999KA88F1$\" 3I>5<)Qf0+\"F-7$$\"3C\"333`@OO\"F1$\"3'[3ehwA2+\"F-7$$\"3&yuuu%)>ST\"F 1$\"3-U0C:**z+5F-7$$\"3:\"33e+>#R9F1$\"3e^\"f=O,3+\"F-7$$\"3Y999k\"=WY \"F1$\"3;M8'Rhy2+\"F-7$$\"3wZZZAth*[\"F1$\"3G)yxG.M2+\"F-7$$\"32\"333[ ;[^\"F1$\"3lT52v8n+5F-7$$\"3H!**)*)*[];h\"F1$\"3uTu()>1O+5F-7$$\"3^**) *)*)\\%[3+5F-7$$\"3aFFFFY;8=F1$\"3))3CdgWR+5F-7$$\"3cbbb bZ%y\">F1$\"3#\\$Rg8\\'3+\"F-7$$\"3#>===jGx'>F1$\"3\\JWD]!p5+\"F-7$$\" 3G3333DhQFBF1$\"34j`mD+q+5F-7$$\"3%=+++vW6N#F1$\"3*QwZ**G12+\"F-7$$\"3)yh hh^2\\P#F1$\"3'oFpBBN2+\"F-7$$\"3#RBBBGq')R#F1$\"34yw7*['y+5F-7$$\"3(* \\[[[IVACF1$\"3F%oD]Xe3+\"F-7$$\"31#333ee*pCF1$\"3)3%\\Y#\\_5+\"F-7$$ \"3:9888T[555S,#GGF1$\"36#*z=*4(f,5F-7$$\"37\\[ [[-#)GHF1$\"3T9u)\\`w8+\"F-7$$\"3_333LP\"R&HF1$\"3%[5r&4UO,5F-7$$\"3Yo nna;F;JT,5F-7$$\"3!yooo=%> HIF1$\"3X$4XJ.v9+\"F-7$$\"39211c6QzIF1$\"3?SG7:Sm,5F-7$$\"3[EDDD\"o&HJ F1$\"3uqHj'\\6>+\"F-7$$\"3_5444'))>=$F1$\"3i>ru=H=-5F-7$$\"3d%HHH44WB$ F1$\"3`#HD'=KT-5F-7$$\"3)o[[[L>1E$F1$\"3l&=!eo%)\\-5F-7$$\"3iywww&HoG$ F1$\"3YRZ>E'eD+\"F-7$$\"3Oqoo=)RIJ$F1$\"3C'[Z%G>f-5F-7$$\"3miggg+DRLF1 $\"3&=s!\\l%)f-5F-7$$\"3P?>>>C\"eV$F1$\"3abg5@6W-5F-7$$\"34yxxxZPKNF1$ \"3N$QKL?>A+\"F-7$$\"3+gff48#=e$F1$\"3%zvN`1'=-5F-7$$\"3%>999%yEJOF1$ \"37a%*Gf^C-5F-7$$\"3'QKKKP92o$F1$\"3+9k(=V*R-5F-7$$\"3z00004;IPF1$\"3 p'>wTEHE+\"F-7$$\"3F*zzzf]$RQF1$\"3Z>LXRb?.5F-7$$\"3w#444HS&[RF1$\"3d7 WBj0^.5F-7$$\"35a___9(z*RF1$\"3\")f;&R&=].5F-7$$\"3W:999ESZSF1$\"3`5Yi *[@M+\"F-7$$\"3ywvvvP$o4%F1$\"335//txI.5F-7$$\"39QPPP\\EYTF1$\"3.updEs ?.5F-7$$\"39544fv\"o>%F1$\"3V_y1#RjJ+\"F-7$$\"3;#333=qtC%F1$\"3nLJSu4@ .5F-7$$\"3;a__-G#zH%F1$\"3.#pLLXfL+\"F-7$$\"3;ECCCaZ[VF1$\"3Rqe\"\\K$f .5F-7$$\"3)\\DDD&oH^WF1$\"3u*)>try;/5F-7$$\"3y$333G=Tb%F1$\"3L#eK(Q3b/ 5F-7$$\"3CMJJJZP,YF1$\"3fPE3Cef/5F-7$$\"3o%====J'[YF1$\"3i&o'))oJc/5F- 7$$\"39NKKKw)ep%F1$\"3B5e3@#zW+\"F-7$$\"3f&GGG3WJu%F1$\"3X_A?o111E?Y/&F1$\"3X,**z_e;05F-7$$\"3daaaa5gW^F1$\"3t@UXyzm05F-7$$\"3 uZZZZ\"Q8<&F1$\"35()[A+.v05F-7$$\"3)3///Cv!)>&F1$\"3z)>B*)p/e+\"F-7$$ \"31MLLLB\"[A&F1$\"3Mt%4jDJe+\"F-7$$\"3AFEEE%\\:D&F1$\"3IEN/`?$e+\"F-7 $$\"3a8777O-0`F1$\"3t;%>Imtd+\"F-7$$\"3')*zzzz(\\e`F1$\"3>4b*=Txc+\"F- 7$$\"3q^]]]ko0aF1$\"3\"Q5Xl(zg05F-7$$\"3a....^(GX&F1$\"3/&[()RL#f05F-7 $$\"3YHHHH%pkZ&F1$\"3'o-*4(R8c+\"F-7$$\"3QbbbbP1+bF1$\"3O>&yqMcc+\"F-7 $$\"3I\"===3eO_&F1$\"37F&=*z;s05F-7$$\"3A2333CDZbF1$\"3Ed&ePm3e+\"F-7$ $\"3-EEEEU9ccF1$\"3fpHJH.T15F-7$$\"3%[WWW/O]w&F1$\"3:q_)\\G6q+\"F-7$$ \"3m777iEg*y&F1$\"3&H;N,X,r+\"F-7$$\"3[!)zzz#pT\"eF1$\"3XrA#eVor+\"F-7 $$\"3I[ZZ(*etQeF1$\"3A\"G/4l6s+\"F-7$$\"37;:::DIjeF1$\"3JHKBY>B25F-7$$ \"3x^]]]dV7fF1$\"3k!)z$4+9s+\"F-7$$\"3S(eee)*o:'fF1$\"3wuzG'GYr+\"F-7$ $\"3s0//aP=5gF1$\"3e!z_E9tq+\"F-7$$\"3/CAAA&)zegF1$\"3tIevJ!Qq+\"F-7$$ \"3=LJJ1f5$3'F1$\"3oU*R?QYq+\"F-7$$\"3NUSS!H8u5'F1$\"3xy*GtMwq+\"F-7$$ \"3^^\\\\u1sJhF1$\"3l/P()e'Hr+\"F-7$$\"3mgeee!Gg:'F1$\"3?7BBB.M^O'F1$\"3Anwf`iS35F-7$$ \"3.&\\\\\\*>iru35F-7$$\" 3X222d@%Gi'F1$\"3KdF>_Vo35F-7$$\"3oJJJJ%)fqmF1$\"3&[&ym!3X'35F-7$$\"3M WVVolZ%p'F1$\"3_yvn[jk35F-7$$\"3#fbbbqa$=nF1$\"33%)oN(em'35F-7$$\"3[nn nUGBUnF1$\"3Nl'=SH3(35F-7$$\"3/zzzz46mnF1$\"3$**Q43ws(35F-7$$\"3Ikjjj5 anoF1$\"3+zCR@sF45F-7$$\"3c\\ZZZ6(*opF1$\"3&*e]'3ZY*45F-7$$\"3BurrrSY= qF1$\"3\"y7P#e(=-,\"F-7$$\"3#yfff*p&z1(F1$\"3)[p7Q%yS55F-7$$\"3-433eMq #4(F1$\"3B$4-Afm/,\"F-7$$\"3Q@???*\\u6(F1$\"3qYOvA>]55F-7$$\"3tLKK#Q'> UrF1$\"3'GGRB)f^55F-7$$\"33YWWWG%p;(F1$\"3Ro*z)GA^55F-7$$\"3xZYY'p$f?s F1$\"3#)*3rzzl/,\"F-7$$\"3Y\\[[[XCusF1$\"3eOD#[$[T55F-7$$\"3K]\\\\u*p5 I(F1$\"375F-7$$\"3SPOOO9^!y(F1$\"3L#z\"zo)*R75F-7$$\"3xGGGGNjIyF1$ \"3G!ybwR\"Q75F-7$$\"3C@???cv!)yF1$\"3N*pbn$enns495F-7$$\"3M\"444\\h#y$)F1$\"3@Ze-tpT95F-7$$\"3'*4444sfz% )F1$\"33U6#[-EW,\"F-7$$\"3gGFFFH$4e)F1$\"3pXIAg!HW,\"F-7$$\"31#4444Z$) o)F1$\"3K1]b2Dy95F-7$$\"3_baaa7w&z)F1$\"3e\\8VM-^:5F-7$$\"3wDCCC&\\U*) )F1$\"3)HtQ'H]>;5F-7$$\"3)fRRRzPF**)F1$\"33(35L<(e;5F-7$$\"3FHGGy![8/* F1$\"3([K^,@^m,\"F-7$$\"3u>> >>h5\"H*F1$\"331g)fdDp,\"F-7$$\"3;3222L.&R*F1$\"3KjNT*4)e<5F-7$$\"3gba aaUt)\\*F1$\"3sn&QL6p$=5F-7$$\"31.---_V-'*F1$\"3;(G`Pm*))=5F-7$$\"31om mTYqD'*F1$\"3cHwa,.&*=5F-7$$\"31LJJ\"3u*['*F1$\"3BM8ZQ;**=5F-7$$\"31)f f4_VAn*F1$\"3FJAc6j,>5F-7$$\"3/jgggH^&p*F1$\"3\"G:yz'y->5F-7$$\"3.$**) *)R=0U(*F1$\"3mN01%=H!>5F-7$$\"3/B>>>2f)y*F1$\"3IKCaKS.>5F-7$$\"3_hfff `H%*)*F1$\"3Cu[6^MC>5F-7$$\"\"\"F*$\"3+KEO4U')>5F--%&COLORG6&%$RGBG$\" \"%F)FeiqFeiq-F$6$7a`l7$F(F_iq7$$!3aFGGGoJ%*)*F1$\"3.e&o^`V#>5F-7$F/$ \"335F-7$$!3lCDDvklU(*F1$\"3dc25Q\"H!>5F-7$F5$\"3&*yiCw\"G!>5F- 7$$!3)*GGG.2pt'*F1$\"3'G\\->P5F-7$$!3miii7@q]'*F1$\"3c$*4&[+%**=5F- 7$$!3K'pp>_8xi*F1$\"3rKUi+Y&*=5F-7$F:$\"3fmS))Rl*)=5F-7$FD$\"3E!oa!*\\ 'Q=5F-7$FN$\"3zK78]*4w,\"F-7$$!3+````D!QH*F1$\"3T,\"F-7$F]s$\"3#Q6 4/wkM,\"F-7$Fgs$\"3k@U5S)\\F,\"F-7$Fat$\"3!*p/n')RO75F-7$$!3kSTTTcKBzF 1$\"3/We)*RkK75F-7$Fft$\"3(Q^H%QRM75F-7$$!3M#>>>*GVIyF1$\"36A4BZ:Q75F- 7$F[u$\"3>Iq\"eD+C,\"F-7$$!37eee3F%yv(F1$\"3r@ci')*)Q75F-7$F`u$\"39xU6 .dN75F-7$$!3>STT\"4bbq(F1$\"3V\"eX49(H75F-7$Feu$\"3q;'fDx6A,\"F-7$Fju$ \"32Lld)oj>,\"F-7$F_v$\"3J0[*Q![j65F-7$Fiv$\"3I*eR()yA4,\"F-7$F]x$\"33 .`0lTZ55F-7$$!3Pyyyy5B9tF1$\"3>]q(4&*3/,\"F-7$Fbx$\"3%)*Q'Q&QA/,\"F-7$ $!3%4111mRI@(F1$\"3Zn+d1SZ55F-7$Fgx$\"3v'**[L/90,\"F-7$F\\y$\"3\"f,c`( 4]55F-7$Fay$\"3!>prx8:/,\"F-7$F[z$\"3<$Q$p.sC55F-7$Fez$\"3jgJ/^]+55F-7 $F_[l$\"3$G[cz0*G45F-7$Fi[l$\"3q;tlRWv35F-7$$!3lAAAAT)pt'F1$\"3=EVA+sp 35F-7$$!3IlkkkI%Qr'F1$\"3S&3!Qy6m35F-7$$!3%zqqq+-2p'F1$\"3/1kjl\\k35F- 7$F^\\l$\"3B$4f*3hk35F-7$$!3nLMMM)y7i'F1$\"3;O\"4BE'o35F-7$Fc\\l$\"3$p \"=(>BZ(35F-7$Fh\\l$\"3n)>y=F$z35F-7$F]]l$\"3*o#3QoBw35F-7$Fg]l$\"3ll, C(eB'35F-7$Fa^l$\"3XNbhWoP35F-7$F[_l$\"3RhO_;Py25F-7$Fe_l$\"3#RjwzKds+ \"F-7$$!3s\"33e&f)G9'F1$\"3/7yfD@;25F-7$$!3/&[[[VVn6'F1$\"3ki1y)*R425F -7$$!3M)))))Q\"4g!4'F1$\"3I=>E5K025F-7$Fj_l$\"3'o6EP=Qq+\"F-7$$!3`+,,^ L<7gF1$\"3\"G%eZw1225F-7$F_`l$\"3nyd1p*[r+\"F-7$$!3L$[[[e+,\"fF1$\"3JG \"f=H;s+\"F-7$Fd`l$\"3n#eU(Q1B25F-7$$!3-Z[[)**=a$eF1$\"36K)\\!)=2s+\"F -7$$!3mNOOO^_5eF1$\"3x\">OF+gr+\"F-7$$!3=BCCu7j&y&F1$\"3!p&G/4%)325F-7 $Fi`l$\"3D#G*R`K*p+\"F-7$F^al$\"3XZ6:&p9k+\"F-7$Fcal$\"3\"G%py)pKe+\"F -7$$!31nnn<'R_]&F1$\"3g?W4p(oc+\"F-7$Fhal$\"3/_DlB[f05F-7$$!3\\:;;m,$) 4aF1$\"3r@ztrRg05F-7$F]bl$\"37bku\"e+\"F-7$$!3aXXXq%H?=&F1$\"3`l)))QWvd+\"F-7$Fgbl $\"3AkDKhtq05F-7$Facl$\"3o$HqbE%>05F-7$F[dl$\"35J%=Ip\"f/5F-7$F`dl$\"3 cnIsM>T/5F-7$Fedl$\"3My1Z9^J/5F-7$Fjdl$\"3MkA*f]-V+\"F-7$F_el$\"3&)QVp kwN/5F-7$$!3Mrrr@-:1ZF1$\"34u)y*zxX/5F-7$Fdel$\"3S(**yQu_X+\"F-7$$!3h( yyy`\\cg%F1$\"3#p![l!y&f/5F-7$Fiel$\"3/sj(Q?`X+\"F-7$Fcfl$\"3$HUo:JpT+ \"F-7$F]gl$\"3E9G=\"\\*e.5F-7$Fbgl$\"3t2)fGtbL+\"F-7$Fggl$\"3-o0qk'3K+ \"F-7$F\\hl$\"3D?T()zM;.5F-7$Fahl$\"3&o#*RP\"*4K+\"F-7$$!3k\"===$oX&4% F1$\"3<*4!3#)4J.5F-7$Ffhl$\"32?]YuPU.5F-7$$!37FFFx#yr*RF1$\"3'3RJBk-N+ \"F-7$F[il$\"3IV*oX>5N+\"F-7$Feil$\"3Bx&3(RN?.5F-7$F_jl$\"3$GyROPFE+\" F-7$Fdjl$\"3!RW%yb'*R-5F-7$Fijl$\"33h)Rz?YA+\"F-7$F^[m$\"3W[2gsi=-5F-7 $Fc[m$\"3))*)*o.H(*>\\zf-5F -7$Fg\\m$\"3d+p$p_%R-5F-7$Fa]m$\"3(Gd)Hjx$>+\"F-7$Ff]m$\"3_76!o:z;+\"F -7$F[^m$\"3@')piA+[,5F-7$F`^m$\"3*[([d.[T,5F-7$Fe^m$\"38RG(y.w8+\"F-7$ Fj^m$\"3zU$y8Ck8+\"F-7$F__m$\"3UND0X\"y8+\"F-7$Fd_m$\"3ESO!p:$f,5F-7$F i_m$\"3PWlS=&>=+\"F-7$Fc`m$\"3]S'R[iN=+\"F-7$F]am$\"3I9,UN:v,5F-7$Fbam $\"3*)o)zz&[d,5F-7$Fgam$\"33y+CKqL,5F-7$Fabm$\"3&Gll7&*y3+\"F-7$F[cm$ \"3UlV4;)*p+5F-7$F`cm$\"3nNP[A0*3+\"F-7$Fecm$\"3p*G2*QS=,5F-7$F_dm$\"3 lcF$GrT7+\"F-7$Fidm$\"3;owh!407+\"F-7$Fcem$\"3v?5*y0t5+\"F-7$F]fm$\"3' \\A\\_Oo3+\"F-7$Fgfm$\"3%QhEXH//+\"F-7$Fagm$\"3uBtNV8>+5F-7$Ffgm$\"3pv v<>FO+5F-7$F[hm$\"3t*e()>E(o+5F-7$$!31kll!p\\b[\"F1$\"3$z&fRmDu+5F-7$F `hm$\"3Pw6tO;y+5F-7$$!3k'zzHx%pQ9F1$\"3h!GXPe,3+\"F-7$Fehm$\"3XeN(\\d+ 3+\"F-7$F_im$\"3\")=Dh\"=M2+\"F-7$Fiim$\"3#)pwBW6f+5F-7$F^jm$\"3!GT2y) *H,+\"F-7$Fcjm$\"3+cK.e-Y)***F17$$!3m;==ozt#3\"F1$\"3!y,M-=,%)***F17$$ !3/\\]]]Dse5F1$\"3qTvef2d)***F17$$!3U\"GGG82Z.\"F1$\"3ep\"Q&eL&*)***F1 7$Fhjm$\"3oq$4_C@&****F17$$!3U&yzzz3mi*F_[n$\"35XOej^5+5F-7$F][n$\"3() fT9rsF+5F-7$Fc[n$\"3z;qe3MV+5F-7$Fh[n$\"3)fAB'\\w^+5F-7$$!3Iv%[[BGg'yF _[n$\"3sI*GBsB0+\"F-7$F]\\n$\"3Dn)Qm(R]+5F-7$$!3E#455N>RN(F_[n$\"3d([* G$=f/+\"F-7$Fb\\n$\"3N`()Qj f9i)o.+\"F-7$Fi_n$\"3kE5cjYS+5F-7$F^`n$\"37'f!poTN+5F-7$Fc`n$\"3QW5%f6 -++\"F-7$Fi`n$\"3!RDm#>W_'***F17$F^an$\"3uSQ9!))[g***F17$Fcan$\"3!3q$H xY^'***F17$Fhan$\"3sPl;S9#y***F17$F]bn$\"3EXVnp]n****F17$Fbbn$\"3(ydh7 C$=+5F-7$Fgbn$\"3=vU**=iN+5F-7$F\\cn$\"3l%G.\\mS/+\"F-7$Facn$\"3x.xG/d T+5F-7$Ffcn$\"3]Wv^0\"f,+\"F-7$F[dn$\"3W>%*RW[Q)***F17$F`dn$\"3IJaS`(e x***F17$Fedn$\"3;Q$=n9gt***F17$Fjdn$\"3e!o>A^@s***F17$F_en$\"3K_:S!ydt ***F17$Fden$\"3K#[rJ6A%)***F17$Fien$\"3DVvJ>9.+5F-7$Fcfn$\"3F)eH<#GX+5 F-7$Fggn$\"3Dk0rfUk+5F-7$F\\hn$\"3ZxE56jY+5F-7$Fahn$\"30*\\oNZI,+\"F-7 $$\"3$zuuCPA%Q8F1$\"3$)>l[Ck1+5F-7$Ffhn$\"3'Q<;a1@++\"F-7$$\"3c999*o?) )Q\"F1$\"2ED,r@\")*****F-7$F[in$\"2a&y$eb#z+ +\"F-7$F_jn$\"3%*fgsilC+5F-7$Fdjn$\"3Rl1+P%y1+\"F-7$Fijn$\"3_A9iVj(4+ \"F-7$$\"3_888jX#3w\"F1$\"3rtWA[')*4+\"F-7$F^[o$\"3'>,G*42#4+\"F-7$$\" 3aTTT\"p/b'=F1$\"30v_A;Mx+5F-7$Fc[o$\"3a+#*==lg+5F-7$$\"3uooo$p'yU>F1$ \"3e8R&[LO0+\"F-7$Fh[o$\"3W'=P?$*z/+\"F-7$$\"33&\\\\*p0n#*>F1$\"3ZTH)z +U/+\"F-7$F]\\o$\"3>=-tKhU+5F-7$$\"3Y@@@YWbU?F1$\"32H-G)Q,5F-7$F__o$\"30[h1y4D,5F-7$Fd_o $\"3_R*G:K16+\"F-7$Fi_o$\"3r9y\\A2-,5F-7$F^`o$\"3Pm^(4\"G,,5F-7$Fc`o$ \"3W&[@4(>.,5F-7$Fh`o$\"3?Yf%p!)y5+\"F-7$F]ao$\"3c-5F-7$F[co$\"3G\\#=m\\'>-5F-7$F`co$\"3$))) oF>-8-5F-7$Feco$\"3Kz9@hu+-5F-7$Fjco$\"3t&oH*o%o=+\"F-7$F_do$\"3]0Hm,G x,5F-7$Fddo$\"3h(*>5%pb<+\"F-7$Fido$\"3ir=jHUw,5F-7$F^eo$\"3!3(GHr,!=+ \"F-7$Fceo$\"3?F<&*GM'=+\"F-7$Fheo$\"3(zgC+m#G-5F-7$F]fo$\"3Mb%Q:%Rx-5 F-7$Fbfo$\"3`2B>2z%H+\"F-7$Fgfo$\"3)oa%)=gJI+\"F-7$F\\go$\"3Yw=2z?-.5F -7$Fago$\"3%**H;y&*QH+\"F-7$$\"3__^^^dv%y$F1$\"3r>\"oH32G+\"F-7$Ffgo$ \"3GETY.Op-5F-7$$\"3-YWWWa%R*QF1$\"3'[(e=Pdl-5F-7$F[ho$\"3Aodlm!HF+\"F -7$Feho$\"3K$fejnMJ+\"F-7$F_io$\"3SF.9?Ln.5F-7$Fdio$\"3g-2Kcg)Q+\"F-7$ Fiio$\"3dOqJF%4S+\"F-7$F^jo$\"3o2Kb`R./5F-7$Fcjo$\"3ZC(\\\"R^(R+\"F-7$ $\"3cSQQQh))*R%F1$\"3JCFLtw'Q+\"F-7$Fhjo$\"3yKlt\")Hw.5F-7$$\"3Qpmmmvq -XF1$\"3#)>4#3p6P+\"F-7$F][p$\"3$e_xa*3v.5F-7$Fg[p$\"3W5L3op3/5F-7$Fa \\p$\"3(e7)e[Ri/5F-7$Ff\\p$\"3iQ#RS:*)[+\"F-7$F[]p$\"3*\\X,lJ$305F-7$$ \"3Uvss(py!p[F1$\"3(\\SA2LX^+\"F-7$F`]p$\"3`b8reB=05F-7$$\"3NrooVDX>\\ F1$\"3Z9iNs^>05F-7$Fe]p$\"30kIk(3'=05F-7$Fj]p$\"3K.uOi?-05F-7$F_^p$\"3 6vz\"G8G\\+\"F-7$Fc_p$\"3r&oh@y+\"F-7$$\"3/kjj8!y8R'F1$\"39NV_Fz#y+\"F-7$F`gp$\"3VrgX*pey+ \"F-7$$\"3-EEEwf'QW'F1$\"3fCU0Uk\"z+\"F-7$Fegp$\"3!\\c%yO=+35F-7$Fjgp$ \"3HVM?J9'e]&45 F-7$Fa[q$\"3Eh9^fyg45F-7$F[\\q$\"3;y#yUb&z45F-7$Fe\\q$\"3>(R(Q,!p+,\"F -7$F_]q$\"3,dn!yP)y55F-7$Fc^q$\"3jhQ:BsJ65F-7$$\"3BIHH/xv/uF1$\"3zC7T! \\w8,\"F-7$$\"3f211c\"pzU(F1$\"3F4N2R]T65F-7$$\"3'\\GGyg!=^uF1$\"3S&zl $3XV65F-7$Fh^q$\"3\\z]`svV65F-7$$\"3\"[JJJ'\\\"3_(F1$\"3!4lv\"\\%49,\" F-7$F]_q$\"3@pPY,iO65F-7$$\"3]644fib?wF1$\"3Av**4'4[8,\"F-7$Fb_q$\"3i# y5fO69,\"F-7$$\"3V&RRR/$>FxF1$\"3#=))G1!Re65F-7$Fg_q$\"38Mx&fdj=,\"F-7 $Fa`q$\"34cvv=Gb75F-7$F[aq$\"3Z[&*f/%[J,\"F-7$$\"3TA@@rxjI!)F1$\"3e^h \\R$=L,\"F-7$F`aq$\"3&\\Jzoc%R85F-7$$\"3/dbb0P\"*H\")F1$\"3'>$)**3Y(R8 5F-7$Feaq$\"3s*fdvLlL,\"F-7$$\"3/GFFF\"H#H#)F1$\"3(e*4Q2_M85F-7$Fjaq$ \"3m+&HUa\"Q85F-7$$\"3kPOOOSeG$)F1$\"3O!f5`R0N,\"F-7$F_bq$\"3fIg2$*os8 5F-7$Fdbq$\"3nh#4QS/W,\"F-7$Fibq$\"3/'*R]ki4:5F-7$$\"3M5444+kM')F1$\"3 v!*\\cZ![`,\"F-7$F^cq$\"3^.U%p\\)[:5F-7$$\"3\"QFFFe^:5F-7$$\"3lSRR*Q0]%))F1$\"3yD]aC5]:5F-7$Fhcq$\"3\"= [$eo:`:5F-7$$\"3(3\"44fO\\V*)F1$\"3t=UgH>k:5F-7$F]dq$\"3?m`'>&y%e,\"F- 7$Fgdq$\"3]lXIz*)[;5F-7$Faeq$\"3x&=Vse0s,\"F-7$F[fq$\"3.krFjD#y,\"F-7$ $\"3!>333y$)oW*F1$\"3UVa7xT\"y,\"F-7$F`fq$\"3XchET*Hy,\"F-7$$\"3MHGGGZ e]&*F1$\"3X7ISow\"z,\"F-7$Fefq$\"3E*)**eTl5=5F-7$Figq$\"3GAV?*G\"p=5F- 7$Fchq$\"3!y4oz8:%>5F-7$F]iqF+Faiq-%)POLYGONSG6%7&7$F]iq$\"$-\"!\"#7$$ \"#6F)F^dv7$Fbdv$\"$***!\"$7$F]iqFedv7&7$F(F^dv7$$!#6F)F^dv7$F\\evFedv 7$F(Fedv-%'COLOURG6&Fdiq$\")#)eqk!\")$\"))eqk\"FeevFfev-%+AXESLABELSG6 $Q\"x6\"Q!F\\fv-%*AXESSTYLEG6#%%NONEG-%%VIEWG6$;F\\evFbdv;FedvF^dv" 1 2 0 1 10 0 2 9 1 1 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 41 "The curve formed in this way is called a " }{TEXT 260 8 " catenary" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 56 "We can show t hat the equation of the curve has the form " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "y = (cosh*k*x-1)/k;" "6#/%\"yG*&,&*(%% coshG\"\"\"%\"kGF)%\"xGF)F)F)!\"\"F)F*F," }{TEXT -1 1 " " }}{PARA 0 " " 0 "" {TEXT -1 26 "with respect to a pair of " }{TEXT 297 1 "x" } {TEXT -1 5 " and " }{TEXT 298 1 "y" }{TEXT -1 15 " axes with the " } {TEXT 299 1 "x" }{TEXT -1 86 " axis horizontal, the y axis vertical an d with the origin located at the lowest point." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 53 "Consider a section of cab le between the lowest point " }{XPPEDIT 18 0 "P[0];" "6#&%\"PG6#\"\"! " }{TEXT -1 24 " and an arbitrary point " }{TEXT 262 1 "P" }{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 453 262 262 {PLOTDATA 2 "6>-%'CURVESG6%7W7$$!\"#\"\"!$\"3SJO3\"p&>iF!#<7$$!3ymmm\" p0k&>F-$\"3-$*QIf1h2EF-7$$!3MLLL$Q6G\">F-$\"3'3\"GeYG))fCF-7$$!31++v3- )[(=F-$\"3\\w\\G%o0nL#F-7$$!3bmm;M!\\p$=F-$\"3N4z,j)H$=AF-7$$!3MLLL))Q j^F-7$$!3ALLL=Kvl;F-$\"3$)4\"RE6_$R'***F_o$\"3B,Thq,MEaF_o7$$! 3E++++0\"*H\"*F_o$\"372)*z9AVlWF_o7$$!35++++83&H)F_o$\"3d2X%z5uAk$F_o7 $$!3\\LLL3k(p`(F_o$\"3O7C^VxKxHF_o7$$!3Anmmmj^NmF_o$\"3Cbm0ndZ$G#F_o7$ $!3)zmmmYh=(eF_o$\"3'3c2LMVSx\"F_o7$$!3+,++v#\\N)\\F_o$\"3?!Q'*3^-xE\" F_o7$$!3commmCC(>%F_o$\"3T'f]xv(\\Q*)!#>7$$!39*****\\FRXL$F_o$\"3G=#z> @#G6cFbr7$$!3t*****\\#=/8DF_o$\"3kk)4(pHMuJFbr7$$!3=mmm;a*el\"F_o$\"3E +`Y\"RITP\"Fbr7$$!3komm;Wn(o)Fbr$\"3EhOW5&ehx$!#?7$$!3IqLLL$eV(>Fgs$\" 31!oAs/Y!\\>!#B7$$\"3)Qjmm\"f`@')Fbr$\"37hiBln%)=PFgs7$$\"3%z****\\nZ) H;F_o$\"3\"o')Gi_W6L\"Fbr7$$\"3ckmm;$y*eCF_o$\"3?fhx$p^&QIFbr7$$\"3f)* *****R^bJ$F_o$\"3M5x9Ro(pa&Fbr7$$\"3'e*****\\5a`TF_o$\"3m]P;5(y1v)Fbr7 $$\"3'o****\\7RV'\\F_o$\"3qBg=)f[xD\"F_o7$$\"3Y'*****\\@fkeF_o$\"3'zm@ SYG&p$)F_o$\"3=#3()fE'GlOF_o7$$\"3M*******pfa<* F_o$\"3%f%*[qU\">8XF_o7$$\"39HLLeg`!)**F_o$\"3W8-m::'zS&F_o7$$\"3w**** \\#G2A3\"F-$\"3s,`2#>m,X'F_o7$$\"3;LLL$)G[k6F-$\"3aL*p))RZaK#RT*))F_o7$$\"3xmmm')fdL8F-$\"3?Q_%y'31H5F-7$ $\"3bmmm,FT=9F-$\"3)4s]$*RGj=\"F-7$$\"3FLL$e#pa-:F-$\"3ADJ!yOSyN\"F-7$ $\"3!*******Rv&)z:F-$\"39f:]LV8I:F-7$$\"3ILLLGUYo;F-$\"3G__.(3wiu\"F-7 $$\"3_mmm1^rZF-7$$\"34++]sI@K=F-$\"3I&=-hYyQ?#F-7$$\" 33+++S2ls=F-$\"3'>iA5$ohHBF-7$$\"34++]2%)38>F-$\"3Q(>PQa+3Y#F-7$$\"3/+ +v.Uac>F-$\"3[!=M2\"343EF-7$$\"\"#F*F+-%'COLOURG6&%$RGBG$\"#5!\"\"$F*F *Fe\\l-%*THICKNESSG6#F]\\l-F$6&7$7$Fe\\lFe\\l7$$\"3/+++++++8F-$\"3*)** *****HU\"4(*F_o-%'SYMBOLG6#%'CIRCLEG-F_\\l6&Fa\\l$\"*++++\"!\")Fe\\lFe \\l-%&STYLEG6#%&POINTG-F$6&F[]l-Fc]l6#%(DIAMONDGFf]lF[^l-F$6&F[]l-Fc]l 6#%&CROSSGFf]lF[^l-F$6%7$7$F^]lFe\\lF]]l-%&COLORG6&Fa\\l$\"\"\"F)F`_lF `_l-%*LINESTYLEGFh\\l-F$6$7S7$$\"3/++++++]:F-F`]l7$$\"3idH'H$f$*\\:F-$ \"3IF`,Notl(*F_o7$$\"3/ni>lfx\\:F-$\"3$>!)='fo&\\\")*F_o7$$\"3gQy*>G![ \\:F-$\"3c6@1E*f-()*F_o7$$\"3W:uQ()*e!\\:F-$\"3,Y*R1m\\e#**F_o7$$\"3Cm *=bs;&[:F-$\"3m3#yUBo5)**F_o7$$\"3'*z?m=Y!za\"F-$\"3h'f5bQ9K+\"F-7$$\" 3M%4w[@gra\"F-$\"3zpX:K&)[35F-7$$\"3c#*=bSBFY:F-$\"3+)RQB4DR,\"F-7$$\" 3s(H6&>$o_a\"F-$\"3FbtDQMK>5F-7$$\"3-)\\<\">@6W:F-$\"3f3#z(4>&[-\"F-7$ $\"3&\\aY6^!*Ha\"F-$\"33&e7%z')pH5F-7$$\"35sCTOFhT:F-$\"3N2f%\\/F^.\"F -7$$\"3QP>/0t5S:F-$\"3ht%)Rs_aS5F-7$$\"34Y[w#eT&Q:F-$\"3%4SKSVLd/\"F-7 $$\"3xLK`DE-P:F-$\"3A+.9(Q9/0\"F-7$$\"3(*4\"[*\\p4N:F-$\"3w#f#G]-%f0\" F-7$$\"3\"*3K!*G[OL:F-$\"3[qUu`]eg5F-7$$\"3>'4i@cM7`\"F-$\"3/E`ZzN%f1 \"F-7$$\"3&G_cIBY#H:F-$\"3=6w))*oW12\"F-7$$\"3?Z`Z5[&p_\"F-$\"35vRg^Yv v5F-7$$\"3(o223'pmC:F-$\"3u!=Q:F-$\"3a22o*=h+4\"F-7$$\"3Xyuv$z,r^\"F-$\"3gYsjsr( [4\"F-7$$\"3$*)3LnS1U^\"F-$\"3#Rg&>-e\")*4\"F-7$$\"3web^hWf6:F-$\"3%\\ !p8a*fS5\"F-7$$\"3n9'3$z!z'3:F-$\"3a2^1xUe36F-7$$\"3ma&f=klb]\"F-$\"3a Y(oyI\">86F-7$$\"3\"H\\pML@C]\"F-$\"3*3?$4&HIw6\"F-7$$\"3Qc,_8yG*\\\"F -$\"3[='f]9f=7\"F-7$$\"31%)e+[^q&\\\"F-$\"3W/D'))4wk7\"F-7$$\"3!>*\\8 \"p%R#\\\"F-$\"3tQ8D/AbI6F-7$$\"3?UmE7kw)[\"F-$\"3R-8]Gl#[8\"F-7$$\"3A r\"3'4mR&[\"F-$\"3*)**H'>cG'Q6F-7$$\"3qmM5n\\i\"[\"F-$\"3sRm\"4)eqU6F- 7$$\"3Ef!R)\\U*zZ\"F-$\"3uG/XE\\YY6F-7$$\"3e=L#G)f6u9F-$\"35e26#o6F-7$$\"3;ien3ma\\9F-$\"3G`7(=Z`7<\" F-7$$\"3B4\\GN!)*[W\"F-$\"3kQA534ku6F-7$$\"3***QP7Cv1W\"F-$\"3c!G=8Ozv <\"F-7$$\"3!e*GN)G2hV\"F-$\"3#)\\-LPfh!=\"F-7$$\"3Yk&\\sxt;V\"F-$\"3U' =JvzFM=\"F-7$$\"3Smfz(oWoU\"F-$\"3qyIw>]M'=\"F--F^_l6&Fa\\l$\"\"%Fd\\l F\\_m$\"\"'Fd\\l-F$6'7$7$$\"#8Fd\\l$\"*IU\"4(*!\"*7$$\"#BFd\\l$\"+nmHp EFh_m7%7$$\"+#4Qa:#Fh_m$\"+t4KADFh_mFi_m7$$\"+3/hTAFh_m$\"+)4$erCFh_m- F\\^l6#%,PATCHNOGRIDG-F_\\l6&Fa\\lFe\\lFe\\lFh]lFf\\l-F$6'7$Fc_m7$Fj_m Ff_m7%7$$\"+LLLL@Fh_m$\"+IUTL5Fh_mFaam7$Fdam$\"++B9%3*!#5Fi`m-F_\\l6&F a\\l$\")#)eqkFj]l$\"))eqk\"Fj]lF`bmFf\\l-F$6'7$FaamFi_m7%7$$\"++++]AFh _m$\"+nmHpCFh_mFi_m7$$\"++++]BFh_mFibmFi`m-F^_l6&Fa\\lF*$\"\"(Fd\\lF*F f\\l-F$6'7$F\\]l7$$!#8Fd\\lFe\\l7%7$$!++++v6Fh_m$!++++]i!#6Fecm7$Fjcm$ \"++++]iF^dmFi`m-F^_l6&Fa\\lF\\_mF*$\"\"*Fd\\lFf\\l-%%TEXTG6%7$$\"#EFd \\l$\"$(=F)Q&T~sin6\"F^cm-Fgdm6%7$F\\em$\"#vF)Q&T~cosF_emF\\bm-Fgdm6%7 $F\\em$\"$l\"F)Q\"TF_emF\\am-Fgdm6%7$$!\"(Fd\\l$F)Fd\\lF[fmFbdm-Fgdm6% 7$$!#jF)$F]tF)Q\"oF_emFbdm-Fgdm6%7$$!$b\"F)$\"#@Fd\\lQ&cableF_emFf]l-F gdm6&7$$\"$j\"F)$\"$<\"F)Q\"qF_emFj^m-%%FONTG6#Fc]l-Fgdm6&7$$\"$:#F)Fc emFhgmF\\bmFigm-Fgdm6&7$$\"$&GF)F\\emFhgmF^cmFigm-Fgdm6%7$$FgcmF)$F]\\ lFd\\lQ\"PF_emF]_l-Fgdm6%7$$\"$D\"F)$\"#6Fd\\lF[imF]_l-Fgdm6%7$Fd_m$!# :F)Q&(x,0)F_emF]_l-Fgdm6%7$$\"$#GF)$Fd\\lFd\\lQ\"xF_emF]_l-Fgdm6%7$F^j m$\"#GFd\\lQ\"yF_emF]_l-Fgdm6%7$$Fj]lF)$\"#:F)FhfmF]_l-%*AXESTICKSG6$F *F*-%+AXESLABELSG6%F_jmQ!F_em-Fjgm6#%(DEFAULTG-%%VIEWG6$;F($\"#HFd\\l; FgfmF\\jm" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Cu rve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Cu rve 8" "Curve 9" "Curve 10" "Curve 11" "Curve 12" "Curve 13" "Curve 14 " "Curve 15" "Curve 16" "Curve 17" "Curve 18" "Curve 19" "Curve 20" "C urve 21" "Curve 22" "Curve 23" "Curve 24" "Curve 25" }}{TEXT -1 2 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 113 "Thre e forces are acting on this piece of cable: the weight of the cable ac ting vertically downwards, the tension " }{XPPEDIT 18 0 "T[0]" "6#&%\" TG6#\"\"!" }{TEXT -1 34 " in the cable at the lowest point " } {XPPEDIT 18 0 "P[0]" "6#&%\"PG6#\"\"!" }{TEXT -1 38 " acting horizonta lly, and the tension " }{TEXT 289 1 "T" }{TEXT -1 27 " in the cable at the point " }{TEXT 300 1 "P" }{TEXT -1 67 " acting in the direction o f the tangent to the cable at this point." }}{PARA 0 "" 0 "" {TEXT -1 77 "Suppose that the weight per unit length of the cable is uniform an d equal to " }{TEXT 263 1 "w" }{TEXT -1 69 " Newtons per metres. Then, if the length of this section of cable is " }{TEXT 264 1 "s" }{TEXT -1 30 " metres, the total weight is " }{XPPEDIT 18 0 "w*`.`*s;" "6#*( %\"wG\"\"\"%\".GF%%\"sGF%" }{TEXT -1 34 " Newtons. If the tangent lin e at " }{TEXT 302 1 "P" }{TEXT -1 16 " makes an angle " }{XPPEDIT 18 0 "theta" "6#%&thetaG" }{TEXT -1 97 " with the horizontal, then the ho rizontal and vertical components of the tension in the cable at " } {TEXT 301 1 "P" }{TEXT -1 5 " are " }{XPPEDIT 18 0 "T*cos*theta;" "6#* (%\"TG\"\"\"%$cosGF%%&thetaGF%" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "T* sin*theta;" "6#*(%\"TG\"\"\"%$sinGF%%&thetaGF%" }{TEXT -1 14 " respect ively." }}{PARA 0 "" 0 "" {TEXT -1 8 "We have:" }}{PARA 256 "" 0 "" {TEXT -1 7 " " }{XPPEDIT 18 0 "PIECEWISE([T[0] = T*cos*theta, `` ],[w*s = T*sin*theta, ``]);" "6#-%*PIECEWISEG6$7$/&%\"TG6#\"\"!*(F)\" \"\"%$cosGF-%&thetaGF-%!G7$/*&%\"wGF-%\"sGF-*(F)F-%$sinGF-F/F-F0" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 5 "Hence" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "tan*theta = w*s/T[0];" "6#/*&%$tanG \"\"\"%&thetaGF&*(%\"wGF&%\"sGF&&%\"TG6#\"\"!!\"\"" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 72 "Taking the origin to be at the lowest poi nt of the cable the derivative " }{XPPEDIT 18 0 "dy/dx" "6#*&%#dyG\"\" \"%#dxG!\"\"" }{TEXT -1 58 " for the curve formed by the cable satisfi es the equation:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "d y/dx=k*s" "6#/*&%#dyG\"\"\"%#dxG!\"\"*&%\"kGF&%\"sGF&" }{TEXT -1 14 " \+ ------- (i)," }}{PARA 0 "" 0 "" {TEXT -1 7 " where " }{XPPEDIT 18 0 " k=w/T[0]" "6#/%\"kG*&%\"wG\"\"\"&%\"TG6#\"\"!!\"\"" }{TEXT -1 6 ", and " }{TEXT 265 1 "s" }{TEXT -1 64 " is the arc length along the cable f rom the origin to the point " }{XPPEDIT 18 0 "P(x,y)" "6#-%\"PG6$%\"xG %\"yG" }{TEXT -1 14 " on the cable." }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 15 "The arc length " }{TEXT 272 1 "s" } {TEXT -1 24 " satisfies the equation:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "ds/dx = sqrt(1+``(dy/dx)^2);" "6#/*&%#dsG\"\"\"%#d xG!\"\"-%%sqrtG6#,&F&F&*$-%!G6#*&%#dyGF&F'F(\"\"#F&" }{TEXT -1 16 " - ------ (ii). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 257 "" 0 "" {TEXT 260 36 "Not e on the arc length formula (ii) " }{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 27 "In general, the arc length " }{TEXT 295 1 "s" }{TEXT -1 15 " along a curve " } {XPPEDIT 18 0 "y = f(x)" "6#/%\"yG-%\"fG6#%\"xG" }{TEXT -1 13 " from a point" }{XPPEDIT 18 0 "``(a,f(a));" "6#-%!G6$%\"aG-%\"fG6#F&" }{TEXT -1 24 " on the curve to a point" }{XPPEDIT 18 0 "``(b,f(b));" "6#-%!G6 $%\"bG-%\"fG6#F&" }{TEXT -1 13 " is given by:" }}{PARA 256 "" 0 "" {TEXT -1 4 " " }{XPPEDIT 18 0 "s = Int(sqrt(1+(dy/dx)^2),x = a .. b );" "6#/%\"sG-%$IntG6$-%%sqrtG6#,&\"\"\"F,*$*&%#dyGF,%#dxG!\"\"\"\"#F, /%\"xG;%\"aG%\"bG" }{TEXT -1 18 " ------- (ii a), " }}{PARA 0 "" 0 " " {TEXT -1 30 " provided that the derivative " }{XPPEDIT 18 0 "dy/dx; " "6#*&%#dyG\"\"\"%#dxG!\"\"" }{TEXT -1 37 " exists throughout the int erval from " }{XPPEDIT 18 0 "x = a" "6#/%\"xG%\"aG" }{TEXT -1 4 " to \+ " }{XPPEDIT 18 0 "x = b" "6#/%\"xG%\"bG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 4 "Note" }{TEXT -1 92 " : A rough explanation of (ii a) can be obtained by starting with the a pproximate equation: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "delta*s^2" "6#*&%&deltaG\"\"\"*$%\"sG\"\"#F%" }{TEXT -1 1 " " } {TEXT 268 1 "~" }{TEXT -1 1 " " }{XPPEDIT 18 0 "delta*x^2 + delta*y^2 " "6#,&*&%&deltaG\"\"\"*$%\"xG\"\"#F&F&*&F%F&*$%\"yGF)F&F&" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 19 "for a small change " }{XPPEDIT 18 0 "delta;" "6#%&deltaG" }{TEXT 291 1 "x" }{TEXT -1 4 " in " }{TEXT 287 1 "x" }{TEXT -1 28 ", and corresponding changes " }{XPPEDIT 18 0 " delta;" "6#%&deltaG" }{TEXT 292 1 "y" }{TEXT -1 4 " in " }{TEXT 286 1 "y" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "delta;" "6#%&deltaG" }{TEXT 293 1 "s" }{TEXT -1 19 " in the arc length " }{TEXT 271 1 "s" }{TEXT -1 51 " measured from some specific point along the curve " }{XPPEDIT 18 0 "y=f(x)" "6#/%\"yG-%\"fG6#%\"xG" }{TEXT -1 41 ", as suggested by \+ the following picture. " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {GLPLOT2D 291 188 188 {PLOTDATA 2 "6/-%'CURVESG6$7%7$$\"\"!F)F(7$$\"\" $F)F(7$F+$\"\"\"F)-%'COLOURG6&%$RGBGF)F)F)-F$6$7%7$$\"3#)************* z#!# 7$$!3Fnmm\"HU,\"*)F>$!3LqDedvy2DF>7$$!3)HL$3FH'='zF>$!3[60))4\"e4D#F>7 $$!35mm;/OU&*oF>$!3'[E0!oF>7$$!3Plm;H_\">#eF>$!3#)Hv0r)oDm\"F>7$$! 3SLL3_!4Nv%F>$!3G]tLd[Bk8F>7$$!3YmmTg(fHw$F>$!3a.&G!3<#\\3\"F>7$$!3^++ vVLIPFF>$!3#[&*>gu=&Hz!#>7$$!3#em;/,oln\"F>$!3q:e.EWX!)[Feo7$$!3Y&*** \\(oWB>'Feo$!3W)ocPU38\"=Feo7$$\"3C%ommTIOo%Feo$\"3w&3(eu(*yw8Feo7$$\" 3eML$3_>jU\"F>$\"3f<(4!fd*4@%Feo7$$\"3E,++D;v/DF>$\"3`=b(>%=#4V(Feo7$$ \"3y+++v=h(e$F>$\"3]>F_Gu_p5F>7$$\"3V+++v$[6j%F>$\"3o%RwN!p1(Q\"F>7$$ \"3EMLe*[z(ybF>$\"3o<#oU-Rzn\"F>7$$\"3knmmTXg0nF>$\"3YQfwT3$p-#F>7$$\" 3OommmJ$\"3$3WA$*z@_K#F>7$$\"3U++D1Mcq()F>$\"3caK&*3BEvEF>7$$\"3' fmmm\"pW`(*F>$\"3-pR1J2&y)HF>7$$\"3K+]i!f#=$3\"F:$\"36**)=c`tPL$F>7$$ \"3?+](=xpe=\"F:$\"3'o+_%zi0mOF>7$$\"37nm\"H28IH\"F:$\"37$$ \"3um;zpSS\"R\"F:$\"3eI@(f(feRVF>7$$\"3GLL3_?`(\\\"F:$\"3g*G@*=Vx\"p%F >7$$\"3fL$e*)>pxg\"F:$\"3w\"*=7B#z21&F>7$$\"33+]Pf4t.7$$\"3uLLe*Gst!=F:$\"3N!et%G87$$\"30+++DRW9>F:$\"3JA'obe\"Q/hF> 7$$\"3:++DJE>>?F:$\"3y)p&GpNemkF>7$$\"3F+]i!RU07#F:$\"3&\\hnKo<)>oF>7$ $\"3+++v=S2LAF:$\"3-;*[[1L_@(F>7$$\"3Jmmm\"p)=MBF:$\"3Sqh7U7$$\" 3B++](=]@W#F:$\"3iquGbN&)ezF>7$$\"35L$e*[$z*RDF:$\"3KtwDOa!3J)F>7$$\"3 e++]iC$pk#F:$\"3oT-4&3-&)p)F>7$$\"3[m;H2qcZFF:$\"396*Qj\")zg1*F>7$$\"3 O+]7.\"fF&GF:$\"3uFr'HI\">`%*F>7$$\"3Ymm;/OgbHF:$\"3P%oa8E;X$)*F>7$$\" 3w**\\ilAFjIF:$\"3**[AGC]nB5F:7$$\"3yLLL$)*pp;$F:$\"3/lrH5\"F:7$$\"3Cn;HdO=yLF:$\"3T=<=Pb4V6F:7$$\" 3a+++D>#[Z$F:$\"3m))[AIHF!=\"F:7$$\"3SnmT&G!e&e$F:$\"3h\"*e#pp)=B7F:7$ $\"3#RLLL)Qk%o$F:$\"3_wV<8-&=E\"F:7$$\"37+]iSjE!z$F:$\"3!44oCrfLI\"F:7 $$\"3a+]P40O\"*QF:$\"3[w\\sP#oLM\"F:7$$\"\"%F)$\"31++!ommmQ\"F:-F16&F3 $\"*++++\"!\")F(F(-%*THICKNESSG6#\"\"#-%%TEXTG6%7$$\"$W\"!\"#$!#:Fb]lQ \"d6\"-%%FONTG6#%'SYMBOLG-F]]l6%7$$\"$D$Fb]l$\"\"&FEFe]lFg]l-F]]l6%7$$ \"$C\"Fb]l$\"#jFb]lFe]lFg]l-F]]l6%7$$\"$h\"Fb]lFc]lQ\"xFf]l-Fh]l6$%*HE LVETICAG\"#5-F]]l6%7$$\"$U$Fb]lF`^lQ\"yFf]lF__l-F]]l6%7$$\"$T\"Fb]lFg^ lQ\"sFf]lF__l-F]]l6%7$$\"#RFE$\"#6FEQ)y~=~f(x)Ff]lFc\\l-%+AXESLABELSG6 %Q!Ff]lFj`l-Fh]l6#%(DEFAULTG-%*AXESSTYLEG6#%%NONEG-%%VIEWG6$;FDF_\\lF] al" 1 2 0 1 10 0 2 9 1 1 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" }}{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 12 " This gives: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``(de lta*s/(delta*x))^2;" "6#*$-%!G6#*(%&deltaG\"\"\"%\"sGF)*&F(F)%\"xGF)! \"\"\"\"#" }{TEXT -1 1 " " }{TEXT 269 1 "~" }{TEXT -1 1 " " }{XPPEDIT 18 0 "1+``(delta*y/(delta*x))^2;" "6#,&\"\"\"F$*$-%!G6#*(%&deltaGF$%\" yGF$*&F*F$%\"xGF$!\"\"\"\"#F$" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 8 "so that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "delta*s/(delta*x)" "6#*(%&deltaG\"\"\"%\"sGF%*&F$F%%\"xGF%!\"\"" } {TEXT -1 1 " " }{TEXT 270 1 "~" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sqrt(1 +``(delta*y/(delta*x))^2);" "6#-%%sqrtG6#,&\"\"\"F'*$-%!G6#*(%&deltaGF '%\"yGF'*&F-F'%\"xGF'!\"\"\"\"#F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 16 "In the limit as " }{XPPEDIT 18 0 "delta;" "6#%&deltaG" } {TEXT 290 1 "x" }{TEXT -1 39 " tends to zero we obtain the equation: \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "ds/dx = sqrt(1+`` (dy/dx)^2);" "6#/*&%#dsG\"\"\"%#dxG!\"\"-%%sqrtG6#,&F&F&*$-%!G6#*&%#dy GF&F'F(\"\"#F&" }{TEXT -1 15 " ------- (ii), " }}{PARA 0 "" 0 "" {TEXT -1 50 "which gives (ii a) by integration with respect to " } {TEXT 288 1 "x" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 "Differentiating (i) with respect to " }{TEXT 294 1 "x " }{TEXT -1 23 " and using (ii) gives: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "d^2*y/(d*x^2) = k*sqrt(1+``(dy/dx)^2);" "6#/*(% \"dG\"\"#%\"yG\"\"\"*&F%F(*$%\"xGF&F(!\"\"*&%\"kGF(-%%sqrtG6#,&F(F(*$- %!G6#*&%#dyGF(%#dxGF,F&F(F(" }{TEXT -1 16 " ------- (iii). " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 266 20 "____________________" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 67 "The 2nd order differential equation (iii) can be solved a s follows." }}{PARA 0 "" 0 "" {TEXT -1 17 "First substitute " } {XPPEDIT 18 0 "u=dy/dx" "6#/%\"uG*&%#dyG\"\"\"%#dxG!\"\"" }{TEXT -1 75 " to obtain the 1st order differential equation (with separable var iables): " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "du/dx=k* sqrt(1+u^2)" "6#/*&%#duG\"\"\"%#dxG!\"\"*&%\"kGF&-%%sqrtG6#,&F&F&*$%\" uG\"\"#F&F&" }{TEXT -1 3 ", " }}{PARA 0 "" 0 "" {TEXT -1 3 "or " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "1/sqrt(1+u^2)" "6#*& \"\"\"F$-%%sqrtG6#,&F$F$*$%\"uG\"\"#F$!\"\"" }{TEXT -1 1 " " } {XPPEDIT 18 0 "du/dx= k" "6#/*&%#duG\"\"\"%#dxG!\"\"%\"kG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 39 "Integrating both sides with resp ect to " }{TEXT 296 1 "x" }{TEXT -1 8 " gives: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(1/sqrt(1+u^2),u) = Int(k,x);" "6#/- %$IntG6$*&\"\"\"F(-%%sqrtG6#,&F(F(*$%\"uG\"\"#F(!\"\"F.-F%6$%\"kG%\"xG " }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 5 "Thus " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "arcsinh*u = k*x+c[1];" "6#/*&%(arc sinhG\"\"\"%\"uGF&,&*&%\"kGF&%\"xGF&F&&%\"cG6#F&F&" }{TEXT -1 2 ". " } }{PARA 0 "" 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 "dy/dx=u" "6#/*&%# dyG\"\"\"%#dxG!\"\"%\"uG" }{TEXT -1 10 " = 0 when " }{XPPEDIT 18 0 "x \+ = 0" "6#/%\"xG\"\"!" }{TEXT -1 18 ", it follows that " }{XPPEDIT 18 0 "c[1] = 0;" "6#/&%\"cG6#\"\"\"\"\"!" }{TEXT -1 10 ", so that " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "arcsinh*u = k*x;" "6# /*&%(arcsinhG\"\"\"%\"uGF&*&%\"kGF&%\"xGF&" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 12 "which gives " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "u = sinh*k*x;" "6#/%\"uG*(%%sinhG\"\"\"%\"kGF'%\"xGF '" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 9 "that is, " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = sinh*k*x;" "6#/*&%# dyG\"\"\"%#dxG!\"\"*(%%sinhGF&%\"kGF&%\"xGF&" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 4 "Then" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "y = Int(sinh*k*x,x);" "6#/%\"yG-%$IntG6$*(%%sinhG\"\"\" %\"kGF*%\"xGF*F," }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "` ` = 1/k" "6#/%!G*&\"\"\"F&%\"kG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 " cosh*k*x+c[2];" "6#,&*(%%coshG\"\"\"%\"kGF&%\"xGF&F&&%\"cG6#\"\"#F&" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 "y =0" "6#/%\"yG\"\"!" }{TEXT -1 6 " when " }{XPPEDIT 18 0 "x=0" "6#/%\"x G\"\"!" }{TEXT -1 18 ", it follows that " }{XPPEDIT 18 0 "c[2] = -1/k; " "6#/&%\"cG6#\"\"#,$*&\"\"\"F*%\"kG!\"\"F," }{TEXT -1 17 ", and we ob tain: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y = (cosh*k *x-1)/k" "6#/%\"yG*&,&*(%%coshG\"\"\"%\"kGF)%\"xGF)F)F)!\"\"F)F*F," } {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 267 9 "______ ___" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 31 "for the equation o f the curve. " }}{PARA 0 "" 0 "" {TEXT -1 8 "Maple's " }{TEXT 0 6 "dso lve" }{TEXT -1 70 " can be used to obtain this result, although the ex traneous solution " }{XPPEDIT 18 0 "y(x) = -cosh*x*k/k+1/k;" "6#/-%\" yG6#%\"xG,&**%%coshG\"\"\"F'F+%\"kGF+F,!\"\"F-*&F+F+F,F-F+" }{TEXT -1 16 " is also given." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "de := diff(y(x),x$2)=k*sqrt(1+diff(y(x),x )^2);\nic := y(0)=0,D(y)(0)=0;\ndsolve(\{de,ic\},y(x));" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xG-%\"$G6$F,\"\"#*&% \"kG\"\"\"-%%sqrtG6#,&F3F3*$)-F'6$F)F,F0F3F3F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG6$/-%\"yG6#\"\"!F*/--%\"DG6#F(F)F*" }}{PARA 11 " " 1 "" {XPPMATH 20 "6$/-%\"yG6#%\"xG,&*&%\"kG!\"\"-%%coshG6#*&F'\"\"\" F*F0F0F+*&F0F0F*F+F0/F$,&F)F0F1F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 65 "Web links to pictures of catenary curves and further in formation " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 45 "1. The catenary and parabola in engineering. " }}{PARA 0 "" 0 " " {URLLINK 17 "Brantacan Bridges" 4 "http://www.brantacan.co.uk/engcur ves.htm" "" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 107 "2. You can get a good idea of what a larger piece of catenary looks like from the Gateway Arch in St Louis." }}{PARA 0 "" 0 "" {URLLINK 17 "Gateway Arch" 4 "http://www.nps.gov/jeff/main.htm " "" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 36 "Comparison of catenary with parabola" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 1" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {PARA 0 "" 0 "" {TEXT 275 8 "Question" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 143 "(a) Find the equation of the catenary with its axis of symmetry vertical, and lowest point at the origin, such that it pa sses through the point" }{XPPEDIT 18 0 " ``(1,1)" "6#-%!G6$\"\"\"F&" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 82 "(b) Compare the graph of the catenary found in (a) with the graph of the parabola " }{XPPEDIT 18 0 "y=x^2" "6#/%\"yG*$%\"xG\"\"#" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 276 8 "Solution" }{TEXT -1 2 ": \+ " }}{PARA 0 "" 0 "" {TEXT -1 41 "The catenary has an equation of the f orm:" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "y = (cosh*k* x-1)/k;" "6#/%\"yG*&,&*(%%coshG\"\"\"%\"kGF)%\"xGF)F)F)!\"\"F)F*F," } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 40 "Since the curve passes \+ through the point" }{XPPEDIT 18 0 " ``(1,1)" "6#-%!G6$\"\"\"F&" } {TEXT -1 9 " we have:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "1 = (cosh*k-1)/k;" "6#/\"\"\"*&,&*&%%coshGF$%\"kGF$F$F$!\"\"F$F) F*" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 6 "Hence " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "cosh*k-1 = k;" "6#/,&*&%%coshG \"\"\"%\"kGF'F'F'!\"\"F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 78 "The following graph indicates that there is exactly one positive solution for " }{TEXT 277 1 "k" }{TEXT -1 6 " with " }{TEXT 304 1 "k" }{TEXT -1 1 " " }{TEXT 303 1 "~" } {TEXT -1 6 " 1.6. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 29 "plot([cosh(k)-1,k],k=0..2.5);" }}{PARA 13 " " 1 "" {GLPLOT2D 399 298 298 {PLOTDATA 2 "6&-%'CURVESG6$7S7$$\"\"!F)F( 7$$\"3WmmmT&)G\\a!#>$\"3]\\6NCZ5&[\"!#?7$$\"3PL$ek`o!>5!#=$\"37$)GzZ') *p>&F07$$\"3omm\"z>)G_:F4$\"3u/!\\V]?s?\"F-7$$\"3-nmT&QU!*3#F4$\"3M&[x -;'***=#F-7$$\"3HL$eRZXKi#F4$\"3Ka?;8X[gMF-7$$\"3xm;z>,_=JF4$\"3Rz'\\+ r>@!\\F-7$$\"3v**\\7G$[8j$F4$\"39g#GrB=hm'F-7$$\"35n;z%*frhTF4$\"3QS[3 \"zacy)F-7$$\"3A+]ilFQ!p%F4$\"3;?Zm)3*H?6F47$$\"3@ML$3_\"=M_F4$\"3Fa!e X%QR,9F47$$\"3HnmTg(fJr&F4$\"3c-))z_k)on\"F47$$\"3k++]7eP_iF4$\"3XY!)e H27>?F47$$\"3Q++]Pf!Qz'F4$\"3c'Q=&\\=$zR#F47$$\"3@++](=ubJ(F4$\"3/Y:(e nptz#F47$$\"37n;zW(*Q*y(F4$\"3y0ssDxD!>$F47$$\"3#QLL3F-GN)F4$\"3mqy(fi kgp$F47$$\"3=MLL$e'3I))F4$\"3c\"zi$[_]eTF47$$\"3?+]7.E;lhB\"f'F47$$\"3cL$ek`1l9\"Fiq$\"3 $4^b,&*>YK(F47$$\"3OLe*[.-d>\"Fiq$\"3I$[e\")eY=/)F47$$\"3km;/Egw[7Fiq$ \"3/B![(ze[k))F47$$\"3zm\"z%*f%)QI\"Fiq$\"3%QO*GJlEv(*F47$$\"3/+voza'= N\"Fiq$\"3wa(**)\\Eph5Fiq7$$\"3(om\"zWho.9Fiq$\"3]X(4_iKz:\"Fiq7$$\"3- ++]i>Ad9Fiq$\"3+nh:s\"[ME\"Fiq7$$\"32+]i:jf4:Fiq$\"3;<3C$[^HP\"Fiq7$$ \"39+DJ&>r-c\"Fiq$\"3;C$)eSb4&[\"Fiq7$$\"3++]P4q`;;Fiq$\"3sTSP#\\.rh\" Fiq7$$\"3;LL$eM%4n;Fiq$\"36+J*)f[xUFiq$\"3ZM/ $yVV^]#Fiq7$$\"3CLL3-=!y(>Fiq$\"350Jf\"G1Eo#Fiq7$$\"3))*\\7G8O;.#Fiq$ \"3AbAF8v$)yGFiq7$$\"3!pmm;*\\[$3#Fiq$\"396\"\\wLa%yIFiq7$$\"3*pmT&Qz] O@Fiq$\"3#[;'>Ew$RH$Fiq7$$\"3iLekG=4*=#Fiq$\"3)p-lSEg&>NFiq7$$\"3F++]i 4TPAFiq$\"380))3r())yt$Fiq7$$\"3qL$3F9!z#H#Fiq$\"3B_\"*)QZc<+%Fiq7$$\" 3'pmm;%>KUBFiq$\"3YF4s7nt]UFiq7$$\"3/+DJqJ8&R#Fiq$\"3$4wqkm4/`%Fiq7$$ \"3G+voa-oXCFiq$\"3+?4To:`7[Fiq7$$\"3++++++++DFiq$\"3w&oj'z%*GK^Fiq-%' COLOURG6&%$RGBG$\"#5!\"\"F(F(-F$6$7SF'7$F+F+7$F2F27$F8F87$F=F=7$FBFB7$ FGFG7$FLFL7$FQFQ7$FVFV7$FenFen7$FjnFjn7$F_oF_o7$FdoFdo7$FioFio7$F^pF^p 7$FcpFcp7$FhpFhp7$F]qF]q7$FbqFbq7$FgqFgq7$F]rF]r7$FbrFbr7$FgrFgr7$F\\s F\\s7$FasFas7$FfsFfs7$F[tF[t7$F`tF`t7$FetFet7$FjtFjt7$F_uF_u7$FduFdu7$ FiuFiu7$F^vF^v7$FcvFcv7$FhvFhv7$F]wF]w7$FbwFbw7$FgwFgw7$F\\xF\\x7$FaxF ax7$FfxFfx7$F[yF[y7$F`yF`y7$FeyFey7$FjyFjy7$F_zF_z7$FdzFdz-Fiz6&F[[lF( F\\[lF(-%+AXESLABELSG6$Q\"k6\"Q!Fh^l-%%VIEWG6$;F($\"#DF^[l%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Cur ve 2" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 26 "We can solve the equation " }{XPPEDIT 18 0 "cosh*k-1 = k;" "6#/,&*&%% coshG\"\"\"%\"kGF'F'F'!\"\"F(" }{TEXT -1 19 " numerically using " } {TEXT 0 6 "fsolve" }{TEXT -1 36 " for the required positive solution \+ " }{XPPEDIT 18 0 "k = k[0];" "6#/%\"kG&F$6#\"\"!" }{TEXT -1 11 " by gi ving " }{TEXT 0 6 "fsolve" }{TEXT -1 27 " the starting approximation" }{XPPEDIT 18 0 "k=1.6" "6#/%\"kG-%&FloatG6$\"#;!\"\"" }{TEXT -1 2 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "k0 := evalf(evalf(fsolve(cosh(k)-1=k,k=1.6),13));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#k0G$\"+9v8;;!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 "We can now plot the graph of " }{XPPEDIT 18 0 "y = (cosh*k[0]*x-1)/k[0];" "6#/%\"yG*&,&*(%%coshG\"\" \"&%\"kG6#\"\"!F)%\"xGF)F)F)!\"\"F)&F+6#F-F/" }{TEXT -1 25 " together \+ with the graph " }{XPPEDIT 18 0 "y=x^2" "6#/%\"yG*$%\"xG\"\"#" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 119 "plot([(cosh(k0*x)-1)/k0,x^2],x=0..1.2,y,thickness=2, \ncolor=[COLOR(RGB,.6,.3,.8),coral],legend=[`catenary`,`parabola`]);" }}{PARA 13 "" 1 "" {GLPLOT2D 416 351 351 {PLOTDATA 2 "6'-%'CURVESG6%7S 7$$\"\"!F)F(7$$\"3h*******\\ech#!#>$\"3lTuP3EOHb!#@7$$\"3-+++v*G:*[F-$ \"3E`Q^jxZM>!#?7$$\"3u******\\L)4X(F-$\"3wIkQw7f\"\\%F67$$\"3)******\\ MSF+\"!#=$\"3i6CB[M$G9)F67$$\"3#)****\\Fy:f7F?$\"3/)G0#QHg&G\"F-7$$\"3 ')****\\d'*)o\\\"F?$\"3p>gi.,Z>=F-7$$\"3w****\\(>ZIu\"F?$\"3oW'Rvtj8Z# F-7$$\"3u****\\xOi(*>F?$\"3oA/Z7Rq_KF-7$$\"3#)****\\FPQ^AF?$\"3/4'=#Q! p79%F-7$$\"3/+++IrS7DF?$\"3-*HT\\yZ6<&F-7$$\"3p*****\\o;Bu#F?$\"3u&*y, VT/xhF-7$$\"3*********QS6+$F?$\"3Gxf()QW&>U(F-7$$\"3[******\\o-hKF?$\" 3E0Z**)4'*Rz)F-7$$\"3(*******4cZ6NF?$\"3K(QOo/;M-\"F?7$$\"3S****\\xq!* QPF?$\"3)**H%pMmUk6F?7$$\"3&********3X$4SF?$\"3ww>;\"QZ]M\"F?7$$\"3s** ****f:WQUF?$\"3e>D=7?I4:F?7$$\"3f****\\<_$\\]%F?$\"3h&zoUViOr\"F?7$$\" 3**)*****fs#3u%F?$\"3BOCl'ptn!>F?7$$\"3!)****\\<#Q'**\\F?$\"3GIl5uU>K@ F?7$$\"33++]_u3Y_F?$\"3W=>H=LPgBF?7$$\"3[*****\\PJK]&F?$\"35=4gpy\"Hh# F?7$$\"3%*****\\n(p$RdF?$\"3kgA]B5@eGF?7$$\"3A*****\\#p2%*fF?$\"3;o$Gm %[dPJF?7$$\"3o****\\xgkeiF?$\"3%fiK\"=a\\WMF?7$$\"3g****\\-V&*)['F?$\" 3%f>>b>*)fs$F?7$$\"3.+++&\\$pPnF?$\"3!Q+Ni.^a/%F?7$$\"37******>am%*pF? $\"3!f^qUshGR%F?7$$\"3?*****\\JigC(F?$\"3;APQMER]ZF?7$$\"3G****\\PnG4O%H)F ?7$$\"39)*****\\'[M\\*F?$\"3k*p=dKK$G))F?7$$\"3y)***\\PM&=v*F?$\"3U[!p EwqHT*F?7$$\"3*******fzs++\"!#<$\"3\")R?y$*f<+5F^x7$$\"3(*****\\5Q_D5F ^x$\"3i:'3k-)4j5F^x7$$\"3-++vxSw]5F^x$\"3539Jm$4$G6F^x7$$\"3'******>Ed R2\"F^x$\"3Us$oOY#z!>\"F^x7$$\"3&*****\\o#R05\"F^x$\"3/c]/Cx`l7F^x7$$ \"3()*****>`9V7\"F^x$\"3h>*zk=L`L\"F^x7$$\"3)****\\<#Rm\\6F^x$\"3H9K4r 0$HT\"F^x7$$\"3%****\\A_ER<\"F^x$\"3I[Eq=OR!\\\"F^x7$$\"3%************ **>\"F^x$\"3s%eOS:dsd\"F^x-%&COLORG6&%$RGBG$\"\"'!\"\"$\"\"$F_[l$\"\") F_[l-%'LEGENDG6#%)catenaryG-F$6%7SF'7$F+$\"3xZAi)Qp;%oF07$F2$\"33bkKrb q#R#F67$F8$\"3CAx>)G:)[05F-7$FC$\"3M>XbM%yae\"F-7 $FH$\"3'oYInky1C#F-7$FM$\"3B.ErKN@QIF-7$FR$\"33B'3pN+0*RF-7$FW$\"3i$z^ %)oG(o]F-7$Ffn$\"3hP[(oe*=7jF-7$F[o$\"37)QH3!3I?vF-7$F`o$\"3K_$*[SO%o+ *F-7$Feo$\"3)=4U;hHM1\"F?7$Fjo$\"38([if4YIB\"F?7$F_p$\"3sezT8E%zR\"F?7 $Fdp$\"3z5220[[2;F?7$Fip$\"3/BNv&oQkz\"F?7$F^q$\"3%p<(QJTWH?F?7$Fcq$\" 3!*4f\"4JWvC#F?7$Fhq$\"3Yd')eI#Q'*\\#F?7$F]r$\"37%zFfNV@v#F?7$Fbr$\"3M Q%ymbb&GIF?7$Fgr$\"3?+$4G`OSH$F?7$F\\s$\"3UWa]_F?7$Fjt$\"3Z]?`^S'*3cF?7$F_u$\"3FuDxVTz?gF?7$Fdu$\"3<: MU(*\\G.kF?7$Fiu$\"3J=W1)>xY#oF?7$F^v$\"3PTck$Q<\"=sF?7$Fcv$\"3_:pNQq' 3m(F?7$Fhv$\"3qC_5\")G\\*3)F?7$F]w$\"3AIk1f:+]&)F?7$Fbw$\"30z'=qscD,*F ?7$Fgw$\"3pa![mak)4&*F?7$F\\x$\"3nd#*H(fX,+\"F^x7$Fbx$\"3wRC!f3*p^5F^x 7$Fgx$\"3+3)*3Z^5/6F^x7$F\\y$\"3aODg+UQ`6F^x7$Fay$\"3,:0^\"o'=67F^x7$F fy$\"3yyj'o;$3k7F^x7$F[z$\"3,!ftH8FkzM5y8F^x7$Fez$ \"3%*************R9F^x-%'COLOURG6&F\\[l$\"*++++\"!\")$\")AR!)\\F`elF(- Fe[l6#%)parabolaG-%*THICKNESSG6#\"\"#-%+AXESLABELSG6$Q\"x6\"Q\"yF^fl-% %VIEWG6$;F($\"#7F_[l%(DEFAULTG" 1 2 0 1 10 2 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "catenary" "parabola" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 72 "The following graph shows the vertical distance between the two curves. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "plot(x^2-(co sh(k0*x)-1)/k0,x=0..1.2,color=blue);" }}{PARA 13 "" 1 "" {GLPLOT2D 444 288 288 {PLOTDATA 2 "6&-%'CURVESG6#7W7$$\"\"!F)F(7$$\"3h*******\\e ch#!#>$\"3(4_W-y1BJ\"!#@7$$\"3-+++v*G:*[F-$\"3KBd7y!yAe%F07$$\"3u***** *\\L)4X(F-$\"3KD8\"=,C,1\"!#?7$$\"3)******\\MSF+\"!#=$\"30;[nY&[?\">F; 7$$\"3#)****\\Fy:f7F?$\"3%QL#\\j\\v)*HF;7$$\"3')****\\d'*)o\\\"F?$\"3V aW/Ja37UF;7$$\"3w****\\(>ZIu\"F?$\"37M'H<&z\\ocF;7$$\"3u****\\xOi(*>F? $\"3I!*>QWW'zP(F;7$$\"3#)****\\FPQ^AF?$\"3im=L-lfu#*F;7$$\"3/+++IrS7DF ?$\"3**RN$>!=/T6F-7$$\"3p*****\\o;Bu#F?$\"3&z\\6ylcKM\"F-7$$\"3******* **QS6+$F?$\"3)3P8;?*)[e\"F-7$$\"3[******\\o-hKF?$\"3v5iU<+IS=F-7$$\"3( *******4cZ6NF?$\"3%R*4E\"\\+j4#F-7$$\"3S****\\xq!*QPF?$\"3![eOsyf^L#F- 7$$\"3&********3X$4SF?$\"3\"fM(3RUPCEF-7$$\"3s******f:WQUF?$\"3wM+rNnO rGF-7$$\"3f****\\<_$\\]%F?$\"3M8Q=rp\"y:$F-7$$\"3**)*****fs#3u%F?$\"3f OZjUhq2MF-7$$\"3!)****\\<#Q'**\\F?$\"3mu7#[cRWn$F-7$$\"33++]_u3Y_F?$\" 3,d(ejP+x\"RF-7$$\"3[*****\\PJK]&F?$\"3Y-_xqoPcTF-7$$\"3%*****\\n(p$Rd F?$\"3g&RqI4b#eVF-7$$\"3A*****\\#p2%*fF?$\"3;oP`;(4Kb%F-7$$\"3o****\\x gkeiF?$\"3Lr)z?a'pDZF-7$$\"3g****\\-V&*)['F?$\"3w)HTZ)fjY[F-7$$\"3.+++ &\\$pPnF?$\"3Iy_@qK+U\\F-7$$\"37******>am%*pF?$\"3-+'Q]4Fn*\\F-7$$\"3? *****\\JigC(F?$\"3zCf-HF\\,]F-7$$\"3G****\\PyF22kY F-7$$\"35******\\/;h#)F?$\"3cf1?l<U?CF-7$$\"3'******>EdR2\"F^x$!3&*)e$e1j#3u$F-7$ $\"3&*****\\o#R05\"F^x$!3(Q5aMD/^V&F-7$$\"3-++D+pU76F^x$!3q')3eG6xeiF- 7$$\"3()*****>`9V7\"F^x$!3q#3a8'>+DrF-7$$\"3$***\\(oA*)p8\"F^x$!3s(**4 w\\Mp4)F-7$$\"3)****\\<#Rm\\6F^x$!30GC'>\"QM?\"*F-7$$\"3')*****>A&zh6F ^x$!3\"4vR#o4'\\,\"F?7$$\"3%****\\A_ER<\"F^x$!3f(o11R,H7\"F?7$$\"3#)** \\7hK'p=\"F^x$!3pGib\\\"fYC\"F?7$$\"3%**************>\"F^x$!3#y%eOS:ds 8F?-%+AXESLABELSG6$Q\"x6\"Q!Fb\\l-%'COLOURG6&%$RGBGF(F($\"*++++\"!\")- %%VIEWG6$;F($\"#7!\"\"%(DEFAULTG" 1 2 0 1 10 0 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 260 4 "Note" }{TEXT -1 130 ": An alternative way to compare the catenary with the parabola is to construct a Taylor po lynomial approximation for the catenary." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 161 "f := x -> (cosh(k0*x )-1)/k0;\nchop := _X -> if abs(coeffs(_X))<1.5*10^(-10) then 0 else _X end if:\ntaylor(f(x),x,9);\np := unapply(map(chop,convert(%,polynom)) ,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operator G%&arrowGF(*&,&-%%coshG6#*&%#k0G\"\"\"9$F3F3F3!\"\"F3F2F5F(F(F(" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#+3%\"xG$\"+qvo!3)!#5\"\"#$\"+b%=vB\"!# >\"\"$$\"+*HH)e.$F2 \"\"($\"+Q&G@9(!#8\"\")-%\"OG6#\"\"\"\"\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"pGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,**$)9$\"\"#\" \"\"$\"+qvo!3)!#5*&$\"+*HH)e " 0 "" {MPLTEXT 1 0 47 "plot((cosh (k0*x)-1)/k0-p(x),x=0..1,color=blue);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6#7Z7$$\"\"!F)F(7$$\"3emmm;arz@!# >$\"3iLYP\"4Iwg(!#K7$$\"3[LL$e9ui2%F-$\"3YJb>KP;&p#!#J7$$\"3nmmm\"z_\" 4iF-$\"3ydW:R\"o([kF67$$\"3[mmmT&phN)F-$\"3FfN/0U>@7!#I7$$\"3CLLe*=)H \\5!#=$\"3UE![1(*He0#FA7$$\"3gmm\"z/3uC\"FE$\"3?ki(*omb2KFA7$$\"3%)*** \\7LRDX\"FE$\"37A8OAB8C_FA7$$\"3]mm\"zR'ok;FE$\"3\"ydff+Dq_*FA7$$\"3w* **\\i5`h(=FE$\"3w+H8nn;g>!#H7$$\"3WLLL3En$4#FE$\"3WpQ(*z>B)[%Ffn7$$\"3 qmm;/RE&G#FE$\"3$*)Q&pq\\o1&*Ffn7$$\"3\")*****\\K]4]#FE$\"3s5%HBG9X<#! #G7$$\"3$******\\PAvr#FE$\"39-wvH(>+![Ffo7$$\"3)******\\nHi#HFE$\"3K>C yozkf)*Ffo7$$\"3jmm\"z*ev:JFE$\"3HmmcE$pt#=!#F7$$\"3?LLL347TLFE$\"3%3p 9Na&)ok$Ffp7$$\"3,LLLLY.KNFE$\"3st'4**exNL'Ffp7$$\"3w***\\7o7Tv$FE$\"3 /D%eJMsB;\"!#E7$$\"3'GLLLQ*o]RFE$\"3hy\"e/X\"yL>Ffq7$$\"3A++D\"=lj;%FE $\"31u<\"eOUvG$Ffq7$$\"31++vV&R*y;`Ffq7$$\"3WLL$e9Ege%F E$\"3I*GZ\"zV2y&)Ffq7$$\"3GLLeR\"3Gy%FE$\"3=wv9>yf,#F`s7$$\"3&em;zRQb@&FE$\"3%>*[x2%\\d5$F`s7$$\"3\\*** \\(=>Y2aFE$\"3)=;jeKn)eWF`s7$$\"39mm;zXu9cFE$\"3-\\joA8O(\\'F`s7$$\"3l ******\\y))GeFE$\"3)3.syr^2X*F`s7$$\"3'*)***\\i_QQgFE$\"3Eg)4AQpeM\"!# C7$$\"3@***\\7y%3TiFE$\"3;knY@&=K(=F_u7$$\"35****\\P![hY'FE$\"3?Q!G\\J ))4n#F_u7$$\"3kKLL$Qx$omFE$\"3,?\"H$e$ygj$F_u7$$\"3!)*****\\P+V)oFE$\" 32UhHklY.]F_u7$$\"3?mm\"zpe*zqFE$\"3)e\"fvp]LDmF_u7$$\"3%)*****\\#\\'Q H(FE$\"3'zQY(zriF*)F_u7$$\"3GKLe9S8&\\(FE$\"3)zeY&4uus6!#B7$$\"3R***\\ i?=bq(FE$\"3Q-P%=**[xa\"Fcw7$$\"3\"HLL$3s?6zFE$\"3-@82DG]:?Fcw7$$\"3a* **\\7`Wl7)FE$\"3%Gm*\\sa@QEFcw7$$\"3#pmmm'*RRL)FE$\"3i*H:IvAmR$Fcw7$$ \"3Qmm;a<.Y&)FE$\"3#p==QPZ,P%Fcw7$$\"3=LLe9tOc()FE$\"3'zMn=WSqd&Fcw7$$ \"3u******\\Qk\\*)FE$\"3.XMW'G2D%pFcw7$$\"31nmT5ASg!*FE$\"3%=t#)pvkU&y Fcw7$$\"3CLL$3dg6<*FE$\"3C9+&3kQD())Fcw7$$\"3y***\\(oTAq#*FE$\"3WZQrlq X#))*Fcw7$$\"3ImmmmxGp$*FE$\"3M([)p'4\"[*4\"!#A7$$\"3sK$eRA5\\Z*FE$\"3 @$H#zigRI7F[[l7$$\"3A++D\"oK0e*FE$\"3c^Tw\"G&>v8F[[l7$$\"3C+++]oi\"o*F E$\"3YP!e%fl)z_\"F[[l7$$\"3A++v=5s#y*FE$\"3#Rqkjo5fp\"F[[l7$$\"3;+D1k2 /P)*FE$\"3G$pj)>o%Gz\"F[[l7$$\"35+]P40O\"*)*FE$\"3!yl])f\"[Z*=F[[l7$$ \"31+voa-oX**FE$\"3!)4v;c*R=+#F[[l7$$\"\"\"F)$\"3*z4p\"yjN9@F[[l-%'COL OURG6&%$RGBGF(F($\"*++++\"!\")-%+AXESLABELSG6$Q\"x6\"Q!F_^l-%%VIEWG6$; F(F`]l%(DEFAULTG" 1 2 0 1 10 0 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Exampl e 2" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT 273 8 "Question" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 112 "(a) Find the equation of the parabola with its axis of symmetry v ertical, such that it passes through the points" }{XPPEDIT 18 0 " ``(- 1,1)" "6#-%!G6$,$\"\"\"!\"\"F'" }{TEXT -1 4 " and" }{XPPEDIT 18 0 " `` (2,2)" "6#-%!G6$\"\"#F&" }{TEXT -1 33 " and has its lowest point on th e " }{TEXT 305 1 "x" }{TEXT -1 14 " axis between " }{XPPEDIT 18 0 "x=- 1" "6#/%\"xG,$\"\"\"!\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "x=2" "6 #/%\"xG\"\"#" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 112 "(b) Find the equation of the catenary with its axis of symmetry vertical, such that it passes through the points" }{XPPEDIT 18 0 " ``(-1,1)" "6#-%!G 6$,$\"\"\"!\"\"F'" }{TEXT -1 4 " and" }{XPPEDIT 18 0 " ``(2,2)" "6#-%! G6$\"\"#F&" }{TEXT -1 33 " and has its lowest point on the " }{TEXT 306 1 "x" }{TEXT -1 14 " axis between " }{XPPEDIT 18 0 "x=-1" "6#/%\"x G,$\"\"\"!\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "x=2" "6#/%\"xG\"\" #" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 97 "(c) Make a graphical comparison between the catenary found in (b) and the parabola found i n (a). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 274 8 "Solution" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 46 "(a) The p arabola has an equation of the form: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y=k*(x-a)^2" "6#/%\"yG*&%\"kG\"\"\"*$,&%\"xGF'%\"a G!\"\"\"\"#F'" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 45 "where t he lowest point or vertex is the point" }{XPPEDIT 18 0 " ``(a,0)" "6#- %!G6$%\"aG\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 43 "Since the parabola passes through the poi nt" }{XPPEDIT 18 0 " ``(-1,1)" "6#-%!G6$,$\"\"\"!\"\"F'" }{TEXT -1 9 " we have:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "1 = k*(- 1-a)^2;" "6#/\"\"\"*&%\"kGF$*$,&F$!\"\"%\"aGF)\"\"#F$" }{TEXT -1 14 " \+ ------- (i), " }}{PARA 0 "" 0 "" {TEXT -1 37 "and since it passes thro ugh the point" }{XPPEDIT 18 0 " ``(2,2)" "6#-%!G6$\"\"#F&" }{TEXT -1 10 " we have: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "2 = k*(2-a)^2;" "6#/\"\"#*&%\"kG\"\"\"*$,&F$F'%\"aG!\"\"F$F'" }{TEXT -1 15 " ------- (ii). " }}{PARA 0 "" 0 "" {TEXT -1 60 "Hence we need to s olve this pair of simultaneous equations. " }}{PARA 0 "" 0 "" {TEXT -1 18 "From (ii) we have " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "2/k = (a-2)^2;" "6#/*&\"\"#\"\"\"%\"kG!\"\"*$,&%\"aGF&F %F(F%" }{TEXT -1 15 " ------- (iii)," }}{PARA 0 "" 0 "" {TEXT -1 21 "a nd from (i) we have " }{XPPEDIT 18 0 "1/k = (a+1)^2;" "6#/*&\"\"\"F%% \"kG!\"\"*$,&%\"aGF%F%F%\"\"#" }{TEXT -1 9 ", so that" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "2/k=2*(a+1)^2" "6#/*&\"\"#\"\"\"% \"kG!\"\"*&F%F&*$,&%\"aGF&F&F&F%F&" }{TEXT -1 14 " ------- (iv)." }} {PARA 0 "" 0 "" {TEXT -1 25 "Thus (iii) and (iv) give " }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "(a-2)^2=2*(a+1)^2" "6#/*$,&%\"aG \"\"\"\"\"#!\"\"F(*&F(F'*$,&F&F'F'F'F(F'" }{TEXT -1 2 ", " }}{PARA 0 " " 0 "" {TEXT -1 8 "so that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "a^2-4*a+4=2*a^2+4*a+2" "6#/,(*$%\"aG\"\"#\"\"\"*&\"\"%F (F&F(!\"\"F*F(,(*&F'F(*$F&F'F(F(*&F*F(F&F(F(F'F(" }{TEXT -1 2 ", " }} {PARA 0 "" 0 "" {TEXT -1 4 "and " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "a^2+8*a-2=0" "6#/,(*$%\"aG\"\"#\"\"\"*&\"\")F(F&F(F(F'! \"\"\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 5 "Then " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a^2+8*a+16=2+16" "6#/ ,(*$%\"aG\"\"#\"\"\"*&\"\")F(F&F(F(\"#;F(,&F'F(F+F(" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 3 "so " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "(a+4)^2=18" "6#/*$,&%\"aG\"\"\"\"\"%F'\"\"#\"#=" } {TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 12 "which gives " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a+4" "6#,&%\"aG\"\"\"\"\"%F %" }{TEXT -1 3 " = " }{TEXT 278 1 "+" }{TEXT -1 1 " " }{XPPEDIT 18 0 " 3*sqrt(2)" "6#*&\"\"$\"\"\"-%%sqrtG6#\"\"#F%" }{TEXT -1 2 ", " }} {PARA 0 "" 0 "" {TEXT -1 8 "so that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a = -4" "6#/%\"aG,$\"\"%!\"\"" }{TEXT -1 1 " " } {TEXT 279 1 "+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "3*sqrt(2)" "6#*&\"\"$ \"\"\"-%%sqrtG6#\"\"#F%" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "When " }{XPPEDIT 18 0 "a = 3*sqrt(2)- 4, k = 1/(a+1)^2" "6$/%\"aG,&*&\"\"$\"\"\"-%%sqrtG6#\"\"#F(F(\"\"%!\" \"/%\"kG*&F(F(*$,&F$F(F(F(F,F." }{XPPEDIT 18 0 "`` = 1/((3*sqrt(2)-3)^ 2);" "6#/%!G*&\"\"\"F&*$,&*&\"\"$F&-%%sqrtG6#\"\"#F&F&F*!\"\"F.F/" } {XPPEDIT 18 0 "`` = 1/(9*(sqrt(2)-1)^2)" "6#/%!G*&\"\"\"F&*&\"\"*F&*$, &-%%sqrtG6#\"\"#F&F&!\"\"F.F&F/" }{XPPEDIT 18 0 "`` = (sqrt(2)+1)^2/9; " "6#/%!G*&,&-%%sqrtG6#\"\"#\"\"\"F+F+F*\"\"*!\"\"" }{TEXT -1 8 ", sin ce " }{XPPEDIT 18 0 "1/(sqrt(2)-1)=sqrt(2)+1" "6#/*&\"\"\"F%,&-%%sqrtG 6#\"\"#F%F%!\"\"F+,&-F(6#F*F%F%F%" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 32 "Thus the corresponding value of " }{TEXT 280 1 "k" } {TEXT -1 5 " is " }{XPPEDIT 18 0 "k = (3+2*sqrt(2))/9" "6#/%\"kG*&,& \"\"$\"\"\"*&\"\"#F(-%%sqrtG6#F*F(F(F(\"\"*!\"\"" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 16 "Similarly, when " }{XPPEDIT 18 0 "a=-4-3* sqrt(2), k = 1/((a+1)^2)" "6$/%\"aG,&\"\"%!\"\"*&\"\"$\"\"\"-%%sqrtG6# \"\"#F*F'/%\"kG*&F*F**$,&F$F*F*F*F.F'" }{XPPEDIT 18 0 "`` = 1/((-3*sqr t(2)-3)^2);" "6#/%!G*&\"\"\"F&*$,&*&\"\"$F&-%%sqrtG6#\"\"#F&!\"\"F*F/F .F/" }{XPPEDIT 18 0 " ``= 1/(9*(sqrt(2)+1)^2)" "6#/%!G*&\"\"\"F&*&\"\" *F&*$,&-%%sqrtG6#\"\"#F&F&F&F.F&!\"\"" }{XPPEDIT 18 0 "`` = (sqrt(2)-1 )^2/9;" "6#/%!G*&,&-%%sqrtG6#\"\"#\"\"\"F+!\"\"F*\"\"*F," }{TEXT -1 8 ", since " }{XPPEDIT 18 0 "1/(sqrt(2)+1)=sqrt(2)-1" "6#/*&\"\"\"F%,&-% %sqrtG6#\"\"#F%F%F%!\"\",&-F(6#F*F%F%F+" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 32 "Thus the corresponding value of " }{TEXT 281 1 "k" } {TEXT -1 4 " is " }{XPPEDIT 18 0 "k = (3-2*sqrt(2))/9" "6#/%\"kG*&,&\" \"$\"\"\"*&\"\"#F(-%%sqrtG6#F*F(!\"\"F(\"\"*F." }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 8 "Maple's " }{TEXT 0 5 "solve" }{TEXT -1 26 " can obtain these results." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "solve(\{1=k*(a+1)^2,2=k*(a-2)^2\}, \{k,a\}):\nallvalues(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$<$/%\"aG,& !\"%\"\"\"*&\"\"$F(-%%sqrtG6#\"\"#F(F(/%\"kG,&#F(F*F(*&#F.\"\"*F(F+F(F (<$/F%,&F'F(*&F*F(F+F(!\"\"/F0,&F2F(*&#F.F5F(*$F+F(F(F:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 103 "We can also illustrate these solution \+ graphically as the points of intersection of the implicit curves " } {XPPEDIT 18 0 "2 = k*(a-2)^2" "6#/\"\"#*&%\"kG\"\"\"*$,&%\"aGF'F$!\"\" F$F'" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "1 = k* (a+1)^2" "6#/\"\"\"*& %\"kGF$*$,&%\"aGF$F$F$\"\"#F$" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 181 "p1:=plots[i mplicitplot](1=k*(a+1)^2,k=0..1.5,a=-8..8,color=red,grid=[50,50]):\np2 :=plots[implicitplot](2=k*(a-2)^2,k=0..1.5,a=-8..8,color=blue,grid=[50 ,50]):\nplots[display]([p1,p2]);" }}{PARA 13 "" 1 "" {GLPLOT2D 340 322 322 {PLOTDATA 2 "6%-%'CURVESG6b[l7$7$$\"3471`Ej\"3/#!#>$!\")\"\"!7 $$\"3V71`Ej\"3/#F*$!3)Q.&eik:\"*y!#<7$7$$\"3Oym.#=:aC#F*$!3o,^v(QpMn(F 3F.7$F57$F6$!3M#yg#G(\\ke(F37$7$$\"3Yh4nCyR#[#F*$!3O.-^v(QpM(F3F;7$F?7 $F@$!3)=ajXE(>&G(F37$7$$\"3M??P0t(*eFF*$!3.0`Ej\"3/-(F3FE7$7$$\"3o??P0 t(*eFF*FL7$FJ$!3EAMfj%o\"))pF37$7$$\"3I=fz*[C71$F*$!3T+05SQ\"\\r'F3FR7 $7$$\"3'z\"fz*[C71$F*$\"3KX2o^g%\\r%F37$$\"3R!)[%>jT2/$F*$\"3+B5bxQpMZ F37$7$F\\o$\"3*Q-^v(QpMZF37$F\\o$\"3'3!=s:(yot%F37$7$$\"3+?iysW%>s#F*$ \"3AAfz*[C71&F3Fdo7$7$$\"3N?iysW%>s#F*F[p7$Fio$\"3?cpFeVT(4&F37$7$$\"3 yI1BX_x]CF*$\"3_?3/-^v(Q&F3Fap7$7$$\"37J1BX_x]CF*Fhp7$Ffp$\"3%QyF\")op GX&F37$7$$\"3E'fk(z#)>=AF*$\"3'zr&G9dG9dF3F^q7$7$$\"3h'fk(z#)>=AF*$\"3 &)=dG9dG9dF37$Fcq$\"3C$RA+^3U!eF37$7$$\"3'pv@CFBs,#F*$\"3<<1`Ej\"3/'F3 F]r7$7$$\"3Jd6=( F37$7$$\"3h@'p$\\%4`V\"F*$\"3Y5-^v(QpM(F3F^u7$7$$\"3F@'p$\\%4`V\"F*Feu 7$Fcu$\"3f#obJrp._(F37$7$$\"36#Q,kNt#H8F*$\"3y3^v(QpMn(F3F[v7$7$$\"3$> Q,kNt#H8F*Fbv7$F`v$\"35gQ)>%3@eyF37$7$$\"3elXB,zcM7F*$\"352++++++!)F3F hv7$FV7$$\"35HG$fU]+3$F*$!3S5](3l&)ep'F37$7$$\"3Uow.[F\\%3$F*$!3q1/-^v (Qp'F3Fbw7$7$$\"31ow.[F\\%3$F*F[x7$$\"3MY<>k_,*Q$F*$!3U'RW!)G6BS'F37$7 $$\"3s'H0-?37Z$F*$!3Q3bxQpMnjF3Fax7$Fgx7$$\"3_zQ4!\\v4v$F*$!3I#Rilw*Q9 hF37$7$$\"3g#=`%G\\[NRF*$!3251`Ej\"3/'F3F]y7$Fcy7$$\"3'4PlsI+\"zTF*$!3 f`C[,e_LeF37$7$$\"3cJs&F*$!3U83/-^v(Q&F3Fez7$F[[l7$$\"3o `j;D#40J&F*$!3)[4-\\>[6I&F37$7$$\"3AI5i2#yH1'F*$!35:fz*[C71&F3Fa[l7$Fg [l7$$\"3!Q]oMnH42'F*$!33fF9(4Hd0&F37$7$$\"3gO=fz*[C7'F*$!3Q/v;hQnS]F3F ]\\l7$7$Fd\\l$\"3%*)e\\hV1%RIF37$$\"3v4f*[?AH%fF*$\"31MlK;3/-JF37$7$$ \"314f*[?AH%fF*$\"3hLlK;3/-JF37$$\"3$e(\\8.wI2fF*$\"3Gl_\")H#*)\\7$F37 $7$$\"3#Q.,!f`&))4&F*$\"3\\J9dG9dGMF3Fg]l7$F]^l7$$\"3`@r^*z&49\\F*$\"3 qngWMCYdNF37$7$$\"3bg0_^qiAWF*$\"3#)Hj\"3/-^v$F3Fc^l7$7$$\"3&)f0_^qiAW F*F\\_l7$$\"3^*RNJby69%F*$\"3m#orHbPk'RF37$7$$\"39d0>&==D(QF*$\"3CF71` Ej\"3%F3Fb_l7$Fh_l7$$\"39Y2bZ6WGNF*$\"3#oF7=#oKeVF37$7$$\"3D+(eRK5!>MF *$\"3dDhIlK;3WF3F^`l7$Fd`l7$FW$\"3@Y2o^g%\\r%F37$7$Fd\\l$!3\\.v;hQnS]F 37$$\"3I721U)e!*z'F*$!3QjV@'flo![F37$7$$\"3yU(3$\\$=&prF*$!3z;5bxQpMZF 3Faal7$Fgal7$$\"3I>8UNyIuwF*$!3i'f%fw[ptXF37$7$$\"3i^YHZ!H\"4')F*$!3Y= hIlK;3WF3F]bl7$Fcbl7$$\"3OKutoKsi()F*$!3%Q3q0\">EjVF37$7$$\"3?axQpMn$= *F*$!3#R*GFN$*='H%F3Fibl7$7$F`cl$\"3klv(Gp0eH#F37$$\"3Ye]Y(RolS)F*$\"3 'ytO=fz*[CF37$Fhcl7$$\"3)p[nPXVd*yF*$\"3a**G]$zeje#F37$7$$\"3X,$))RmR` ,(F*$\"3uN;3/-^vFF3F^dl7$FddlFi\\l7$F_cl7$$\"3W>?Mdl;&*)*F*$!3![n4\"\\ __dTF37$7$$\"3Gse\"\\qgV6 b$F3Fbhl7$7$Fihl$\"3#R$\\)Q\"44_:F37$$\"3im'*zRqBz7Fdel$\"3)=%pMn$=fz \"F37$FailFdfl7$Fhhl7$$\"3!*piX*y!>\"f\"Fdel$!3ybpwU&)=$\\$F37$7$$\"3A 7Q5I<]&p\"Fdel$!3QC9dG9dGMF3Fhil7$F^jl7$$\"3q=%R^,!3uElmXrF8F37$$\"3aI:L1`\"*R;Fdel$\"3+W?5bxQp9F37$7$$\"3FI:L1`\"*R;Fdel$ \"3yV?5bxQp9F3F^il7$7$F[[m$!3KG8ZG8ZGLF37$$\"3)oScAhqU'>Fdel$!3zsREH\" z!QKF37$7$$\"3,Vr&G9dG9#Fdel$!3!ol*zc'*zcJF3Fd\\m7$7$F[[mFc[m7$$\"3%zi ))*\\!yn5#Fdel$\"3MiNe3:M\"=\"F37$7$F[]m$\"3M.xkJ83f6F3Fa]m7$Fj\\m7$$ \"3?az$R6lb<#Fdel$!3TDF\")=$Hp8$F37$7$$\"3-`,$yxsJE#Fdel$!3^ElK;3/-JF3 F[^m7$7$$\"3I`,$yxsJE#Fdel$!3&p_Ej\"3/-JF37$$\"3wF;50W;(Q#Fdel$!3?7C9< Y5OIF37$7$$\"3kMn$=fz*[CFdel$!3k(zh=g$e:IF3F\\_m7$7$$\"3?Nn$=fz*[CFdel $\"3K/.u;2U95F37$$\"3)4xxxxxx<#Fdel$\"38Yr&G9dG9\"F37$F]`m7$$\"3HVr&G9 dG9#Fdel$\"3c.xkJ83f6F37$7$Fc_m$!3>(zh=g$e:IF37$$\"31!R3p:)o0EFdel$!3X FZ#o+mE%HF37$7$$\"3#oK;3/-^v#Fdel$!31U3f/ed)*GF3F\\am7$Fh_m7$$\"3.EIj' e(yDEFdel$\"3o&oxS[>Ea*Fdel7$7$Fcam$\"35J%=P^a#)**)FdelFham7$Fbam7$$\" 3&[JLN9R#QGFdel$!3o#38.y*=kGF37$7$$\"3W=fz*[C71$Fdel$!3sxSxm&p\\!GF3Fb bm7$7$$\"3)y\"fz*[C71$Fdel$\"3'GOLhrR`1)Fdel7$$\"3>Q![9$H=JIFdel$\"3Y# [AhIlK;)Fdel7$FccmF^bm7$Fhbm7$$\"35k6KXyp\"3$Fdel$!3qB_(*\\%[tz#F37$7$ $\"3U>_Sd&\\@<$Fdel$!3kG;3/-^vFF3Fjcm7$7$$\"3)*>_Sd&\\@<$FdelFcdm7$$\" 331gz`*pBK$Fdel$!3RCoV`Z`FFF37$7$$\"315bxQpMnLFdel$!3TBh\"G#y$)=FF3Fid m7$7$FibmFacm7$$\"3I$R?a&4ysIFdel$\"3B=9YRj**R!)Fdel7$7$$\"3i5bxQpMnLF del$\"3#G\"R6S!)4nrFdelFfem7$7$F`em$!3'H7;G#y$)=FF37$$\"3C1gjA2@mNFdel $!3SEzGh45hEF37$7$$\"3o,^v(QpMn$Fdel$!3g\\:=D$*3UEF3Fefm7$7$F`emF_fm7$ $\"3arK%)G]?#[$Fdel$\"37&p*RXB6QpFdel7$7$F\\gm$\"3U-x45Lc=kFdelFbgm7$F [gm7$$\"3-!G$p,!f$=QFdel$!3](yqL^GNg#F37$7$$\"3'QpMn$=fzRFdel$!3-E:s)G [rd#F3F\\hm7$7$F\\gm$\"3`.x45Lc=kFdel7$$\"35[b_]D&3(QFdel$\"31>xB.\\%y 0'Fdel7$7$$\"3I$pMn$=fzRFdel$\"3wbZJpx=&y&FdelF[im7$7$Fbim$!3eD:s)G[rd #F37$$\"3'4oN\\bcs2%Fdel$!3G<6D\"HbJb#F37$7$$\"3#\\G9dG9dG%Fdel$!3WiV= VX[@DF3Fjim7$Faim7$$\"377Ij;)=WC%Fdel$\"3-#32s=W%Q`Fdel7$7$Fajm$\"3cfO k[WHU_FdelFfjm7$7$Fajm$!3*>O%=VX[@DF37$$\"3#)G%=\"=>rTVFdel$!3&R[M(*R5 (3DF37$7$$\"33xQpMn$=f%Fdel$!3w([&QIECtCF3Fc[n7$7$Fj[n$\"3n\"od@6H2u%F del7$$\"30j]zZd`0XFdel$\"3o.Nn$=fz*[Fdel7$Fb\\nF\\[n7$7$$\"3_wQpMn$=f% FdelF\\\\n7$$\"3i\"yOT[)z5YFdel$!3oi\"4zX0#pCF37$7$$\"3!z>X.'Q%Hw%Fdel $!3wIn$=fz*[CF3F\\]n7$Fb]n7$$\"3/k\"y,+o'y[Fdel$!3%)ReX!H3YFdel$\"3 #Gi**\\#GXAZFdel7$7$F_^n$\"3d%\\BT#\\pFUFdelFe^n7$F^^n7$$\"3N4e\"3>eW9 &Fdel$!3]'*p(QD\"Q&Q#F37$7$$\"3MgIlK;3/_Fdel$!3mCz7!yn(zBF3F__n7$F[_n7 $$\"3Oxh&*4Zfn\\Fdel$\"3oUzlO-=bTFdel7$7$Ff_n$\"3OZuQ*p<]x$FdelF[`n7$F e_n7$$\"3UB96=\\Q9aFdel$!3y`23%[snM#F37$7$$\"3__Ej\"3/-^&Fdel$!3j$>Z&3 CDQBF3Fe`n7$Fa`n7$$\"3>(y$>G#4+K&Fdel$\"33=Cdk\")RhOFdel7$7$F\\an$\"3i ;l1WzjsLFdelFaan7$F[an7$$\"3+NS))\\G%yo&Fdel$!3#pI$RK.$>J#F37$7$$\"3pW AhIlK;eFdel$!3ZB-3wq5,BF3F[bn7$7$F\\an$\"31;l1WzjsLFdel7$$\"3/U([gT\"z mcFdel$\"3z6_B$G%pFKFdel7$7$Fbbn$\"3bxNY_Rh7IFdelFjbn7$Fabn7$$\"3y$4&H r#RV'fFdel$!3C0-KO#H.G#F37$7$$\"3(o$=fz*[C7'Fdel$!3))\\*foFwwE#F3Fdcn7 $F`cn7$$\"3u&)>))\\-\"*3gFdel$\"34lizWGtVGFdel7$7$F[dn$\"3EL4-gBf)o#Fd elF`dn7$Fjcn7$$\"3p#4C$f!pMC'Fdel$!3#=dRY1P:D#F37$7$$\"3%zUr&G9dGkFdel $!3K.(zKOHuB#F3Fjdn7$Ffdn7$$\"3eeb$H\\CrM'Fdel$\"3GDAF37$7$$\"3A@5bxQpMnFdel$!3XUe6p@$*4AF3Fefn7$F \\fn7$$\"3cR$4\"on.#o'Fdel$\"3]>\"p_*)GV>#Fdel7$7$F\\gn$\"3)RFZ.q=*G@F delFagn7$F[gn7$$\"3JM\"G\"\\*Q$3oFdel$!3s>ugAP+,AF37$7$$\"3H71`Ej\"3/( Fdel$!3GcSJNg#[=#F3F[hn7$7$F\\gn$\"3rtsM+(=*G@Fdel7$$\"3WoX`c<89qFdel$ \"3Yyc^u%)G<>Fdel7$7$$\"3S81`Ej\"3/(Fdel$\"3!z6wlK#e&)=FdelFjhn7$7$Fai nFdhn7$$\"3K@!3]ggN4(Fdel$!3qn5IV&4(y@F37$7$$\"3e0-^v(QpM(Fdel$!3#e3H$ z?\"=;#F3Fgin7$7$FbhnFcin7$$\"3Mb6C0P!QM(Fdel$\"3'f0h2\"Q4m;Fdel7$7$F^ jn$\"3_eD&R)R_i;FdelFdjn7$F]jn7$$\"3W4#4(\\'f.Q(Fdel$!3iq227z4e@F37$7$ $\"3k'z*[C71`wFdel$!3Q7H#)R#R19#F3F^[o7$7$$\"3w(z*[C71`wFdel$\"3PzFdel $!3(G$=fz*[C7#F3Fi\\o7$F_]o7$$\"33*Ggo*fadzFdel$!3qC@aC@q?@F37$7$$\"3% **QpMn$=fzFdel$!3tQ\\QyTj?@F3Fe]o7$Fj[o7$$\"3'H&*=*QU%en(Fdel$\"3giokS Uj*Q\"Fdel7$7$F\\^o$\"3s\"*\\8\"[t([6FdelFa^o7$F[^o7$$\"3a,Vb,#zXF&)Fdel$!3yg(G:&pav?F37$7$$\" 3Gu&G9dG9d)Fdel$!3'G\\La%H&Q2#F3Fd`o7$F``o7$$\"3!G%pg:,EN$)Fdel$\"3')o F1U8xk))F*7$7$F[ao$\"3&=WqTKb0+(F*F`ao7$Fj`o7$$\"3n'=)GL(HX\"))Fdel$!3 5@0LtiAb?F37$7$$\"3Ml\"3/-^v())Fdel$!3MMN!z/#)G0#F3Fjao7$Ffao7$$\"3wA5 dI/Mi')Fdel$\"3Kr20PcEHmF*7$7$$\"3Ym\"3/-^v())Fdel$\"3E&)oHr51*)\\F*Ff bo7$7$F]coFcbo7$$\"37A[vr,$G5*Fdel$!3bnLyGh@O?F37$7$$\"3kexQpMn$=*Fdel $!3J1*3-@4L.#F3Fcco7$F\\co7$$\"3_-`@#*)>z)*)Fdel$\"3gNn$=fz*[*Fdel$!3s7lBO\"**\\,#F3F[eo7$7$Fjco$\"3Rl:GN km6JF*7$$\"3wU\">![T:7$*Fdel$\"3<&>k:*[#>i#F*7$7$$\"3!3Nn$=fz*[*Fdel$ \"3Kxy%Q.%Rb8F*Fjeo7$Faeo7$$\"3uNGs/.n#o*Fdel$!3)y6Eh/^;+#F37$7$$\"3)H %pMn$=fz*Fdel$!33$R()=WLy*>F3Fffo7$F`fo7$$\"3vO:\\U!y^j*Fdel$\"3)oJ()* *QS5>)!#?7$7$F]go$!3)H+i\\U86\"HFggoFbgo7$7$$\"35WpMn$=fz*Fdel$!3K$R() =WLy*>F37$$\"3GJk2=&>S(**Fdel$!3Eb]7GH*e)>F37$7$$\"3T`Ej\"3/-,\"F3$!3) =M]E43<)>F3Fbho7$7$F]go$!3Bv>'\\U86\"HFggo7$$\"3WbsI!)*3r&**Fdel$!3_a8 )=qZ8n)Fggo7$7$Fiho$!3CjL@9&Gy$=F*Faio7$Fhho7$$\"3\\&GyVE@m-\"F3$!3mW. I$)\\.r>F37$7$$\"3^71`Ej\"3/\"F3$!3;6%oLHJl'>F3F[jo7$7$Fiho$!3'=O8U^Gy $=F*7$$\"3FkJul\\!y-\"F3$!3C9(*HsduZCF*7$7$Fbjo$!3;Q$))GKiNH$F*Fjjo7$F ajo7$$\"3;d')yPz\"f0\"F3$!3Aygw?A+d>F37$7$$\"3ir&G9dG92\"F3$!3oL(f(RF3Fd[p7$7$$\"3t71`Ej\"3/\"F3$!3%4M))GKiNH$F*7$$\"3aY!f%H!4)f5F3$!3S. xT6DNKRF*7$7$F[\\p$!3uvh\\U86mYF*Fe\\p7$Fj[p7$$\"3%3&QX,dG&3\"F3$!3iXK GaxsV>F37$7$$\"3tIlK;3/-6F3$!3JQ:NhrqQ>F3F_]p7$7$F[\\p$!3/vh\\U86mYF*7 $$\"3'G3wv/J<4\"F3$!3J!***4XRWH`F*7$7$Ff]p$!3!GY#zQwSifF*F^^p7$Fe]p7$$ \"3f\\UlW*=Z6\"F3$!3QPfF37$7$$\"3$)*[C71`E8\"F3$!3F3Fh ^p7$Fd^p7$$\"3G$>%3n!yN7\"F3$!3w<]fio`YmF*7$7$F__p$!3_\"o=]'Qj)=(F*Fd_ p7$F^_p7$$\"3YD>'>j7U9\"F3$!3%o\"H@b/A>>F37$7$$\"3%*[C71`Ej6F3$!3UTkL& )=\"Q\">F3F^`p7$7$F__p$!3I%o=]'Qj)=(F*7$$\"3WMQS]jNb6F3$!3#yzb!z6J!*yF *7$7$Fe`p$!31W>\")o=K]$)F*F]ap7$Fd`p7$$\"3)>\\\\8BiP<\"F3$!3ZG(o(pb)y! >F37$7$$\"3/3/-^v(Q>\"F3$!39e[%zv@B!>F3Fgap7$Fcap7$$\"3)e!eh\"[rq=\"F3 $!3_GcLZKsm!*F*7$7$F^bp$!3KE4#)GhV_%*F*Fcbp7$F]bp7$$\"3U9n\"GljL?\"F3$ !3#R?uJX.r*=F37$7$$\"3:n$=fz*[C7F3$!3IaG#p819*=F3F]cp7$7$F^bp$!3sF4#)G hV_%*F*7$$\"3Fig)eXG(=7F3$!3+i_WC,6=5Fdel7$7$Fdcp$!35ZHenW%*\\5FdelF\\ dp7$Fccp7$$\"3W)Q#y&>8IB\"F3$!3'>;jDjMo)=F37$7$$\"3EEj\"3/-^D\"F3$!3e' **=_)H-\")=F3Ffdp7$Fbdp7$$\"3[!R2^vJ.D\"F3$!3'yi/ceAQ7\"Fdel7$7$F]ep$! 3A1z^jz`\\6FdelFbep7$7$F]ep$!3!o**=_)H-\")=F37$$\"3Wc\\\"Q]2FE\"F3$!3g !**)*fIVq(=F37$7$$\"3O&G9dG9dG\"F3$!3#3c5$pU8r=F3F_fp7$7$F]ep$!331z^jz `\\6Fdel7$$\"3[dV<7a)=G\"F3$!3/V@j$))RUA\"Fdel7$7$Fffp$!3r!RF?\"*)QW7F delF^gp7$7$Fffp$!3/h0JpU8r=F37$$\"3#yjMb_VCH\"F3$!3)\\pIa#ppn=F37$7$$ \"3ZWAhIlK;8F3$!3;Opn$\\0<'=F3F[hp7$7$Fffp$!3e!RF?\"*)QW7Fdel7$$\"3n0c $e1$R88F3$!3F6jr^,v>8Fdel7$7$Fbhp$!3DyJ^)=G[L\"FdelFjhp7$Fahp7$$\"3J)H F2[=AK\"F3$!3EU3CNewe=F37$7$$\"3e.-^v(QpM\"F3$!3wIRv)H0F&=F3Fdip7$F`ip 7$$\"3xOmp2!e[M\"F3$!3V')RW%\\/2T\"Fdel7$7$F[jp$!3om]z)ec6U\"FdelF`jp7 $7$F[jp$!3)4$Rv)H0F&=F37$$\"3mjr(3$)H?N\"F3$!3'>%y#z&HA]=F37$7$$\"3Yi \"3/-^vP\"F3$!33qT\\7^5W=F3F][q7$Ffjp7$$\"3#z0`EA$Gw8F3$!3iV*z1I@u\\\" Fdel7$7$$\"3oi\"3/-^vP\"F3$!3'*[mGr\"[O]\"FdelFi[q7$7$F`\\q$!3IqT\\7^5 W=F37$$\"3%3TgMDv=Q\"F3$!3')[U!*>N/U=F37$7$$\"3,AhIlK;39F3$!3#y8wt%)ye $=F3Fh\\q7$F_\\q7$$\"31e6!QUrwS\"F3$!3+p;On**=!e\"Fdel7$7$$\"3z@hIlK;3 9F3$!3k=^BjJb#e\"FdelFd]q7$7$F[^qFa]q7$$\"3E.8X>Ev69F3$!30/Ac6[?M=F37$ 7$$\"3!43/-^v(Q9F3$!3e:H4OE+G=F3Fa^q7$7$Fh^q$!31UD7Y'Q/n\"Fdel7$$\"3:A nGykJG9F3$!3(\\XC71`Ej\"Fdel7$F`_qFj]q7$Fg^q7$$\"3nT?et*f;W\"F3$!3j[=r wfoE=F37$7$$\"3-S?5bxQp9F3$!3JG)z-ga/#=F3Fg_q7$F]_q7$$\"3&Gl#ynU8R9F3$ !35R@FO0#4n\"Fdel7$7$F^`q$!3o]&pV(4%zx\"FdelFc`q7$F]`q7$$\"3)p*3(p^&fr 9F3$!3V59GFyY>=F37$7$$\"3M*************\\\"F3$!3p^6cZY@8=F3F]aq7$Fi`q7 $$\"3'[U*yLowq9F3$!3]U)yyya(z(>4)e(zX#F*FLFfdq7$Fjdq7$ F[eq$!3Kx^)\\khg&pF37$7$$\"3)Q)RTA*zgk#F*F[xF^eq7$Fbeq7$Fceq$!3;+I$QY& f\\mF37$7$$\"3KpMdcHjcGF*FjxFfeq7$7$$\"3mpMdcHjcGF*Fjx7$F[fq$!3)pW)*=% Q_XjF37$7$FW$!3QjT<#3'F3Fafq7$Fefq7$$\"3GS3zHmy)3$F*$!3Qo+8)GcP/'F3 7$7$$\"3]GVBMYN$4$F*FfyFifq7$F_gq7$$\"3u'=\\Pq7mJ$F*$!3$31uP7F:u&F37$7 $$\"3o#*Hev\"o2O$F*FbzFcgq7$Figq7$$\"31_*eOE,@d$F*$!3S.ieC'[AW&F37$7$$ \"3OQo0e!=Wm$F*F^[lF]hq7$Fchq7$$\"3-L\\xaU**fQF*$!3UV.1)fEk9&F37$7$$\" 3$f>HZuf6,%F*Fj[lFghq7$7$$\"3i'>HZuf6,%F*Fj[l7$$\"3%48%3(3.h=%F*$!3wD% Gz#4oa[F37$7$$\"3-(o8f/]&4WF*FjalFdiq7$7$$\"3r(o8f/]&4WF*Fjal7$$\"37_j S>JhdXF*$!3![ckC0yxc%F37$7$$\"3mV_7P,Qq[F*FfblFajq7$Fgjq7$$\"3?6@2rY\\ $)\\F*$!3%4oId![n'G%F37$7$$\"3*Q$oW!39uS&F*FeelF[[r7$Fa[r7$$\"3i/Co?b1 vaF*$!3%Q@WuTyD,%F37$7$$\"3L2.Z]'G%QgF*FjglFe[r7$F[\\r7$$\"3W\"\\5VWsn /'F*$!3kyH0x)Hqu$F37$7$Fd\\l$!3U4.V1&fWr$F3F_\\r7$7$Fd\\l$\"3u9AX9AX9x F37$$\"33UbbbbbbbF*F_w7$Fe\\r7$$\"36t-IMbo1mF*$!3GC.uFLA![$F37$7$$\"3S V!36g.ny'F*FajlF`]r7$Ff]r7$$\"31QMYm[BSsF*$!39kh#4Jr7A$F37$7$$\"3E7+++ +?$o(F*$!31ElK;3/-JF3Fj]r7$7$Fa^rFd^m7$$\"3]h?(=,'=\")zF*$!3y_[_([vP(H F37$7$$\"350;2!=:)p()F*FcdmFg^r7$F]_r7$$\"3]$*pYXPHc))F*$!3>Zn>+(*eSFF 37$7$F`cl$!3cEEh(3)4kEF3Fa_r7$7$F`cl$\"3w]\\i3ywlmF37$$\"35'*y*es/v2*F *F[t7$F^`r7$$\"3+NG?LFdM!*F*$\"3N0oaH;y4nF37$7$$\"3v#RPga\"4NzF*FhtFb` r7$Fh`r7$$\"3!fH'G'42lZ(F*$\"39o%H(4f]-sF37$7$$\"356#))>38b*pF*FeuF\\a r7$Fbar7$$\"37^yT>pOfiF*$\"3G+q]2[')ewF37$7$$\"3E_)z*3lW8iF*FbvFfar7$F \\brFi\\r7$Fg_r7$$\"3;A+3\"p%*fv*F*$!3>Rs\"))=F+^#F37$7$$\"3=2l\">$oV5 5FdelFe]nFabr7$Fgbr7$$\"3)=#H:@N>u5Fdel$!31c-UlNm)G#F37$7$$\"3m5IJ2a%o <\"FdelFb]oF[cr7$7$$\"3!3,8tSXo<\"FdelFb]o7$$\"3q+-Gplx\">\"Fdel$!3[)y x6([b(3#F37$7$F_fl$!3e*R&R=.%*Q?F3Fhcr7$7$F_fl$\"3Za=^]sUTgF37$$\"3#y) >H$>j&[5FdelFas7$FedrF[`r7$F^dr7$$\"3#*4\\6\\co58Fdel$!31!ec2hgy)=F37$ 7$$\"3c_phO<-)Q\"Fdel$!3+NpMn$=fz\"F3Fjdr7$F`er7$$\"3O!p\"Rc/kW9Fdel$! 3&=eti7:Uq\"F37$7$Fihl$!3Sqx)*>&f6h\"F3Ffer7$7$Fihl$\"3^\\&[>s.;h&F37$ $\"3!4$=,UTq\\9FdelF[r7$Fcfr7$$\"3\")H0@d^?E7Fdel$\"3*o(*=yg')*QgF37$7 $$\"3K(z`F,v[A\"FdelFdrFgfr7$7$F^gr$\"3H;1`Ej\"3/'F37$$\"3Yn$=fz*[C7Fd el$\"3Ob=^]sUTgF37$F\\fr7$$\"3)**fTQ2C1f\"Fdel$!3J?>ks.SL:F37$7$$\"3uL <]&p\"fh;Fdel$!37P?5bxQp9F3Fjgr7$7$$\"3YL<]&p\"fh;FdelFchr7$$\"32j5\">Fdel $!3fvC:x[()3#Fdel$!3e]KwY:9&3\"F37$7$F[]m$!3!3!R%>%H`^5F3Fgjr7$7$F[]m$\" 3;7ni2#4Z0&F37$$\"3_;AAAAAM@FdelF[p7$Fd[s7$F[[mFdir7$F][s7$$\"3?(RMk-u 1F#Fdel$!3,EJh(RXm_*Fdel7$7$Fi_m$!3)4QhB'ocd&)FdelFj[s7$Fa[s7$$\"3k8V( o'H)o:#Fdel$\"3msU%3hji/&F37$7$$\"3#\\tO=fz*[CFdel$\"351X\"y7=M&[F3Fd \\s7$7$F[]s$!3')z8Oiocd&)Fdel7$$\"3%\\gw')>&frCFdel$!3c%pZdW)\\/%)Fdel 7$7$$\"3(p?F04H:_#Fdel$!3k8C71`Ej\")FdelFc]s7$7$$\"3U1s_!4H:_#FdelF\\^ s7$$\"3s:Q;H,4qEFdel$!3Mf*G=erkD(Fdel7$7$Fcam$!39_PbCB)f*oFdelFb^s7$7$ Fcam$\"3c_z'Q*Q0#p%F37$$\"3scJ?#*\\JuEFdelFbo7$F__sFj\\s7$7$Fcam$!3E`P bCB)f*oFdel7$$\"3GPRjy@EzGFdel$!3\\\"Ghq)RLAiFdel7$7$Fibm$!3A$eik:\"*y _&FdelFg_s7$F\\_s7$$\"3%R0+8wl*>GFdel$\"3oZ].4f]lYF37$7$Fibm$\"3unM\\I ;!4b%F3Fa`s7$F]`s7$$\"3kUu]__t,JFdel$!31$pHf)R2I`Fdel7$7$$\"3gO*)=dKGE KFdel$!3IMMn$=fz*[FdelF[as7$Faas7$$\"3@bu&*=-ZELFdel$!3e>v%*Q3%>Y%Fdel 7$7$F`em$!3/t8\\9 UCLFdel$\"3')GNDa/&RX%F37$7$F`em$\"3*>D0]T8aV%F3Fdbs7$F]bs7$$\"3C]00)3 zCb$Fdel$!3%R)[#o)fR2OFdel7$7$F\\gm$!3+Uj*frYhF$FdelF^cs7$7$F\\gm$\"3X ka*>KQ\"HVF37$$\"3+;]rJ;s[MFdelFg`l7$F[dsFjbs7$Fdcs7$$\"3!)fFd0BX)y$Fd el$!3YTh%z^P\"fGFdel7$7$Fbim$!3k#**))pt'3wBFdelF`ds7$Fhcs7$$\"3O?b%z_m #Fdel7$7$$\"3%QR[\\!HwtUFdelFc _qFges7$F]fs7$$\"3k(4kn1KJG%Fdel$!3@bCU#R4^g\"Fdel7$7$Fajm$!3QH&3iwF+g \"FdelFafs7$F`es7$$\"3qgjdV1m)=%Fdel$\"3SY$3!Ql:&=%F37$7$Fajm$\"3;UJw) *f!f:%F3F[gs7$Fgfs7$$\"32L()\\D@-FXFdel$!3]ge4)HsHT*F*7$7$Fj\\n$!3,-vI %))*\\A#)F*Fegs7$Fags7$$\"3Ia@Nu_\\&e%Fdel$\"3mIdUxoR)3%F37$7$Fj\\n$\" 3_7-ZpIh'3%F3F_hs7$7$Fj[n$!3y/vI%))*\\A#)F*7$$\"3S,D#Q**4wx%Fdel$!3=: \"3[p$G*[$F*7$7$F_^n$!35S>vGV%pT\"F*F\\is7$7$F_^n$\"3J!yzKE4l,%F37$$\" 3s$z/v[Kbh%FdelF[`l7$FiisFehs7$Fbis7$$\"3Ey@.-'pQ.&Fdel$\"3[e&\\1K>%H= F*7$7$Ff_n$\"3s'RK?jdze%F*F^js7$Ffis7$$\"3Gs&GHA)4q\\Fdel$\"3\")\\C_WV o/SF37$7$Ff_n$\"3)f\"=Cm\\!R&RF3Fhjs7$Fdjs7$$\"3aXVq#Gu\\H&Fdel$\"3Y#y ung98j'F*7$7$F\\an$\"3W$oSQE[c#**F*Fb[t7$F^[t7$$\"3*fGe2)R2W`Fdel$\"3q ')*[<[2B$RF37$7$F\\an$\"3Iq!3Aqc#)*QF3F\\\\t7$Fh[t7$$\"3'>32id]-c&Fdel $\"3)>%Rw&yB))4\"Fdel7$7$Fbbn$\"3WpzQsw9q9FdelFf\\t7$Fb\\t7$$\"3UaiW'p <$4dFdel$\"3)=0gszW#pQF37$7$Fbbn$\"3_wy<]hY[QF3F`]t7$F\\]t7$$\"31)3<&4 V9HeFdel$\"3GI&Ri3If\\\"Fdel7$7$$\"3d<%G<1&RGfFdelFb\\oFj]t7$F`^t7$$\" 3a![D:xF`4'Fdel$\"3g[cEZf%>#>Fdel7$7$F[dn$\"3SG6Ms?8_>FdelFd^t7$Ff]t7$ $\"3aPDNdITngFdel$\"3Q\\#QX-2Q\"QF37$7$F[dn$\"3&=q]LlaO!QF3F^_t7$7$F[d n$\"3oG6Ms?8_>Fdel7$$\"3MG:*)*4%phjFdel$\"3gJ,\"QB6gM#Fdel7$7$F]fn$\"3 B2w.?kFdel$\"3s'>;x5!pkPF37$7$F]fn$ \"3w:h7=46jPF3Fe`t7$7$F]fn$\"3&pgP+s,pT#Fdel7$$\"3#e#o1([h8j'Fdel$\"3w u#*QE_'[t#Fdel7$7$F\\gn$\"3>h)z:e>%RGFdelFbat7$7$F\\gn$\"3z>_L?_L?PF37 $$\"3m\\V3c()p#\\'Fdel$\"3QHj\"3/-^v$F37$7$F`btF\\_l7$$\"3;I9dG9dGkFde l$\"3?;h7=46jPF37$Fhat7$$\"3,T(3!y^$R!pFdel$\"3;ByXX'=F4$Fdel7$7$Fain$ \"3;W(4&fn>DKFdelF\\ct7$F\\bt7$$\"3W8$fp?\"[qnFdel$\"3!R[Z%*))Gpr$F37$ 7$Fain$\"3?F+/e\"z(zOF3Ffct7$Fbct7$$\"3oj\\$p%43zrFdel$\"3q0r-ml8BMFde l7$7$F^jn$\"3/.)y#Re#)yNFdelF`dt7$7$Fain$\"3kF+/e\"z(zOF37$$\"3]LUAx7- =rFdel$\"3I-)4M4]Fn$F37$7$F^jn$\"3DnFgnFgUOF3F]et7$Ffdt7$$\"3C@#eT#R^c uFdel$\"3HxY4)fd\"HPFdel7$7$F[\\o$\"3vQjm#fkT!RFdelFget7$Fcet7$$\"3Me# \\pS-=Y(Fdel$\"3d1![R$3eKOF37$7$F[\\o$\"35)[+%)[+%3OF3Faft7$7$$\"3'))z *[C71`wFdel$\"3?Qjm#fkT!RFdel7$$\"3N,Qw4)*)ft(Fdel$\"39luT2wQ8SFdel7$7 $F\\^o$\"3k&)45)GxW?%FdelF`gt7$Fgft7$$\"3+-?D3TJ-yFdel$\"3Q7L!['*)*ef$ F37$7$F\\^o$\"3aIAv`\"Hod$F3Fjgt7$7$F\\^o$\"34&)45)GxW?%Fdel7$$\"3ymkr **yH7Z1BpAc$F37$7$Fb_o$\"3kD`[OkfZNF3Fait7$F]it7$ $\"3:))yr-bD+$)Fdel$\"312&QTzl^_%Fdel7$7$$\"3;t&G9dG9d)Fdel$\"3.)**>&e *\\2u%FdelF[jt7$Fgit7$$\"3[n-*3C'4v%)Fdel$\"3Fd*yW\"fKJNF37$7$F[ao$\"3 QqlMMq%e)Fdel$\"3[WTiU1OcZFdel7$7$$ \"3[Ja)*zX,#eyb'\\\"*Fdel $\"3m`;1vW\"3E&Fdel7$7$Fjco$\"3_P6l3(z+G&FdelF`]u7$F\\]u7$$\"3!y(*>47A *Q\"*Fdel$\"3;)Rq-?1jZ$F37$7$Fjco$\"35r8fr8frMF3Fj]u7$Ff]u7$$\"3RdI\"Q [pCV*Fdel$\"3')RED_6W4bFdel7$7$Fafo$\"3]Th2]&f3a&FdelFd^u7$F`^u7$$\"3e L$)zEa2o%*Fdel$\"35;<\"H&*RpJ)[!p()G5F3$\"3'f7MF\\(\\rhFdel7$7$Fbjo$\"3?&z=.8f6B'FdelF ^bu7$Ffau7$$\"3E?S!='*pD,\"F3$\"3m&=5MPNLS$F37$7$Fbjo$\"3-A^$RN\\9Q$F3 Fhbu7$Fdbu7$$\"3[mjK!QYw0\"F3$\"3'ft3zcf!ojFdel7$7$F[\\p$\"3!GHLN'='\\ V'FdelFbcu7$F^cu7$$\"3'*>F8=OVX5F3$\"31^*[\"=.KzLF37$7$F[\\p$\"3k_.FCU vfLF3F\\du7$Fhcu7$$\"3KDo=%*Q^'3\"F3$\"3Ue(3>@kTb'Fdel7$7$$\"3&4`Ej\"3 /-6F3$\"3\"4K!oAAWFmFdelFfdu7$Fbdu7$$\"3W$[pI)H;y5F3$\"3#[q,WqPnN$F37$ 7$Ff]p$\"3lfpezVERLF3Fbeu7$7$Ff]p$\"3!)>.oAAWFmFdel7$$\"3X/vZD=Z:6F3$ \"3\\2'=+LC1t'Fdel7$7$F__p$\"3%zm*3Q#=&4oFdelF_fu7$Fheu7$$\"3Bp$**Qsp2 6\"F3$\"3I?y7\"3jaL$F37$7$F__p$\"3LF3Fifu7$Fefu7$$\"3&4qq\")*oFdel7$7$$\"3;\\C71`Ej6F3$\"3ex&>`I6?)pFdelFcgu7$F _gu7$$\"34BFS%>kK9\"F3$\"3%*4O+3gQ:LF37$7$Fe`p$\"3;%Ra)G*>:I$F3F_hu7$7 $Fe`pF\\hu7$$\"3%3xZ7@KO<\"F3$\"3_,[v#4iu0(Fdel7$7$F^bp$\"3s-+Bu&ec9(F delFjhu7$Fehu7$$\"3kr$QYhbc<\"F3$\"3q@\\tP\"3kH$F37$7$F^bp$\"3Ur;88%*4 %G$F3Fdiu7$F`iu7$$\"30S1x2H#G?\"F3$\"3px5V>R44sFdel7$7$Fdcp$\"3r3\\pzM 7,tFdelF^ju7$Fjiu7$$\"3b%o%)>A_z?\"F3$\"3;$[_\\gT%yKF37$7$Fdcp$\"3k%3& >B*\\vE$F3Fhju7$Fdju7$$\"3j\"*3$\\N!3K7F3$\"3w())>\"ycg`tFdel7$7$F]ep$ \"3a)pM7t/!\\uFdelFb[v7$F^[v7$$\"3yV-m)Rh,C\"F3$\"3u89*fm29E$F37$7$F]e p$\"3mE0XDx!=D$F3F\\\\v7$7$F]ep$\"3U(pM7t/!\\uFdel7$$\"3g)[O!f**Rh7F3$ \"3Y,&QEO([\"\\(Fdel7$7$$\"39&G9dG9dG\"F3$\"3)pW:%=S%)*e(FdelFi\\v7$Fb \\v7$$\"3k1\"*))z(*Gs7F3$\"3ES%G^aN_C$F37$7$Fffp$\"3r%**ph::oB$F3Fe]v7 $F_]v7$$\"3+/AXQvx!H\"F3$\"3bMy29]=BwFdel7$7$Fbhp$\"3rlh2TE8CxFdelF_^v 7$F[^v7$$\"3_QN0YLM/8F3$\"3U)R>;Xh)HKF37$7$Fbhp$\"3R4Kp:*>DA$F3Fi^v7$F e^v7$$\"3#*>3Ty#4-K\"F3$\"3]#4T4a0\"\\xFdel7$7$F[jp$\"33BTVNsJ_yFdelFc _v7$F__v7$$\"3EjJ`,vKO8F3$\"3f)H[(QxA:KF37$7$F[jp$\"35fW()eW()3KF3F]`v 7$Fi_v7$$\"3UGNk\"p\"p\\8F3$\"33Iyc=4ipyFdel7$7$F`\\q$\"3u)\\4nt/[(zFd elFg`v7$7$$\"3!Q?5bxQpM\"F3Fd`v7$$\"37%z@C9Z#o8F3$\"3'QY9:=#G,KF37$7$F `\\q$\"3s3K!oYNe>$F3Fdav7$F]av7$$\"3OMO(Re@#z8F3$\"35=)Q\"f=2&)zFdel7$ 7$F[^q$\"3+n6X%pm>4)FdelF^bv7$Fjav7$$\"3)>]5\\r1,S\"F3$\"3i7)zQNxz=$F3 7$7$F[^q$\"3M32V(QjL=$F3Fhbv7$Fdbv7$$\"3I\"eeT-'z39F3$\"3W\"yE&G5x&4)F del7$7$$\"3j]B$3InuU\"F3FfcmFbcv7$Fhcv7$$\"3mKAaU7GQ9F3$\"3()f@ )Fdel7$7$Fh^q$\"3&G#4%3\"RA<#)FdelF\\dv7$7$F[^q$\"3!zqIuQjL=$F37$$\"3q _T!eE5>V\"F3$\"3?qd#**3q_<$F37$7$Fh^q$\"3G<+%[.A9<$F3Fidv7$Fbdv7$$\"35 <0?!y#fn9F3$\"3=Y#=cHHZN)Fdel7$7$F^`q$\"3kwp7%)p7f$)FdelFcev7$F_ev7$$ \"3E^3Q'[hOY\"F3$\"3!Q@>g,@J;$F37$7$F^`q$\"3YrsgQ#y*fJF3F]fv7$Fiev7$$ \"3q'f3(Ri&p\\\"F3$\"3OA(Q02Lz[)Fdel7$7$F`bq$\"3-*3Q_4Q_\\)FdelFgfv7$7 $$\"3CS?5bxQp9F3Fdfv7$$\"33zdf]PO&\\\"F3$\"3U!)[IVT\\^JF37$7$F`bq$\"3! 3J)4O:+\\JF3Fdgv-Febq6&FgbqFjbqFjbqFhbq-%+AXESLABELSG6%%\"kG%\"aG-%%FO NTG6#%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 67 "Now, since we want the lowest point of the parabol a to lie between " }{XPPEDIT 18 0 "x=-1" "6#/%\"xG,$\"\"\"!\"\"" } {TEXT -1 5 " and " }{XPPEDIT 18 0 "x=2" "6#/%\"xG\"\"#" }{TEXT -1 34 " , we reject the solution given by " }{XPPEDIT 18 0 "a = -4-3*sqrt(2)" "6#/%\"aG,&\"\"%!\"\"*&\"\"$\"\"\"-%%sqrtG6#\"\"#F*F'" }{TEXT -1 26 ", and retain the solution " }{XPPEDIT 18 0 "a = 3*sqrt(2)-4, k = (3+2*s qrt(2))/9" "6$/%\"aG,&*&\"\"$\"\"\"-%%sqrtG6#\"\"#F(F(\"\"%!\"\"/%\"kG *&,&F'F(*&F,F(-F*6#F,F(F(F(\"\"*F." }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 "The required parabola is \+ now " }{XPPEDIT 18 0 "y = k[0]*(x-a[0])^2;" "6#/%\"yG*&&%\"kG6#\"\"!\" \"\"*$,&%\"xGF*&%\"aG6#F)!\"\"\"\"#F*" }{TEXT -1 8 ", where " } {XPPEDIT 18 0 "a[0]=3*sqrt(2)-4" "6#/&%\"aG6#\"\"!,&*&\"\"$\"\"\"-%%sq rtG6#\"\"#F+F+\"\"%!\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "k[0]=(3+ 2*sqrt(2))/9" "6#/&%\"kG6#\"\"!*&,&\"\"$\"\"\"*&\"\"#F+-%%sqrtG6#F-F+F +F+\"\"*!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "a0 := evalf(evalf(3*sqrt(2)- 4,13));\nk0 := evalf(evalf((3+2*sqrt(2))/9,13));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#a0G$\"+roSEC!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %#k0G$\"+R,.wk!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 4 "Note" }{TEXT -1 58 ": This solution can also be obtained \+ numerically by using " }{TEXT 0 6 "fsolve" }{TEXT -1 39 " with suitabl e starting approximations." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 95 "sol := fsolve(\{1=k*(a+1)^2,2=k*(a- 2)^2\},\{a=0.25,k=0.65\});\na0 := subs(sol,a);\nk0 := subs(sol,k);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%$solG<$/%\"aG$\"+roSEC!#5/%\"kG$\"+R ,.wkF*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#a0G$\"+roSEC!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#k0G$\"+R,.wk!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 45 "(b) The catenary has an equation of the form:" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "y = (cosh(k*(x-a))-1)/k;" "6#/%\" yG*&,&-%%coshG6#*&%\"kG\"\"\",&%\"xGF,%\"aG!\"\"F,F,F,F0F,F+F0" } {TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 25 "where the lowest point \+ is" }{XPPEDIT 18 0 " ``(a,0)" "6#-%!G6$%\"aG\"\"!" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 43 "Since the catenary passes through the point" }{XPPEDIT 18 0 " ``(-1,1)" "6#-%!G 6$,$\"\"\"!\"\"F'" }{TEXT -1 9 " we have:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "1 = (cosh(k*(-1-a))-1)/k;" "6#/\"\"\"*&,&-%%c oshG6#*&%\"kGF$,&F$!\"\"%\"aGF-F$F$F$F-F$F+F-" }{TEXT -1 2 ", " }} {PARA 0 "" 0 "" {TEXT -1 37 "and since it passes through the point" } {XPPEDIT 18 0 " ``(2,2)" "6#-%!G6$\"\"#F&" }{TEXT -1 10 " we have: " } }{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "2 = (cosh(k*(2-a))-1 )/k;" "6#/\"\"#*&,&-%%coshG6#*&%\"kG\"\"\",&F$F,%\"aG!\"\"F,F,F,F/F,F+ F/" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 58 "Hence we need to solve the pair of simultaneous equatio ns " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "PIECEWISE([co sh(k*(-1-a))-1 = k, ``],[cosh(k*(2-a))-1 = 2*k, ``]);" "6#-%*PIECEWISE G6$7$/,&-%%coshG6#*&%\"kG\"\"\",&F.!\"\"%\"aGF0F.F.F.F0F-%!G7$/,&-F*6# *&F-F.,&\"\"#F.F1F0F.F.F.F0*&F:F.F-F.F2" }{TEXT -1 3 ". " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 91 "The following impl icit plot of the two equations indicates that there is a solution with k " }{TEXT 282 1 "~" }{TEXT -1 12 " 1.05 and a " }{TEXT 283 1 "~" } {TEXT -1 6 " 0.3. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 196 "p1:=plots[implicitplot](cosh(k*(-1-a))-1=k, k=0..1.5,a=-8..8,color=red,grid=[75,75]):\np2:=plots[implicitplot](cos h(k*(2-a))-1=2*k,k=0..1.5,a=-8..8,color=blue,grid=[75,75]):\nplots[dis play]([p1,p2]);" }}{PARA 13 "" 1 "" {GLPLOT2D 405 356 356 {PLOTDATA 2 "6%-%'CURVESG6dgl7$7$$\"\"!F)$!\")F)7$F($!3A%y$y$y$y$y(!#<7$F,7$F($!3W ovcnvcnvF/7$F17$F($!3k_8N^8N^tF/7$F57$F($!3'o8N^8N^8(F/7$F97$F($!33@*= *=*=*=pF/7$F=7$F($!3I0Fq-Fq-nF/7$FA7$F($!3_*['['['['['F/7$FE7$F($!3st- Fq-FqiF/7$FI7$F($!3%z0aS0aS0'F/7$FM7$F($!3;Uy$y$y$y$eF/7$FQ7$F($!3QE;i @;i@cF/7$FU7$F($!3f5aS0aS0aF/7$FY7$F($!3![>*=*=*=*=&F/7$Fgn7$F($!3-zH( H(H(H(\\F/7$F[o7$F($!3CjnvcnvcZF/7$F_o7$F($!3XZ0aS0aSXF/7$Fco7$F($!3nJ VKCVKCVF/7$Fgo7$F($!3*e63\"3\"3\"3TF/7$F[p7$F($!3k**=*=*=*=*QF/7$F_p7$ F($!3U$ovcnvcn$F/7$Fcp7$F($!3?n%f%f%f%fMF/7$Fgp7$F($!3(4DVKCVKC$F/7$F[ q7$F($!3vMq-Fq-FIF/7$F_q7$F($!3_=3\"3\"3\"3\"GF/7$Fcq7$F($!3G-Yf%f%f%f #F/7$Fgq7$F($!31'Qy$y$y$yBF/7$F[r7$F($!3%)p@;i@;i@F/7$F_r7$F($!3g`f%f% f%f%>F/7$Fcr7$F($!3PP(H(H(H(HF/7$F]w7$ F($\"3ua@;i@;i@F/7$Faw7$F($\"3'4Py$y$y$yBF/7$Few7$F($\"3=(e%f%f%f%f#F/ 7$Fiw7$F($\"3U.3\"3\"3\"3\"GF/7$F]x7$F($\"3l>q-Fq-FIF/7$Fax7$F($\"3)eB VKCVKC$F/7$Fex7$F($\"35_%f%f%f%fMF/7$Fix7$F($\"3KocnvcnvOF/7$F]y7$F($ \"3a%)=*=*=*=*QF/7$Fay7$F($\"3z+\"3\"3\"3\"3TF/7$Fey7$F($\"3Y$\"3'e_%fA%3'**fF /7$$\"3*Ht=$feHMRFd^lFj[l7$Fg^l7$$\"3WEYFZ`mERFd^l$\"3#H=xF=Uw1'F/7$7$ $\"3KD\"o(>vgPNFd^l$\"3ke-Fq-FqiF/F[_l7$7$Fb_lF^\\l7$$\"3DdOzb+;4NFd^l $\"3]n8v'H#RGjF/7$7$$\"3.c?oTm5RKFd^lFb\\lFh_l7$F^`l7$$\"3)e'fHQQZ*>$F d^l$\"3Sr1]4a4'HFd^l$\"3_BN@@+I>oF/7$7$$\"3dd*\\6()=2#GFd^lFj \\lF_al7$Feal7$$\"3e??_8h&=x#Fd^l$\"3;CBC*p'obqF/7$7$$\"3k(>+3?-)oEFd^ lF^]lFial7$F_bl7$$\"3X-vr`NY=EFd^l$\"3+bV])yk#)G(F/7$7$$\"3M$\\z#\\%*[ UDFd^lFb]lFcbl7$Fibl7$$\"33]`\\kwp\"\\#Fd^l$\"3qL4d*p-!=vF/7$7$$\"3#f] 8PzFfV#Fd^lFf]lF]cl7$7$$\"3E1Nr$zFfV#Fd^lFf]l7$$\"3-3y]//J&Q#Fd^l$\"3M ?e=!*ocXxF/7$7$$\"3qxGaNt\"\\M#Fd^lFj]lFjcl7$F`dl7$$\"3=N!4QZh[H#Fd^l$ \"3]QKM#*4VrzF/7$7$$\"3kg@`%RmjE#Fd^lF^^lFddl7$7$$\"33wi8JnCaSFd^lF*7$ $\"3g_e(yE#p.UFd^l$!3m4\"eK>X(*z(F/7$7$$\"3TB_T)4PSB%Fd^lF-Fael7$Fgel7 $$\"3kC6\\Z^/!Q%Fd^l$!37Bv'3&*RBg(F/7$7$$\"3kH)QzO$)=X%Fd^lF2F[fl7$Faf l7$$\"3wx9s$zy3f%Fd^l$!3*y(e\"pl7'3uF/7$7$$\"3@+'3w:q1s%Fd^lF6Fefl7$7$ F\\gl$!3w^8N^8N^tF/7$$\"3s!HZdJYp%[Fd^l$!37\\dr4.r>sF/7$7$$\"3shgW!Q*z f]Fd^lF:Fbgl7$Fhgl7$$\"3)zmG!od'R;&Fd^l$!3h!)Gu,&4t.(F/7$7$$\"3e$*o._6 !**\\&Fd^lF>F\\hl7$Fbhl7$$\"3;z*4E`bec&Fd^l$!3Y/?3F:'R'oF/7$7$$\"3W\"3 \"3\"3\"3\"3'Fd^l$!3@WJ/_`91nF/Ffhl7$7$F]il$\"3rLhnXP:yZYFd^lF^[lF`[m7$Ff[m7$$\"3-d.'e67D[%Fd^l$\"3] tAOG%>fd&F/7$7$$\"3'*)y\"4ihB$R%Fd^lFb[lFj[m7$F`\\m7$$\"3$zCmHFEt@%Fd^ l$\"3f$>([M@U?eF/7$7$$\"3s')G)yaBf=%Fd^lFf[lFd\\m7$7$$\"3-')G)yaBf=%Fd ^lFf[lFa^l7$F\\il7$$\"3C#GovF*G&3'Fd^l$!3YtY*pd^Jq'F/7$7$$\"3o;Uckee'3 'Fd^lFBFb]m7$Fh]m7$$\"3[BqB#4a8P'Fd^l$!3+!z45%*[u^'F/7$7$$\"37h^!3c!Gq kFd^lFFF\\^m7$Fb^m7$$\"3#Ro?*eBm@nFd^l$!3p(\\1*R*)fQjF/7$7$$\"3Y=?pg*R m'pFd^lFJFf^m7$F\\_m7$$\"3_7po&=Z&frFd^l$!3WY`aQ/4phF/7$7$$\"3V:L)zr27 j(Fd^lFNF`_m7$Ff_m7$$\"3\"o/6B%p9@xFd^l$!3KQ!pj>yF,'F/7$7$$\"3c3\"3\"3 \"3\"3\")Fd^l$!317y2eqOKfF/Fj_m7$7$Fa`m$\"3e%*\\'>:GG$RF/7$$\"3m3!*Q@] %\\W(Fd^lFfy7$Fi`m7$$\"3\"*zp&>tt>C(Fd^l$\"35bv*[x&\\+UF/7$7$$\"3`>Z)4 Gm%HoFd^lFjyF]am7$Fcam7$$\"3-.q\\f)y\")\\'Fd^l$\"3oNG%36]g\\%F/7$7$$\" 3)eN1J6!QljFd^lF^zFgam7$F]bmFbil7$F``m7$$\"3G!=;[:*Q'G)Fd^l$!3iO+#G[ao &eF/7$7$$\"3%\\)\\eP!=*p$)Fd^lFRFbbm7$Fhbm7$$\"3)*>ge\"3s!e()Fd^l$!31P f776&4p&F/7$7$$\"3'z&4'y\\Ja5*Fd^lFVF\\cm7$Fbcm7$$\"3)GJ\"H2Y\"eN*Fd^l $!3Y=;l)4%\\QbF/7$7$$\"3d8N^8N^85Fjs$!3GZdaf-`0aF/Ffcm7$7$F]dm$\"3k=lp PFY/MF/7$$\"3H5eQ<]mT)*Fd^l$\"3a_%f%f%f%fMF/7$7$FfdmFjx7$$\"3AK%Q$Q]si '*Fd^l$\"3\"QV%yp)\\)4NF/7$7$$\"3#>U*>8#)H**))Fd^lF^yF\\em7$Fbem7$$\"3 4)=[0_=BI)Fd^l$\"3gU\\c9h\"=#)Fd^lFbyFfem7$7$F]fm $\"35%)=*=*=*=*QF/Ff`m7$F\\dm7$$\"3&\\)QBjMa85Fjs$!3e1Qx^tV0aF/7$7$$\" 3a\"eP4OhN,\"FjsFZFdfm7$Fjfm7$$\"3#4:q&)>0-2\"Fjs$!3Cvi\"*>.m\\_F/7$7$ $\"3\">]M()R%G46FjsFhnF^gm7$7$$\"3x,Xt)R%G46FjsFhn7$$\"3Jf[*o#*>B9\"Fj s$!3_Z(z0[m.6&F/7$7$$\"3G;i@;i@;7Fjs$!3Nc3*=))*=9]F/F[hm7$7$Fbhm$\"3J` +/Ly#[,$F/7$$\"3.J)[u'[x27Fjs$\"3?>q-Fq-FIF/7$7$F[imFbx7$$\"3'f$>[MH,) >\"Fjs$\"3m=p(3'QWYIF/7$7$$\"3_$))zC1)=#3\"FjsFfxFaim7$FgimFbdm7$Fahm7 $$\"3)3rm9WS&G7Fjs$!3Et,C8)=h)\\F/7$7$$\"3)p--=vU#Q7FjsF\\oF\\jm7$7$$ \"3%o--=vU#Q7FjsF\\o7$$\"31&)*zLY79I\"Fjs$!3k\\9t`FjZ[F/7$7$$\"3))*\\c rxN2Q\"FjsF`oFijm7$F_[n7$$\"3k(zg#4Nc&R\"Fjs$!3h+\")y>V%=t%F/7$7$$\"3' )=*=*=*=*=9Fjs$!3\"ppuiwr,r%F/Fc[n7$7$Fj[n$\"3*)4r`N(>'4FF/7$$\"3IG\"= .U!fR8FjsF^x7$Fb\\nFghm7$Fi[n7$$\"3WjgqH*G9[\"Fjs$!3un,Q&)y@2YF/7$7$$ \"372Tt;'Q#\\:FjsFdoFg\\n7$F]]n7$$\"3)fD@SPe1e\"Fjs$!3gCow4k%o\\%F/7$7 $$\"3r@;i@;i@;Fjs$!3%Q_7-M%QkWF/Fa]n7$7$Fh]n$\"3o$zA#f]HkCF/7$$\"37ZY& GenK]\"FjsFjw7$F`^nF_\\n7$Fg]n7$$\"3od\"f<(*=#z;Fjs$!3upV!y\\hdQ%F/7$7 $$\"3Gvx1dBK`*R4U7wy\"Fjs$!33+Vrg\\;&G%F/7$7$ $\"3dCVKCVKC=Fjs$!3]sRg1,,hUF/F__n7$7$Ff_n$\"3]9qmXnngAF/7$$\"3IJw'Hu> &)p\"FjsFfw7$7$$\"3eJw'Hu>&)p\"FjsFfwF]^n7$Fe_n7$$\"3k1UY!=DS*=Fjs$!3[ c1*zobC=%F/7$7$$\"3;tog0@C-?FjsF\\pFf`n7$7$$\"3)G(og0@C-?FjsF\\p7$$\"3 8ezP.b\\:?Fjs$!3gvd@;y!e4%F/7$7$$\"3UFq-Fq-F?Fjs$!3q^w%He)o*3%F/Fcan7$ 7$Fjan$\"3?\"[p%[qV)3#F/7$$\"3sr-HbU\"H$>FjsFbw7$FbbnF[`n7$Fian7$$\"3v \"4%G?8UE@Fjs$!3=,hOrC\"z*RF/7$7$$\"3FI(H(H(H(HAFjs$!3?SzK?KLSRF/Fgbn7 $7$F^cn$\"38!*4\"z\\W<%>F/7$$\"3Vk&)RHx?BAFjsF^w7$FfcnF_bn7$F]cn7$$\"3 U4OG:bGaAFjs$!3sPg\")HZ3=RF/7$7$$\"3t`;IFjsFfv7$7$$\"3^HFN>t`;IFjsFfvFfin7$F`jn7$$\"3[di'[0'QmIFjs$!3ij N23!Gq[$F/7$7$$\"3V7rn.7\\UJFjs$!3kn%f%f%f%fMF/Fc[o7$7$Fj[oFhp7$$\"3@& [k'4(GF@$Fjs$!31C@kB1\"pU$F/7$7$$\"3aWKCVKCVKFjs$!3O+LpAq))=MF/F`\\o7$ 7$Fajn$\"3D$)y@JKK.:F/7$$\"3M8-e%e6W0$Fjs$\"3%H_/Kt<()\\\"F/7$7$Fg\\o$ \"3'*H@'*4[y<9F/F_]o7$7$$\"34XKCVKCVKFjsFi\\o7$$\"3)4'\\mfj2eLFjs$!3Q[ (f2!>tlLF/7$7$$\"3QZf%f%f%fW$Fjs$!3J*\\3;#*G=M$F/F\\^o7$Fe]o7$$\"3!*38 (*\\$yXP$Fjs$\"3UqN+LFUt8F/7$7$Fc^o$\"3cu#)pn3!GM\"F/Fh^o7$Fb^o7$$\"35 x:lAy`5NFjs$!3y*ei]CT@J$F/7$7$$\"3C]'['['['[OFjs$!3?_#GyS2QF$F/Fb_o7$7 $Fi_o$\"3UOW@Mx;u7F/7$$\"3k[vm5!y(zNFjsFbv7$7$$\"33[vm5!y(zNFjsFbv7$Fc ^o$\"3yu#)pn3!GM\"F/7$Fh_o7$$\"3)))p\"3&HT(oOFjs$!3G-&Q%*RvYE$F/7$7$$ \"3c#\\%*ymd^u$FjsF\\qF\\ao7$7$$\"3+#\\%*ymd^u$FjsF\\q7$$\"3Do!*orhxCQ Fjs$!3S9o$\\Q'*[@$F/7$7$$\"35`8N^8N^QFjs$!3*zV@6#\\o4KF/Fiao7$F^`o7$$ \"3S&='o]Y\"on$Fjs$\"3e&e!zhNDn7F/7$7$F`bo$\"3p&z6FVP'*HF/Feho7$F]go7$$\"37Y&*Q4,4,XFjs$\"3y$)*)[a]nO5F/ 7$7$F\\io$\"3?LU!Hyqf%**FjsFaio7$7$F\\io$!3EP2/>VP'*HF/7$$\"3h3%p&p29, [Fjs$!3!*y=reY0fHF/7$7$$\"3On['['['['[Fjs$!3k:JO^=w\\HF/F^jo7$Fgio7$$ \"3+CX(4RX'fZFjs$\"3;1*yMq*)4x*Fjs7$7$Fejo$\"35W_:UU#e\\*FjsFjjo7$Fdjo 7$$\"33Pkpa<'*o\\Fjs$!3AEyH$\\Z=#HF/7$7$$\"3?qvcnvcn]Fjs$!3'Gu$QOk82HF /Fd[p7$F`[p7$$\"3;b:U\"QXM,&Fjs$\"3<%RRv'>&fA*Fjs7$7$F[\\p$\"3ow#f(*p# o%3*FjsF`\\p7$Fj[p7$$\"3TO/'G5)yR^Fjs$!3!Hac%[g%y)GF/7$7$$\"31t-Fq-Fq_ Fjs$!3#oK3xMj!oGF/Fj\\p7$Ff\\p7$$\"3'z.Q)>_(GE&Fjs$\"3<8Cf'=Hvs)Fjs7$7 $Fa]p$\"31()[6m!3$3()FjsFf]p7$F`]p7$$\"3())o)y;lC8`Fjs$!3yi/VS@lcGF/7$ 7$$\"3#f(H(H(H(HZ&Fjs$!3,yf(eCr@$GF/F`^p7$7$Fg^p$\"3+28(362cJ)Fjs7$$\" 3g?Z_jN!GI&FjsFju7$F__p7$Fa]p$\"3<))[6m!3$3()Fjs7$7$Fg^p$!3dxf(eCr@$GF /7$$\"3EU&4Z(4.*[&Fjs$!3+*['*fMRz#GF/7$7$$\"3)o\\X7\">/+cFjsFdqFj_p7$F ``p7$$\"3Ed!f823Rm&Fjs$!3eb(R&*pe#)z#F/7$7$$\"3))zcnvcnvcFjs$!3G)[^j=J rz#F/Fd`p7$F\\_p7$$\"3s$*z#>PJF)Fjs7$7$$\"3wycnvcn vcFjs$\"3b5!>k[1Q%zFjsF`ap7$Fj`p7$$\"3\"zRHU!p\"f$eFjs$!3_p&=*=*=*=*o'Fjs$!3m399>3]TEF/Fggp7$7$F^hp$ \"3E5595[TXkFjs7$$\"35E#4ml/.m'Fjs$\"3Y7k['['['['Fjs7$7$FghpFfuFcgp7$F ]hp7$$\"3gT9['3hjs'Fjs$!3h+IP=YCMEF/7$7$$\"3)e*=*=*=*=*oFjs$!3n_0O.VW; EF/F^ip7$Fchp7$$\"3w\"p_f$eL$p'Fjs$\"3ykdn(ebAW'Fjs7$7$Feip$\"3-c4B0O: ehFjsFjip7$7$Feip$!37`0O.VW;EF/7$$\"3Iy=%\\zA#4pFjs$!3J\\7=\"*338EF/7$ 7$$\"3#*e^Nr3;!3(FjsFhqFgjp7$F][q7$$\"3a7J*3l@H4(Fjs$!3-\\jui))eFjsF`\\q7$Fg [q7$$\"3Y-3j<$R@F(Fjs$!3))pOG;%fxc#F/7$7$$\"3e,tH(H(H(H(Fjs$!3oy$)fH`< mDF/Fj\\q7$Ff\\q7$$\"3)>,k#e#*=brFjs$\"3Y.gM2M9SeFjs7$7$$\"3[+tH(H(H(H (Fjs$\"3kfK/Y[XNcFjsFf]q7$F`]q7$$\"3mZ-75p!GX(Fjs$!3FhGs)HbUa#F/7$7$$ \"3W/++++++vFjs$!3m,v#RF&>TDF/Fb^q7$7$Fa]qF_^q7$$\"3!4Da#*)en$Q(Fjs$\" 3w@c%*Q[6lbFjs7$7$Fi^q$\"3yt5&Q;+uR&FjsF__q7$7$$\"3a0++++++vFjs$!36-v# RF&>TDF/7$$\"3D5xjB'*zMwFjs$!3$*Q>fVY;ADF/7$7$$\"3H2Fq-Fq-xFjs$!3v;7.5 Rl9XNt^FjsF]aq7$7$$\"3=1Fq-Fq-xFjsFg`q7$$\"3EJhXf\"3!= yFjs$!3eQ?el3P,DF/7$7$$\"3:5aS0aS0zFjs$!3)31qWH[a\\#F/Fjaq7$7$$\"3S3Fq -Fq-xFjsFdaq7$$\"3CXx/0DEOyFjs$\"3O/g![f[=1&Fjs7$7$Fabq$\"3mzXE\"R>B' \\FjsFibq7$7$Fabq$!3Mh+Z%H[a\\#F/7$$\"3iQ^SiiL-!)Fjs$!3eG9^#Hr<[#F/7$7 $$\"3+8\"3\"3\"3\"3\")Fjs$!3g)*pa&\\&[uCF/Ffcq7$F_cq7$$\"3#RJ2!\\1ag!) Fjs$\"3@$yKZb5<$[Fjs7$7$F]dq$\"3W=<@]GRjZFjsFbdq7$F\\dq7$$\"3_X=P@()p( =)Fjs$!3Enj_K]FjCF/7$7$$\"3&e\"3\"3\"3\"3J)Fjs$!3%['ezo8oaCF/F\\eq7$7$ F]dq$\"3)*=<@]GRjZFjs7$$\"3#)[%fw4%e$G)Fjs$\"3XQ)QzfUZh%Fjs7$7$Fceq$\" 3'z)*o`qddd%FjsF[fq7$7$Fceq$!3Ilezo8oaCF/7$$\"3q')[UL(>SP)Fjs$!3qFn;H7 !eW#F/7$7$$\"3#)>N^8N^8&)Fjs$!3+8-<3&ffV#F/Fhfq7$7$Fceq$\"3_))*o`qddd% Fjs7$$\"3aN[<-uY0&)Fjs$\"3*Q'o$>^\\,T%Fjs7$7$$\"3r=N^8N^8&)Fjs$\"3!=D( RB9n)R%FjsFggq7$7$F^hqFagq7$$\"3OZ9+c:Bh&)Fjs$!3:V))pBwFHCF/7$7$$\"3c@ i@;i@;()Fjs$!3S%42VM]#=CF/Fdhq7$7$$\"3X?i@;i@;()Fjs$\"3mlsue=J3UFjs7$$ \"3qiiP=&p1g)FjsFbu7$Feiq7$F^hq$\"3O_sRB9n)R%Fjs7$Fjhq7$$\"35Om7*Qt#\\ ()Fjs$!3(=;;GOROT#F/7$7$$\"3TC*=*=*=*=*)Fjs$!3beHWm.\\,CF/F]jq7$7$F[iq Fciq7$$\"3t')z'Rhyvs)Fjs$\"3D*3s\\TDJ?%Fjs7$7$$\"3IB*=*=*=*=*)Fjs$\"3* >=2m#e-5SFjsFjjq7$7$Fdjq$!3**eHWm.\\,CF/7$$\"34Q>qU/4Q*)Fjs$!3EMpu.!G) )R#F/7$7$$\"3:E;i@;i@\"*Fjs$!3w$=@([9i&Q#F/Fi[r7$7$Fdjq$\"3a#=2m#e-5SF js7$$\"3%p9v3GS(\\*)Fjs$\"3cPX7IJc&*RFjs7$7$$\"3PG;i@;i@\"*Fjs$\"3_\"R O([hU@QFjsFh\\r7$7$$\"3EF;i@;i@\"*FjsFb\\r7$$\"3'))pMP:Lw7*Fjs$!3Y&**) Hz1z%Q#F/7$7$$\"3o+Q.!\\]k@*FjsF\\rFg]r7$F]^r7$$\"3g9E'[x(Q:$*Fjs$!31d D&*eg%)oBF/7$7$$\"37IVKCVKC$*Fjs$!3C4.1HU_oBF/Fa^r7$7$Fe]rFa]r7$$\"3I \"*Q$o#>*4<*Fjs$\"3SPMRNxq(z$Fjs7$7$Fh^r$\"38'=.XFG>k$FjsF^_r7$7$Fh^r$ !3!)3.1HU_oBF/7$$\"3&G;@>%yp,&*Fjs$!3UB@$4eg8N#F/7$7$$\"3(H.Fq-Fq_*Fjs $!3m7R\"Gu;/N#F/F[`r7$Fd_r7$$\"3[Pt#RG!Q\"R*Fjs$\"3>(\\D_vg!4OFjs7$7$F b`r$\"316QR/e*4Z$FjsFg`r7$Fa`r7$$\"3)Q)H:nFz)o*Fjs$!3s%=j\\G7ZL#F/7$7$ $\"3rM(H(H(H(H(*Fjs$!3zRVmN89LBF/Faar7$F]ar7$$\"3'pbiBI\\4h*Fjs$\"3;') >T(Q`\"HMFjs7$7$$\"3$ptH(H(H(H(*Fjs$\"3\"*o)4+YN\"3LFjsF]br7$Fgar7$$\" 3]KH3xCiw)*Fjs$!3T8*Q$fx%)=BF/7$7$$\"3oQCVKCVK**Fjs$!3@I]xv4l;BF/Fibr7 $Fcbr7$$\"3C_CK4.uH)*Fjs$\"3S5'o'3[adKFjs7$7$F`cr$\"3WdpKtE*G:$FjsFecr 7$7$$\"3dPCVKCVK**FjsFbcr7$$\"3OOZ_]T^15F/$!32!4)\\1&=PI#F/7$7$$\"3/9N ^8N^85F/$!35&R5RE-4I#F/Fbdr7$F[dr7$$\"3:(p*e;#zZ+\"F/$\"35_Dq+0#Q4$Fjs 7$7$Fidr$\"3_\"=<%3x%[+$FjsF^er7$Fhdr7$$\"3$y99l-\"F/$\"3 ]zy5@1fPHFjs7$7$F_fr$\"3M$4ZCC7O'GFjsFdfr7$7$F_fr$!3%=dSj2beG#F/7$$\"3 t40SY&3W/\"F/$!3sUsSjH\\vAF/7$7$$\"3SaS0aS0a5F/$!35#[%>\\AZrAF/Fagr7$F jfr7$$\"3?k9CXA=[5F/$\"3KR]]N%)[)y#Fjs7$7$Fhgr$\"3,Is,Wj#)GFFjsF]hr7$F ggr7$$\"3_q,j#zVM1\"F/$!3)HuSOP>BE#F/7$7$$\"3cCVKCVKu5F/$!3Gn$z()H>xD# F/Fghr7$Fchr7$$\"3_$y$*=w'yp5F/$\"3O0)>9#)ohk#Fjs7$7$F^ir$\"3***R`izb, g#FjsFcir7$F]ir7$$\"3Kf)\\8SLD3\"F/$!3mSXV]cs\\AF/7$7$$\"3u%f%f%f%f%4 \"F/$!3K)p'>tTcWAF/F]jr7$Fiir7$$\"3]lg9Q2L\"4\"F/$\"3\"y?0+&oI5DFjs7$7 $Fdjr$\"3S0t%)>(*GxCFjsFijr7$Fcjr7$$\"3vfEWVWn,6F/$!3)R'[()\\0oPAF/7$7 $$\"3\"\\'['['['[6\"F/$!3eNLEJq(>B#F/Fc[s7$F_[s7$$\"3101VVq\"G6\"F/$\" 3'=7$[[kf!Q#Fjs7$7$Fj[s$\"374zvd%Q*fBFjsF_\\s7$Fi[s7$$\"3U$**Qd@k37\"F /$!3kuG:Q_:EAF/7$7$$\"34N^8N^8N6F/$!3$Rm;`/I*>AF/Fi\\s7$Fe\\s7$$\"3r/> =#Q[U8\"F/$\"3!G,SQ9\\nD#Fjs7$7$F`]s$\"3Y2Cv)oJyC#FjsFe]s7$F_]s7$$\"37 CG(y@+,9\"F/$!3J=vpWI7:AF/7$7$$\"3E0aS0aSb6F/$!3k5C'\\A(R3AF/F_^s7$7$F f^s$\"3Kpq#zECP8#Fjs7$$\"3gLmuK7>^6F/F^u7$F^_s7$$\"3JN^8N^8N6F/$\"3u2C v)oJyC#Fjs7$Fe^s7$$\"3)pK+?8!Qf6F/$!3cM8%eCfX?#F/7$7$$\"3Uvcnvcnv6F/$! 3cN$H#fUN(>#F/Fh_s7$F[_s7$$\"3Sb$)ojznb6F/$\"3[@uky#*3L@Fjs7$7$F_`s$\" 3Cr]p(4]u*>FjsFd`s7$F^`s7$$\"3m#F/7$7$$\"3gXf %f%f%f>\"F/$!3G<(Q\\Pyn=#F/F^as7$Fj`s7$$\"3/E5BPbDx6F/$\"3;=!eifVO*>Fj s7$7$Feas$\"3+;*[!QCKm=FjsFjas7$Fdas7$$\"3!)e)>EBj!)>\"F/$!3aw0N`lu%=# F/7$7$$\"3y:i@;i@;7F/$!3G3()Q4#[m<#F/Fdbs7$F`bs7$$\"3m%>>&=by)>\"F/$\" 3))4b,@WFf=Fjs7$7$F[cs$\"3x*[izO0,u\"FjsF`cs7$Fjbs7$$\"3dk!eeaiu@\"F/$ !3me&3]Mca<#F/7$7$$\"3'f['['['[O7F/$!3ArFq&pVp;#F/Fjcs7$Ffcs7$$\"3Rzin T)p-A\"F/$\"3,q>UYaxH\\tlgoY7F/$!3Qp@;i@ ;i@F/Fces7$7$$\"39?\\tlgoY7F/F`r7$$\"3s`&))e'G>c7F/$!3avYBEt9c@F/7$7$$ \"39cnvcnvc7F/$!3UNSSFo/c@F/Fbfs7$F\\es7$$\"3xVH&QM5^9>/L8F/$!3>7 coYc+6@F/7$7$$\"3$o$y$y$y$yL\"F/$!3zd2.t,.5@F/F`\\t7$7$F^[t$\"3OW3\\iU Rv6Fjs7$$\"36KZ9$znqK\"F/$\"3uBWEnZ!)[6Fjs7$7$Fg\\t$\"3S3*\\r%zVu5FjsF _]t7$Ff\\t7$$\"3W;Q+OkM_8F/$!3DqIQEVq+@F/7$7$$\"3+2\"3\"3\"3\"e8F/$!3' G*z.iI]*4#F/Fi]t7$7$Fg\\t$\"3a3*\\r%zVu5Fjs7$$\"3-yvWCPJ[8F/$\"3@4I7dU tW5Fjs7$7$F`^t$\"3wyRd(fd4x*Fd^lFh^t7$F_^t7$$\"3KVd_LYor8F/$!3Q2u10#F/F`et7$7$F^dt$\"3A-*40mg2@'Fd^l7$$\"3a\\*RpE]SX\"F/$\"3)G[TTN0)p dFd^l7$7$Fget$\"3;]`e!)Qt(R&Fd^lF_ft7$Ffet7$$\"3s]0ZVL$)o9F/$!3KxU1US$ f/#F/7$7$$\"31G(H(H(H(z9F/$!3c%zo,'QTV?F/Fift7$7$Fget$\"3[\\`e!)Qt(R&F d^l7$$\"3;U/^Hs5v9F/$\"3rO5d+pmI\\Fd^l7$7$F`gt$\"3#)Ru^n2?8YFd^lFhgt7$ F_gt7$$\"3SVj7@iM)[\"F/$!3S\\)4v=by.#F/7$7$$\"3A)************\\\"F/$!3 '\\8KV9G^.#F/Fbht7$7$F`gt$\"3_Su^n2?8YFd^l7$$\"3ktxr*eOh\\\"F/$\"3O3aq n4(47%Fd^l7$7$Fiht$\"3)3W!*\\./h&QFd^lFait-%'COLOURG6&%$RGBG$\"*++++\" F+F(F(-F$6fflF&F0F4F8F)[-&QFd^lF*7$$ \"3iT*z(e_2eQFd^l$!3sg2<&f&4zzF/7$7$Fb^l$!3i#34)H,gmyF/Ffjt7$F\\[u7$$ \"3%)\\Id[I&Q4%Fd^l$!3E5e')4!H!)y(F/7$7$$\"3)=$*yY0R#*4%Fd^lF-F`[u7$Ff [u7$$\"3#38Jc\"['e?%Fd^l$!3yB\"pargPe(F/7$7$$\"3#H&=@7&QvA%Fd^lF2Fj[u7 $F`\\u7$$\"3:\"\\y^&Q_KVFd^l$!3]EY!oza5Q(F/7$7$$\"3A))oAR%HYP%Fd^lF6Fd \\u7$Fj\\u7$$\"3m[sc_otwWFd^l$!3#y()4.I@-=(F/7$7$$\"3L[72W(GZa%Fd^lF:F ^]u7$7$$\"3-\\72W(GZa%Fd^lF:7$$\"3Y(>F6cIAk%Fd^l$!3mc+MYxl\")pF/7$7$$ \"3%4zP1w5Mu%Fd^lF>F[^u7$7$$\"3C!zP1w5Mu%Fd^lF>7$$\"3xpW!\\_iQ$[Fd^l$! 3s[n#HF#)ey'F/7$7$$\"3GDpX,0?y\\Fd^lFBFh^u7$F^_u7$$\"3'QB#\\;Y3e]Fd^l$ !3mGK\")zIe$f'F/7$7$$\"3/Fv+fx]f_Fd^lFFFb_u7$Fh_u7$$\"3*[()HT*[lB`Fd^l $!3Lh$y)fVp0kF/7$7$$\"38)o\"p3Z:-cFd^lFJF\\`u7$Fb`u7$$\"3bVY:#=oFk&Fd^ l$!3s'y%Q%*o^BiF/7$7$$\"36CZWZ!*)z-'Fd^lFNFf`u7$F\\au7$$\"3B)=LX%R#H.' Fd^l$!3I#GAGH<*[gF/7$7$F]il$!3!3reT,7+.'F/F`au7$7$F]il$!3p6(eT,7+.'F/7 $$\"3?w>1[G>liFd^l$!37/eJEkZdeF/7$7$$\"3M9Sj?&zvI'Fd^lFRF]bu7$7$$\"3t: Sj?&zvI'Fd^lFR7$$\"3Z+nQJ[U.lFd^l$!3+DUl@;nmcF/7$7$$\"35(QQdzq#3mFd^lF VFjbu7$F`cu7$$\"3u,DEpa#Gy'Fd^l$!3mhZ9S\"e-[&F/7$7$$\"3QXL**o=:spFd^lF ZFdcu7$Fjcu7$$\"3\\t>$yq([9rFd^l$!3ycfPH#>%*H&F/7$7$$\"3/x!pu)zS?uFd^l FhnF^du7$Fddu7$$\"3.rP45<(Q^(Fd^l$!3O)*4StR!e7&F/7$7$$\"3m_!***z;&[)zF d^lF\\oFhdu7$F^eu7$$\"3Im\\Ejk=.!)Fd^l$!3=xj&3L\"yh\\F/7$7$Fa`m$!3$fgn =O$eJ\\F/Fbeu7$7$$\"3<2\"3\"3\"3\"3\")Fd^l$!3/0w'=O$eJ\\F/7$$\"3sHe>c 6#R3t)Fd^l$!3XCcd\">lpg%F/7$7$$\"3'[4dpIe\"f*)Fd^lFdoF[gu7$Fagu7$$\"3] HtQ)=F:<*Fd^l$!3)zW()GpbxV%F/7$7$$\"3(y#pR;jR/'*Fd^lFhoFegu7$F[hu7$$\" 3#*)G)>=Ey3(*Fd^l$!37Nx^An%)yUF/7$7$F]dm$!3Sa@R8;vzTF/F_hu7$7$$\"3r8N^ 8N^85FjsFfhu7$$\"3E=t=R$*>F5Fjs$!33u,;AjqATF/7$7$$\"3F\"=Kd^:J.\"FjsF \\pF\\iu7$Fbiu7$$\"3!y'piR&>K2\"Fjs$!3%3DY(>$yb&RF/7$7$$\"3!>$G#=P7=5 \"FjsF`pFfiu7$F\\ju7$$\"3C'3D_9Y*G6Fjs$!3wF!oh\"Q!))z$F/7$7$$\"3[\\0)H 116>\"FjsFdpF`ju7$7$$\"3i\\0)H116>\"FjsFdp7$$\"3/'=Qr\\8w>\"Fjs$!3m%yD `xKel$F/7$7$Fbhm$!3E5c4H6qBOF/F][v7$7$Fbhm$\"3VRYtw4lBwF/7$$\"3kO*fN7p L9\"FjsFj]l7$7$$\"3OO*fN7pL9\"FjsFj]l7$$\"3\\'zp!4eK06Fjs$\"3B]*RZ)o1- zF/7$7$$\"3-#[JW@W`1\"FjsF^^lFa\\v7$Fc[v7$$\"37.q#p0=nD\"Fjs$!3\\1V)GU hE]$F/7$7$$\"3%RAL!Q3F!G\"Fjs$!3wm%f%f%f%fMF/F[]v7$Fa]v7$$\"3!)Ro\\JRr >8Fjs$!35\\#4i!3k`LF/7$7$$\"3K4y%\\8CyQ\"Fjs$!3`]KCVKCVKF/Fg]v7$7$F^^v F\\q7$$\"3'[8>!GpZ(R\"Fjs$!3`\"[$G'yr.A$F/7$7$Fj[n$!3!3R]#3OL!>$F/Fd^v 7$7$Fj[n$\"3/Vp(F/7$$\"3!\\)R$ee$4I8FjsFb]l7$Fa_v7$$\"3SCBU%H_( f7Fjs$\"3)\\qZT3H6_(F/7$7$$\"3Nx'*=M'*)oB\"FjsFf]lFe_v7$F[`vFg[v7$Fj^v 7$$\"3u3'=5bkoY\"Fjs$!3+P+gM!p\"yIF/7$7$$\"3ndb(*p&f:]\"FjsF`qF``v7$Ff `v7$$\"3tq4@1**GU:Fjs$!3.2!*eIlSUHF/7$7$Fh]n$!3UE$**RSw0%GF/Fj`v7$7$Fh ]n$\"3Hfq5o(G*RoF/7$$\"37E+!o&fJn:Fjs$\"3)f!*=*=*=*=pF/7$7$$\"3SE+!o&f Jn:FjsFj\\l7$$\"3w->4yY8$\\\"Fjs$\"3sSRoe;(f0(F/7$7$$\"3g@l&eo\\^W\"Fj sF^]lF`bv7$FfbvF^_v7$F`av7$$\"3ekOH*R;7j\"Fjs$!3CkwKS_/@GF/7$7$$\"3*o3 !['=t*Q;FjsFdqF[cv7$Facv7$$\"3P1xb(e)p2lw/5Fd>FjsF\\rFgev7$F]fv7$$\"3QzS&)*)*Ry)>Fjs $!38v01s)ylL#F/7$7$Fjan$!3AeWiGY&GI#F/Fafv7$7$Fjan$\"3\\\"*zi[#*\\.jF/ 7$$\"3SN6H2pop=FjsFb\\l7$F^gvFidv7$Fgfv7$$\"3!3=_*RL(\\3#Fjs$!3kmchi:( RA#F/7$7$$\"3M^#p4mwV:#FjsF`rFcgv7$Figv7$$\"3kT$o4w#>!>#Fjs$!3IbL[&R*) *>@F/7$7$F^cn$!39h:22'p/4#F/F]hv7$7$F^cn$\"3U80K$yJ54'F/7$$\"3'=^PrO\" )[0#FjsF^\\l7$FjhvF[gv7$Fchv7$$\"3i\"3(Hh2@%H#Fjs$!3BG6&HrDZ,#F/7$7$$ \"33JZ85US$Q#FjsFdrF_iv7$7$$\"3!3tM,@/MQ#FjsFdr7$$\"3&Q1E]#\\i3CFjs$!3 :g\"z8F^0#>F/7$7$Faen$!3H%)3F/F\\jv7$7$Faen$\"3.<^FI(z[!fF/7$$ \"3W&y*)HpffE#FjsFj[l7$FijvFghv7$7$Faen$!3^%)3F/7$$\"3oJNTmq2pKFjs$!3YgTD\"[%)\\K\"F/7$7$$ \"3ib;8]O%oK$FjsF`sF[bw7$7$$\"31b;8]O%oK$FjsF`s7$$\"3I4Ji&QVFS$Fjs$!3[ xiGmX@^7F/7$7$Fc^o$!37FJed(*fM7F/Fhbw7$F`aw7$$\"3'p:u29WEN$Fjs$\"31ddq 96r)G&F/7$7$Fc^o$\"34\">j>>`\\B&F/Fbcw7$F^cw7$$\"3D@meBNdPNFjs$!3)o!y* Q;<)y6F/7$7$Fi_o$!3TkW7>***\\8\"F/F\\dw7$7$Fi_o$\"3M.')Q,j.N^F/7$$\"3( 3-wNEim`$FjsFjz7$FidwFhcw7$Fbdw7$$\"3EM%\\d%y'3o$Fjs$!3he_lC#[a6\"F/7$ 7$$\"3CX5'*RmtmPFjsFdsF^ew7$Fdew7$$\"3&RK%=UEACQFjs$!3MCHI\"zV@0\"F/7$ 7$F`bo$!3Z&[Qi#)QO/\"F/Fhew7$Ffdw7$$\"3+1(*RP!p2t$Fjs$\"3+AZ&y5%f,^F/7 $7$F`bo$\"3x'\\**[^tF/&F/Fbfw7$F^fw7$$\"3S\\j.+q.lRFjs$!3C[`&zc&Hh)*Fj s7$7$Ffco$!3c$*y)))[5Yd*FjsF\\gw7$7$Ffco$\"3)\\S#zt#3'f\\F/7$$\"3E*f`& *)p!4-%FjsFfz7$FigwFhfw7$Fbgw7$$\"3F'G$=LQ%H6%Fjs$!3/^Qp#H1oF*Fjs7$7$F _eo$!3b$Rr*[_p0))FjsF^hw7$Ffgw7$$\"3)43T'3<1sSFjs$\"3_P,33[w`\\F/7$7$F _eo$\"3Qp%e&)z%[y[F/Fhhw7$Fdhw7$$\"3-P**e4#enE%Fjs$!35CE?73Lb()Fjs7$7$ $\"3qYXA.2,*H%FjsFhsFbiw7$7$$\"3EZXA.2,*H%FjsFhs7$$\"3o=!QgqQXT%Fjs$!3 imm[XR\\p\")Fjs7$7$Fhfo$!3m_sJp*>Y0)FjsF_jw7$F^iw7$$\"3)f:*)*y1'\\Q%Fj s$\"3bS1)>lXFjs$ !3(Ry+F\"\\O9wFjs7$7$F\\io$!3'*RD'pU12O(FjsFc[x7$7$F\\io$\"3JkZ04tF\"46YF/Ff_x7$Fa_x7$$\"3)**RMT\"*fb>&Fjs$!3qVs.@gd*o&Fj s7$7$Fa]p$!3'*H]9lnfNbFjsF``x7$F\\`x7$$\"3aB.6G$=*[_Fjs$\"31Y6^XfJjXF/ 7$7$Fa]p$\"3](zt\"fH0cXF/Fj`x7$Ff`x7$$\"3ok*)yYWI)41+adFjsFatFjcx7$F`dx7$$\"3yD8OSkF`eFjs$!3WQ\"4)=.dcSFjs7$7$Fcbp$! 3O9:RK,[#)e#=6abVF/7$Fbgx7$$\"3Y'\\isL9ON'Fjs$ !3qq%o#*R@q!HFjs7$7$Fhfp$!3u>YYEnX)o#FjsFchx7$Ffgx7$$\"3O6#H8&H:'H'Fjs $\"3<)>4caI6J%F/7$7$Fhfp$\"3agTu2%o)oUF/F]ix7$Fihx7$$\"3`MTv:fzDlFjs$! 3E9r,TFY\"e#Fjs7$7$F^hp$!3Pn$=SU$45BFjsFgix7$Fcix7$$\"3sTvR'4Pra'Fjs$ \"3U'>>/MI'fUF/7$7$F^hp$\"3+(z$fqI!zA%F/Fajx7$F]jx7$$\"3t!e:&>NC+nFjs$ !3=Pq(*=]2!G#Fjs7$7$$\"3'z))H!)\\y2x'FjsFetF[[y7$Fa[y7$$\"3w3ad]NuroFj s$!3-OIX@lCZ>Fjs7$7$Feip$!3)eB,x*)eO#>FjsFe[y7$Fgjx7$$\"3#\\e!>48*[z'F js$\"31LGK'Rv:@%F/7$7$Feip$\"3efh*y\\I)*=%F/F_\\y7$F[\\y7$$\"3*=a$y9;3 UqFjs$!3S'HW3rB?g\"Fjs7$7$Fh[q$!3pz!)))oI5R:FjsFi\\y7$7$F^\\qFf\\y7$$ \"3+4$z(RFfRqFjs$\"3a4WC`nxmTF/7$7$Fh[q$\"3%p^W;Q(FjsFfy7$Fb`yF^ _y7$7$Fi^q$!3)y:2\"zggY%)Fd^l7$$\"3Al%4c+qec(Fjs$!3t@lnJR8EqFd^l7$7$Fh aq$!3nQoVNl#=J&Fd^lFj`y7$F_`y7$$\"3s;ytrjc@vFjs$\"31MWDyR5&3%F/7$7$Fha q$\"3OZv\\rbA`SF/Fday7$7$Fhaq$!3ORoVNl#=J&Fd^l7$$\"3&=U\\)[,8WxFjs$!3& >p$)*)eF*=WFd^l7$7$Fabq$!3wcdT;P!4Q#Fd^lFaby7$7$Fe`qF[by7$$\"3#4xrh4)> gxFjs$\"3EmxS)ozn/%F/7$7$Fabq$\"3Fd^l7$7$$\"3#\\6(p8Tyz!)FjsFitF icy7$F_dy7$$\"3J%Q#>%HCV5)Fjs$\"30@*pn/e%Qd%)Fjs$\"3'3_9n>'3()fFd^l 7$7$F_gq$\"3gH8:p')Q!['Fd^lFhgy7$Fagy7$$\"3em(RgvyQY)Fjs$\"3')>s>Lc$[% RF/7$7$F^hq$\"35o$4i_\"*f$RF/Fbhy7$F^hy7$$\"3w*z,cBSij)Fjs$\"3%GaPbEr2 `)Fd^l7$7$F[iq$\"3mB/`Gg'eC*Fd^lF\\iy7$7$F_gq$\"3ao$4i_\"*f$RF/7$$\"3M iCvuXr%p)Fjs$\"3%R*)>h(p#[\"RF/7$7$F[iq$\"33\"=_Lk&*4\"RF/Fiiy7$7$F[iq $\"3GA/`Gg'eC*Fd^l7$$\"3Oz:U\"H%\\;))Fjs$\"3u%fr*f$HD4\"Fjs7$7$Fdjq$\" 3Ko6GV(>a=\"FjsFfjy7$7$Fdjq$\"3?6.wBG]')QF/7$$\"3m+a**fc,z))FjsFafm7$F c[z7$FaiqF`jy7$F\\[z7$$\"3'[x2S#z/)**)Fjs$\"3lZV@T%>\"=8Fjs7$7$Fe]r$\" 3'[N9GD&\\J9FjsFi[z7$F`[z7$$\"3Hj'\\AV%pj;FjsF]]z7$Fi\\z7$$\"3AJDI>1la\"*Fjs$\"3_ `A.h4mcQF/7$7$Fh^r$\"3F9!\\vTUM$QF/Fi]z7$7$Fh^r$\"3;>'\\AV%pj;Fjs7$$\" 3W-^*o]2ZO*Fjs$\"3@:0u9#39t\"Fjs7$7$Fb`r$\"3%=$zu13$G)=FjsFf^z7$7$$\"3 ,HVKCVKC$*FjsF`^z7$$\"3!pV,_TWNQ*Fjs$\"3W/j&\\zB(GQF/7$7$Fb`r$\"3*)e,# *)e***3QF/Fc_z7$F\\_z7$$\"3)))olxzg'\\&*Fjs$\"3$\\4$Gu_t?>Fjs7$7$Fdbr$ \"3w.hX8ql*3#FjsF]`z7$Fi_z7$$\"3Vr,@f%36h*Fjs$\"3sOl*3.0A!QF/7$7$$\"3# etH(H(H(H(*Fjs$\"3A(=HmdUey$F/Fg`z7$Fc`z7$$\"3/>;,NigN(*Fjs$\"3'*Q`:R% z%*4#Fjs7$7$$\"31aRw\"=b@!)*FjsF^uFcaz7$Fiaz7$$\"3='eVgHV(=**Fjs$\"3a* \\wphz\"3BFjs7$7$F`cr$\"30V?na4C;BFjsF]bz7$F]az7$$\"3/c6jxTSP)*Fjs$\"3 =/PYZ\"Rqx$F/7$7$F`cr$\"3[[-zA%**Qw$F/Fgbz7$Fcbz7$$\"3AR\"[*p0355F/$\" 3V0a84GMGDFjs7$7$Fidr$\"3#RWgWhC!\\DFjsFacz7$F]cz7$$\"3VHvrZ!\\i+\"F/$ \"3]PG$3JjJv$F/7$7$Fidr$\"3*Rc!\\pL5VPF/F[dz7$7$Fidr$\"3QV/Y9Y-\\DFjs7 $$\"3%**o\\#p;RG5F/$\"3)Rz)zX5KPFFjs7$7$F_fr$\"3+e-3J$[/x#FjsFhdz7$Fad z7$$\"35RcnvBkG5F/$\"3q[D;it^IPF/7$7$F_fr$\"3.%GT7O#RBPF/Fbez7$F^ez7$$ \"3%3')o@C,o/\"F/$\"3,mg\"*HazNHFjs7$7$Fhgr$\"3]ukL>&46)HFjsF\\fz7$Fhe z7$$\"3aH=W%zD40\"F/$\"3,ng?yP/4PF/7$7$Fhgr$\"3\"o$Hr0'3Zq$F/Fffz7$Fbf z7$$\"3=XF9eMIl5F/$\"3,Ks>))yQCJFjs7$7$F^ir$\"3?B12%zk:=$FjsF`gz7$F\\g z7$$\"3wv)o_X/J2\"F/$\"3gArE7wo)o$F/7$7$F^ir$\"3os5&\\=)*po$F/Fjgz7$Ff gz7$$\"3EwmSwH*Q3\"F/$\"3-En?-\"oOI$Fjs7$7$Fdjr$\"3s9HakOLsLFjsFdhz7$7 $Fdjr$\"3yI`t1UmoOF/7$$\"3w3K<^?u(3\"F/F^y7$FaizF`hz7$Fjhz7$$\"3.j7B!* [c-6F/$\"3IhHTc%fTZ$Fjs7$7$Fj[s$\"33p\"y%>=!Rb$FjsFfiz7$F^iz7$$\"3ki)R I6&G&4\"F/$\"3?5GfX,JoOF/7$7$Fj[s$\"3!QRp:TY\"[OF/F`jz7$F\\jz7$$\"3im& y)yYJ@6F/$\"3c8F'G;Wjj$Fjs7$7$F`]s$\"3vp$GY0Cns$FjsFjjz7$Ffjz7$$\"3#=4 [p))yv6\"F/$\"3k9zy1hsYOF/7$7$F`]s$\"35=aP.xaGOF/Fd[[l7$F`[[l7$$\"3QLD 0m\"Q,9\"F/$\"3w!4NJpn1z$Fjs7$7$Ff^s$\"3i._S()oA\"*QFjsF^\\[l7$Fj[[l7$ $\"3B%f^@j&yR6F/$\"3#*pn;u.2EOF/7$7$Ff^s$\"3_Qo\"f'R#)4OF/Fh\\[l7$Fd\\ [l7$$\"3'*)HOQ[J!f6F/$\"3h=*G]iUv$RFjs7$7$F_`s$\"3W(*o'p*3\"y/%FjsFb][ l7$F^][l7$$\"3Qpa*H%)3>;\"F/$\"34&)\\rucI1OF/7$7$F_`s$\"3Mui]=Q$>f$F/F \\^[l7$Fh][l7$$\"3?*)QGT5*z<\"F/$\"3c]9y#*>NxSFjs7$7$Feas$\"3%y+9H#G&o >%FjsFf^[l7$Fb^[l7$$\"3+\"RGq(=&R=\"F/$\"3yM+e%>'R(e$F/7$7$Feas$\"35(= E-RQ[d$F/F`_[l7$7$Feas$\"3S3S\"H#G&o>%Fjs7$$\"3S?_(p\\8q>\"F/$\"3MR\\` 9AX5UFjs7$7$$\"3JZOtNF197F/FbuF]`[l7$Fc`[l7$$\"3f5]SYZ/;7F/$\"3$3\\>vV 9EM%Fjs7$7$F[cs$\"3ApXOJ*fKM%FjsFg`[l7$7$Feas$\"3a(=E-RQ[d$F/7$$\"3hd) H&ez\"f?\"F/$\"3#RFZ9a2$pNF/7$7$F[cs$\"3%*GA4<6]eNF/Fda[l7$7$F[cs$\"3y pXOJ*fKM%Fjs7$$\"3'*GQ<*H+ZB\"F/$\"3vcw+r4&[^%Fjs7$7$Fads$\"3S2F&oSW<_ %FjsFab[l7$Fja[l7$$\"3cpDU#=5yA\"F/$\"3a%*y!Gq1?b$F/7$7$Fads$\"3Mdm[fv )Ga$F/F[c[l7$7$Fads$\"3'zq_oSW<_%Fjs7$$\"3%*f\"*GNF/7$7$F_hs$\"3WM\"*R#[.P^$F/Ffe[l7$Fbe[l7$$\"3WJ%Qa[k5H\"F/$\"3) p&)p#*ed\"*)\\Fjs7$7$Fhis$\"3-XE0M\\m9]FjsF`f[l7$F\\f[l7$$\"3Wd\\)f!G2 $H\"F/$\"3U,W7+2_/NF/7$7$Fhis$\"3'[-/*\\aFjsF_j[l7$F[j[l7$$\"37#zA\"41xd8F/$\"3!f9O'[%fIY$F/7$7$F`^t $\"3Q]P=X\\qiMF/Fij[l7$Fej[l7$$\"3![PH\"zB.n8F/$\"3wZy@WUcMbFjs7$7$Fi_ t$\"3#)Q@B/6T$e&FjsFc[\\l7$7$Fi_t$\"3uu,Fl/S[MF/7$$\"3O[n/ZAqj8F/Fjx7$ 7$Fa\\\\l$\"3m^%f%f%f%fMF/F_[\\l7$Fi[\\l7$$\"3))*fHF\"G:'Q\"F/$\"3I`E- *[8sl&Fjs7$7$F_at$\"3\"QyUrx'\\6dFjsFh\\\\l7$F]\\\\l7$$\"3cm-3>qWz8F/$ \"3-mfID11[MF/7$7$$\"37Z'['['[')R\"F/$\"3!Qln,c6NV$F/Fb]\\l7$F^]\\l7$$ \"3yl$*[p,K09F/$\"3Q!z4ofd[x&Fjs7$7$Fhbt$\"3qAsN(H%RMeFjsF^^\\l7$7$F_a t$\"3Caw;g:^LMF/7$$\"3'f`&**[w:,9F/$\"3vQE4*y'pKMF/7$7$$\"3I<*=*=*=*=9 F/$\"3Kk%*\\cR<>MF/F[_\\l7$Fd^\\l7$$\"3^TV4XC`C9F/$\"3IYy5h.r()eFjs7$7 $F^dt$\"3!*Q\"HTN5B&fFjsFg_\\l7$7$FhbtFd_\\l7$$\"3m8pR!f;GU\"F/$\"3>$ \\FqH%)yT$F/7$7$F^dt$\"35I_uviO0MF/Fb`\\l7$F]`\\l7$$\"3?*[#*yu(yV9F/$ \"3[XU8eN(f*fFjs7$7$Fget$\"3uYkb.'Fjs7$7$Fiht$\" 3e(y\"z'Q\"4yiFjsFgc\\l7$Fcc\\l7$$\"3w%*\\su0]([\"F/$\"3'fgwhWqlP$F/7$ 7$Fiht$\"3u*[TFHCpO$F/Fad\\l-F[jt6&F]jtF(F(F^jt-%+AXESLABELSG6%%\"kG% \"aG-%%FONTG6#%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 14 "The procedure " }{TEXT 0 6 "fsolve" } {TEXT -1 113 " can be used to solve the pair of simultaneous equations numerically when given suitable starting approximations." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 110 "sol \+ := fsolve(\{cosh(k*(-1-a))-1=k,cosh(k*(2-a))-1=2*k\},\{k=1.05,a=0.3\}) ;\na1 := subs(sol,a);\nk1 := subs(sol,k);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$solG<$/%\"aG$\"+[%H6%G!#5/%\"kG$\"+[7\"f/\"!\"*" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#a1G$\"+[%H6%G!#5" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%#k1G$\"+[7\"f/\"!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 26 "The required catenary is " } {XPPEDIT 18 0 "y= (cosh(k[1]*(x-a[1]))-1)/k[1]" "6#/%\"yG*&,&-%%coshG6 #*&&%\"kG6#\"\"\"F.,&%\"xGF.&%\"aG6#F.!\"\"F.F.F.F4F.&F,6#F.F4" } {TEXT -1 8 ", where " }{XPPEDIT 18 0 "a[1]" "6#&%\"aG6#\"\"\"" }{TEXT -1 1 " " }{TEXT 284 1 "~" }{TEXT -1 18 " 0.2841129448 and " }{XPPEDIT 18 0 "k[1]" "6#&%\"kG6#\"\"\"" }{TEXT -1 1 " " }{TEXT 285 1 "~" } {TEXT -1 14 " 1.045911248. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 65 "(c) We can now compare the graphs of the parabo la and catenary. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 135 "plot([(cosh(k1*(x-a1))-1)/k1,k0*(x-a0)^2],x =-1.2..2.2,y,thickness=2,\ncolor=[COLOR(RGB,.6,.3,.8),coral],legend=[` catenary`,`parabola`]);" }}{PARA 13 "" 1 "" {GLPLOT2D 416 351 351 {PLOTDATA 2 "6'-%'CURVESG6%7S7$$!3%**************>\"!#<$\"3Wwu*esNDS\" F*7$$!3ALL$en*)e7\"F*$\"3oD3yzqKU7F*7$$!3_m;/zmSh5F*$\"3*oD&f!*Rr86F*7 $$!3ALL$30))))))*!#=$\"3>TC'*Hj*H!)*F:7$$!3cJL$eN-*e\"*F:$\"3/u]Z?Zbs& )F:7$$!3plmTbhQK%)F:$\"3#z+([ip(HX(F:7$$!37KL3PE\")exF:$\"3OkmI'p?H]'F :7$$!3E)**\\PFm81(F:$\"3'frHr@*=.cF:7$$!3LJL3Zi1SjF:$\"3-6eA)))[vv%F:7 $$!3\\)**\\(Q%z5i&F:$\"3=Ut;Naw&*RF:7$$!3!\\mm;88:)[F:$\"3EghOq?;#H$F: 7$$!3CJL$es-,B%F:$\"3OK#eR4)GOFF:7$$!3a(****\\*)on\\$F:$\"3!*eS*)yjry@ F:7$$!3W)****\\#RUgFF:$\"3uaAAN_P)o\"F:7$$!3)o****\\5>30#F:$\"3C+(H%p_ .z7F:7$$!3\">L$3Z*HkS\"F:$\"3%*z3*37V7f*!#>7$$!3'pjmm6*)=S'Fcp$\"3))\\ IZIcL3kFcp7$$\"3+5^LLLv<*)!#@$\"3;z'G\"zDhDUFcp7$$\"3[@+]i6$)RwFcp$\"3 =!o>jv*>lAFcp7$$\"3#*F:$\"3D!\\2$ePz2AF:7$$\"3U0+]FL!\\)**F:$\"3?$4\\w@?`z#F:7$$\"3o LLL5$[s1\"F*$\"3'H`cOHn1R$F:7$$\"3S++]F@mS6F*$\"3ut\\%GI#Q+TF:7$$\"3+n ;Hdf=27F*$\"3H5&*H-F'H\"[F:7$$\"3%3++X29*z7F*$\"3mpNZH))yrcF:7$$\"3!QL e\\cX$[8F*$\"3!e)4P1NIglF:7$$\"3K+]7!>w)>9F*$\"3isnB-RXxvF:7$$\"3'QLL3 X5)*[\"F*$\"3#)Qn0DYkk')F:7$$\"3U+]iS^-j:F*$\"3j/V&QWiu!**F:7$$\"3\\nm m)eRNj\"F*$\"3p%4:B$HB@6F*7$$\"3RnmT'z]cq\"F*$\"3%G#)4#[;Pm7F*7$$\"33M $ep)[;x=>F*$\"3Q2z@eF*$\"3CW#z?4$3e>F*7$$\"3E+]i66Qd? F*$\"3)[%=\"fe-L<#F*7$$\"30,]PY^7E@F*$\"3)p%eRce#eR#F*7$$\"3;+++++++AF *$\"3+$3c-7;Vl#F*-%&COLORG6&%$RGBG$\"\"'!\"\"$\"\"$Fb[l$\"\")Fb[l-%'LE GENDG6#%)catenaryG-F$6%7S7$F($\"3sb*\\AE*zZ8F*7$F.$\"3YnX3C(zG@\"F*7$F 3$\"3\"=A=p_u75\"F*7$F8$\"3\"z+j\"Rv'>#)*F:7$F>$\"3+6\")4T\")3#p)F:7$F C$\"3GgPm9k5OwF:7$FH$\"3]:FZY#\\\"=nF:7$FM$\"3vCI1+CeHeF:7$FR$\"35q+P? s*o(\\F:7$FW$\"3ITv(Q'*3S>%F:7$Ffn$\"3Ol![*)))o&eMF:7$F[o$\"3I\\.Nm>Zp GF:7$F`o$\"3OmLOO60sAF:7$Feo$\"3*pw\")zAgAu\"F:7$Fjo$\"3S-Wb@sG\\,P^*Fcp7$Fep$\"3w*R`2-l+4'Fcp7$Fjp$\"3sE#)Rniv%y$Fcp7$F `q$\"3')R#GS%)\\(*y\"Fcp7$Feq$\"3q8@u2;O*R'F^r7$Fjq$\"31FqCDXg.WF\\q7$ F`r$\"3bv]o$e&fR7F^r7$Ffr$\"3w!Q_EUkd!))F^r7$F[s$\"3!f4E'>(f4=#Fcp7$F` s$\"3\\;NZ=TcLUFcp7$Fes$\"3ck(Rf^z)zqFcp7$Fjs$\"3]$y\"onO,:5F:7$F_t$\" 3.Ud^+yb39F:7$Fdt$\"3YX?_Cyo#)=F:7$Fit$\"3#y'oV-O(HT#F:7$F^u$\"3C;-71> g))HF:7$Fcu$\"3%\\hyXF7)*p$F:7$Fhu$\"3_M#)=)GcNS%F:7$F]v$\"3k\"H5M)faA _F:7$Fbv$\"3ZNp[c*e\\-'F:7$Fgv$\"3!>HmI[%znpF:7$F\\w$\"3oV%p([h[F*7$F^z$\"3e\\ #[B()RF8#F*7$Fcz$\"3EgjOp0Q(H#F*7$Fhz$\"3i8qt&fK6[#F*-%'COLOURG6&F_[l$ \"*++++\"!\")$\")AR!)\\Ffel$\"\"!Fjel-Fh[l6#%)parabolaG-%*THICKNESSG6# \"\"#-%+AXESLABELSG6$Q\"x6\"Q\"yFffl-%%VIEWG6$;$!#7Fb[l$\"#AFb[l%(DEFA ULTG" 1 2 0 1 10 2 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "catenar y" "parabola" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "; " }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 5 "Tasks" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q1" }}{PARA 0 "" 0 "" {TEXT -1 112 "(a) Fi nd the equation of the parabola with its axis of symmetry vertical, su ch that it passes through the points" }{XPPEDIT 18 0 " ``(-1,1)" "6#-% !G6$,$\"\"\"!\"\"F'" }{TEXT -1 4 " and" }{XPPEDIT 18 0 " ``(4,4)" "6#- %!G6$\"\"%F&" }{TEXT -1 33 " and has its lowest point on the " }{TEXT 307 1 "x" }{TEXT -1 14 " axis between " }{XPPEDIT 18 0 "x=-1" "6#/%\"x G,$\"\"\"!\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "x = 4;" "6#/%\"xG \"\"%" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 112 "(b) Find the eq uation of the catenary with its axis of symmetry vertical, such that i t passes through the points" }{XPPEDIT 18 0 " ``(-1,1)" "6#-%!G6$,$\" \"\"!\"\"F'" }{TEXT -1 4 " and" }{XPPEDIT 18 0 " ``(4,4)" "6#-%!G6$\" \"%F&" }{TEXT -1 33 " and has its lowest point on the " }{TEXT 308 1 " x" }{TEXT -1 14 " axis between " }{XPPEDIT 18 0 "x=-1" "6#/%\"xG,$\"\" \"!\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "x = 4;" "6#/%\"xG\"\"%" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 97 "(c) Make a graphical com parison between the catenary found in (b) and the parabola found in (a ). " }}{PARA 0 "" 0 "" {TEXT -1 32 "________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 32 "________________________________" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 2 "Q2" }}{PARA 0 "" 0 "" {TEXT -1 108 "Find the equation of the parabola with its axis of symmetry vertical, such that it passes \+ through the points" }{XPPEDIT 18 0 " ``(-1,1)" "6#-%!G6$,$\"\"\"!\"\"F '" }{TEXT -1 4 " and" }{XPPEDIT 18 0 " ``(4,4)" "6#-%!G6$\"\"%F&" } {TEXT -1 70 " and also passes through the lowest point of the catenary found in Q1." }}{PARA 0 "" 0 "" {TEXT -1 32 "________________________ ________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 32 "_______________________________ _" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 257 "" 0 "" {TEXT -1 18 "Code for pictures " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 31 "Code for hanging cha in picture " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 320 "p1 := plot([cosh(x/5)+0.0004*sin(100*x),cosh(x/5)- 0.0004*sin(100*x)],x=-1..1,\n color=COLOR(RGB,.4,.4,.4),numpoints=100 ,axes=none):\np2 := plots[polygonplot]([[[1,1.02],[1.1,1.02],[1.1,0.99 9],[1,0.999]],\n [[-1,1.02],[-1.1,1.02],[-1.1,0.999],[-1,0.999]]],colo r=brown):\nplots[display]([p1,p2],view=[-1.1..1.1,0.999..1.02]);" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 31 "Code for cable section \+ picture " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1705 "p1 := plot(cosh(x)-1,x=-2..2,thickness=2):\npurple \+ := COLOR(RGB,.4,0,.9):\ndarkgreen := COLOR(RGB,0,.7,0):\ndarkgrey := C OLOR(RGB,.01,.01,.01):\ngrey := COLOR(RGB,.4,.4,.6):\nx0 := 1.3: y0 := cosh(1.3)-1: \nx1 := 1:\ny1 := sinh(1.3):\na1 := plots[arrow]([x0,y0] ,[x1,y1],shape=arrow,color=blue,\n thickness=2,width=1/20,head_len gth=1/5):\na2 := plots[arrow]([x0,y0],[x1,0],shape=arrow,color=brown, \n thickness=2,width=1/16,head_length=1/6):\na3 := plots[arrow]([x 0+x1,y0],[0,y1],shape=arrow,color=darkgreen,\n thickness=2,width=1 /20,head_length=1/5):\na4 := plots[arrow]([0,0],[-1.3,0],shape=arrow,c olor=purple,\n thickness=2,width=1/16,head_length=1/8):\np2 := plo t([[[0,0],[x0,y0]]$3],style=point,\n symbol=[circle,diamond,c ross],color=red):\np3 := plot([[x0,0],[x0,y0]],linestyle=2,color=darkg rey):\np4 := plot([x0+.25*cos(t),y0+.25*sin(t),t=0..arctan(y1)],color= grey):\nt1 := plots[textplot]([2.6,1.87,`T sin`],color=darkgreen):\nt2 := plots[textplot]([1.87,0.75,`T cos`],color=brown):\nt3 := plots[tex tplot]([1.87,1.65,`T`],color=blue):\nt4 := plots[textplot]([-0.7,-.2,` T`],color=purple):\nt5 := plots[textplot]([-0.63,-.23,`o`],color=purpl e):\nt6 := plots[textplot]([-1.55,2.1,`cable`],color=red):\nt7 := plot s[textplot]([1.63,1.17,`q`],font=[SYMBOL],color=grey):\nt8 := plots[te xtplot]([2.15,0.75,`q`],font=[SYMBOL],color=brown):\nt9 := plots[textp lot]([2.85,1.87,`q`],font=[SYMBOL],color=darkgreen):\nt10 := plots[tex tplot]([[-0.13,.2,`P`],[1.25,1.1,`P`],\n [x0,-0.15,`(x,0)`],[2.82,-.1, `x`],[-.1,2.8,`y`]],color=darkgrey):\nt11 := plots[textplot]([-0.08,.1 5,`o`],color=darkgrey):\nplots[display]([p1,p2,p3,p4,a1,a2,a3,a4,t1,t2 ,t3,t4,t5,t6,t7,t8,t9,t10,t11],\n tickmarks=[0,0],view=[-2..2.9,-.23. .2.82]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 27 "Code for a rclength picture " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 476 "p1 := plot([[[0,0],[3,0],[3,1]],\n \+ [[2.8,0],[2.8,.1],[3,.1]]],color=black):\np2 := plot(.1333333335e-1 *x^2+.2933333333*x,x=-1..4,\n color=red,thickness=2 ):\nt1 := plots[textplot]([[1.44,-.15,`d`],[3.25,.5,`d`],\n [ 1.24,.63,`d`]],font=[SYMBOL]):\nt2 := plots[textplot]([[1.61,-.15,`x`] ,[3.42,.5,`y`],\n [1.41,.63,`s`]],font=[HELVETICA,10]):\nt3 := pl ots[textplot]([3.9,1.1,`y = f(x)`],color=red):\nplots[display]([p1,p2, t1,t2,t3],axes=none);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}}{MARK "4 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }