{VERSION 5 0 "IBM INTEL NT" "5.0" } {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 "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 "" -1 257 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 261 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 263 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 266 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 268 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 0 1 0 0 255 1 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 270 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 271 "" 0 1 0 0 255 1 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 272 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 18 273 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 274 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" 18 275 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 276 "" 0 1 0 0 0 0 0 1 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 "Text Outp ut" -1 2 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 0 0 0 0 0 1 3 0 3 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 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 "Warning" 2 7 1 {CSTYLE "" -1 -1 "" 0 1 0 0 255 1 0 0 0 0 0 0 1 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 12 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Plot" 0 13 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Title" -1 18 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 1 2 2 2 1 1 1 1 }3 1 0 0 12 12 1 0 1 0 2 2 19 1 }{PSTYLE "Author" -1 19 1 {CSTYLE "" -1 -1 "T imes" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 8 8 1 0 1 0 2 2 0 1 } {PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 18 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 "Normal" -1 257 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 258 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Normal" -1 259 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 }} {SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 40 "Partial Differential Equa tions PowerTool" }}{PARA 19 "" 0 "" {TEXT -1 16 "by Dr. Jim Herod" }}} {PARA 256 "" 0 "" {TEXT -1 0 "" }{TEXT 259 37 "Section 7.4: Recipe for a Cheese Cake" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 292 "Here is a recipe for a cheese cake. If you have to worry about cholesterol, just stay away from this one. If you don't, this o ne is really great. Fix it a day before you want to impress guests and serve it with the best coffee you have. Wow! Guaranteed a success if \+ you follow the directions." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG } {EXCHG }{PARA 0 "" 0 "" {TEXT -1 25 "The Mathematical Question" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 269 55 "Now, as \+ an applied mathematician, here is your question" }{TEXT 270 173 ": Eg gs congeal at about 140 degrees. All the ingredients of the cheese cak e before cooking are about 46 degrees. When the ingredients are mixed \+ and placed in the hot oven, " }{TEXT 271 74 "how long does it take to \+ get the center of the cooking cake to 140 degrees" }{TEXT 272 1 "?" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 23 "The Mat hematical Model." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "This is a heat diffusion problem. We model it as" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 265 6 "PDE: " } {TEXT -1 13 " " }{XPPEDIT 273 0 "diff(u(t,r,theta,z),t);" "6#-%%diffG6$-%\"uG6&%\"tG%\"rG%&thetaG%\"zGF)" }{TEXT 274 5 " = c " } {XPPEDIT 275 0 "Delta;" "6#%&DeltaG" }{TEXT 276 2 " u" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 261 14 "side boundary:" } {TEXT -1 10 " u(t, 5, " }{XPPEDIT 18 0 "theta;" "6#%&thetaG" }{TEXT -1 21 ", z) = 550, t > 0, " }{XPPEDIT 18 0 "-Pi;" "6#,$%#PiG!\"\"" } {XPPEDIT 18 0 "` ` < ` `;" "6#2%\"~GF$" }{XPPEDIT 18 0 "theta;" "6#%&t hetaG" }{XPPEDIT 18 0 "` ` < ` `;" "6#2%\"~GF$" }{XPPEDIT 18 0 "Pi;" " 6#%#PiG" }{TEXT -1 14 ", 0 < z < 1.5," }}{PARA 0 "" 0 "" {TEXT 262 25 "top and bottom boundary: " }{TEXT -1 9 " u(t, r, " }{XPPEDIT 18 0 "th eta;" "6#%&thetaG" }{TEXT -1 21 ", 0) = 550 = u(t, r, " }{XPPEDIT 18 0 "theta;" "6#%&thetaG" }{TEXT -1 7 ", 1.5)," }}{PARA 0 "" 0 "" {TEXT 263 18 "initial condition:" }{TEXT -1 10 " u(0, r, " }{XPPEDIT 18 0 " theta;" "6#%&thetaG" }{TEXT -1 11 ", z) = 46. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 204 "That is, we assume the h eat diffuses into the cheese cake as modeled by the diffusion equation for a cylinder, that the cylinder is initially at 46 degrees, and the surrounding temperature is 550 degrees." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 93 "We ask: at what time is the temper ature in the middle 140 degrees? That is, compute t so that" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 " \+ u(t, 0, 0, 3/4 ) = 140." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 " " 0 "" {TEXT -1 9 "Solution:" }}{PARA 258 "" 0 "" {TEXT 264 85 "Recall that when the Laplacian operator is written out for a cylinder the eq uation is" }}{PARA 258 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 " " }{XPPEDIT 18 0 "diff(r*diff(u,r),r)/r+di ff(u,`$`(theta,2))/(r^2);" "6#,&*&-%%diffG6$*&%\"rG\"\"\"-F&6$%\"uGF)F *F)F*F)!\"\"F**&-F&6$F--%\"$G6$%&thetaG\"\"#F**$F)F6F.F*" }{TEXT -1 4 " + " }{XPPEDIT 18 0 "diff(u,`$`(z,2));" "6#-%%diffG6$%\"uG-%\"$G6$% \"zG\"\"#" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "1/c;" "6#*&\"\"\"F$%\"cG! \"\"" }{TEXT -1 2 " " }{XPPEDIT 18 0 "diff(u,t);" "6#-%%diffG6$%\"uG% \"tG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 287 "The number c above is the rate of diffusion for heat t hrough the cream cheese/egg mixture. For this problem, since the init ial condition and boundary condition are independent of theta, the sol ution will be independent of theta. Thus, we can rewrite the partial d ifferential equation as" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 " " }{XPPEDIT 18 0 "1/c;" "6#*&\"\"\"F$%\"cG !\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "diff(u,t);" "6#-%%diffG6$%\"uG% \"tG" }{TEXT -1 5 " = " }{XPPEDIT 18 0 "diff(r*diff(u,r),r)/r;" "6#* &-%%diffG6$*&%\"rG\"\"\"-F%6$%\"uGF(F)F(F)F(!\"\"" }{TEXT -1 5 " + \+ " }{XPPEDIT 18 0 "diff(u,`$`(z,2));" "6#-%%diffG6$%\"uG-%\"$G6$%\"zG\" \"#" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 303 "Agree that the steady state solution for the equation \+ will be 550 degrees. (As a cook, you must not let the cake achieve thi s state!) Thus, we solve the problem with homogeneous boundary conditi ons and add 550. For ease of typing, take a = 5, the radius of the cak e, and b = 1.5, the height of the cake." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 102 "Perform a separation of variables o n u. That is, assume that u(t, r, z) = T(t) R(r) Z(z). We get that" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "1/r;" "6#*&\"\"\"F$%\"rG!\"\"" }{TEXT -1 29 " (r R ') \+ ' Z T + Z '' R T = " }{XPPEDIT 18 0 "1/c;" "6#*&\"\"\"F$%\"cG!\"\"" } {TEXT -1 9 " T ' R Z." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 23 "Dividing by R Z T gives" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 " " }{XPPEDIT 18 0 "1/r;" "6#*&\" \"\"F$%\"rG!\"\"" }{TEXT -1 35 " (r R ') ' / R + Z '' / Z = " } {XPPEDIT 18 0 "1/c;" "6#*&\"\"\"F$%\"cG!\"\"" }{TEXT -1 10 " T ' / T. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 135 "As u sual, we make the argument that each of the terms of the sum on the l eft side of the above equation must be constant. We have that" }} {PARA 0 "" 0 "" {TEXT -1 10 "(1) " }{XPPEDIT 18 0 "1/r;" "6#*&\" \"\"F$%\"rG!\"\"" }{TEXT -1 16 " (r R ') ' = " }{XPPEDIT 18 0 "-mu^ 2;" "6#,$*$%#muG\"\"#!\"\"" }{TEXT -1 15 " R , R(a) = 0." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 "(2) Z '' = \+ " }{XPPEDIT 18 0 "-lambda^2;" "6#,$*$%'lambdaG\"\"#!\"\"" }{TEXT -1 21 " Z, Z(0) = Z(b) = 0." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 3 "and" }}{PARA 0 "" 0 "" {TEXT -1 24 "(3) T ' = - c ( " }{XPPEDIT 18 0 "mu^2+lambda^2;" "6#,&*$%#muG\"\"#\"\"\"*$%' lambdaGF&F'" }{TEXT -1 5 " ) T." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 31 "We handle these one at a time. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 25 "Analysis of e quation (1):" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 26 "We rewrite equation (1) as" }}{PARA 0 "" 0 "" {TEXT -1 24 " \+ (r R ') ' = " }{XPPEDIT 18 0 "-mu^2;" "6#,$*$%#muG\"\"#!\"\"" }{TEXT -1 17 " r R , R(a) = 0." }}{PARA 0 "" 0 "" {TEXT -1 2 "or" }} {PARA 0 "" 0 "" {TEXT -1 10 " " }{XPPEDIT 18 0 "r^2;" "6#*$% \"rG\"\"#" }{TEXT -1 16 " R '' + r R ' + " }{XPPEDIT 18 0 "mu^2;" "6#* $%#muG\"\"#" }{TEXT -1 1 " " }{XPPEDIT 18 0 "r^2;" "6#*$%\"rG\"\"#" } {TEXT -1 21 "R = 0, with R(a) = 0." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 81 "We verify that the solution for this last equation is among the Bessel functions." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "R:=r->BesselJ(0,mu*r);\nr^2*diff(R(r),r,r)+r*diff(R(r ),r)+mu^2*r^2*R(r);\nsimplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %\"RGf*6#%\"rG6\"6$%)operatorG%&arrowGF(-%(BesselJG6$\"\"!*&%#muG\"\" \"9$F2F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*()%\"rG\"\"#\"\"\", &-%(BesselJG6$\"\"!*&%#muGF(F&F(F(*(F/!\"\"F&F1-F+6$F(F.F(F1F()F/F'F(F 1*(F&F(F2F(F/F(F1*(F4F(F%F(F*F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# \"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 25 "Analysis of equation (2):" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 102 "Equation (2) is recognized as an equatio n, with boundary conditions, which defines the sine functions." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "Z:=z->sin(lambda*z);\ndiff(Z (z),z,z)+lambda^2*Z(z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"ZGf*6#% \"zG6\"6$%)operatorG%&arrowGF(-%$sinG6#*&%'lambdaG\"\"\"9$F1F(F(F(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 258 "" 0 "" {TEXT -1 25 "Analysis of equation (3):" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 71 "Equation (3) is recognized as an equation for the exponential function." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "T:=t->exp(-c*(mu^2+lambda^2)*t);\nd iff(T(t),t)+c*(mu^2+lambda^2)*T(t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%\"TGf*6#%\"tG6\"6$%)operatorG%&arrowGF(-%$expG6#,$*(%\"cG\"\"\",&*$ )%#muG\"\"#F2F2*$)%'lambdaGF7F2F2F29$F2!\"\"F(F(F(" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 " " 0 "" {TEXT -1 43 "Products of solutions of (1), (2), and (3)." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 103 "Products of solutions for equations (1), (2), and (3) should form a solution f or the original equation." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "u:=(t,r,z)->BesselJ(0,mu*r)*sin(lambda*z)*exp(-c*(mu^2+lambda^2)*t); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"uGf*6%%\"tG%\"rG%\"zG6\"6$%)op eratorG%&arrowGF**(-%(BesselJG6$\"\"!*&%#muG\"\"\"9%F5F5-%$sinG6#*&%'l ambdaGF59&F5F5-%$expG6#,$*(%\"cGF5,&*$)F4\"\"#F5F5*$)F;FFF5F5F59$F5!\" \"F5F*F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "1/r*diff(r*di ff(u(t,r,z),r),r)+diff(u(t,r,z),z,z)-1/c*diff(u(t,r,z),t);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,(*&%\"rG!\"\",&**-%(BesselJG6$\"\"\"*&%#muG F,F%F,F,F.F,-%$sinG6#*&%'lambdaGF,%\"zGF,F,-%$expG6#,$*(%\"cGF,,&*$)F. \"\"#F,F,*$)F3F>F,F,F,%\"tGF,F&F,F&*,F%F,,&-F*6$\"\"!F-F,*(F.F&F%F&F)F ,F&F,F=F,F/F,F5F,F&F,F,**FDF,F/F,F@F,F5F,F&**FDF,F/F,F;F,F5F,F," }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 17 "G eneral solution " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 121 "We add constants times products of solutions for (1), (2 ), and (3) to make the general solution of the original equation:" }} {PARA 0 "" 0 "" {TEXT -1 28 " u(t, r, z) = " }{XPPEDIT 18 0 "sum(sum(A[n,m]*BesselJ(0,mu[m]*r)*sin(lambda[n]*z)*exp(-c*(mu[m] ^2+lambda[n]^2)*t),m = 1 .. infinity),n = 1 .. infinity);" "6#-%$sumG6 $-F$6$**&%\"AG6$%\"nG%\"mG\"\"\"-%(BesselJG6$\"\"!*&&%#muG6#F-F.%\"rGF .F.-%$sinG6#*&&%'lambdaG6#F,F.%\"zGF.F.-%$expG6#,$*(%\"cGF.,&*$&F56#F- \"\"#F.*$&F=6#F,FJF.F.%\"tGF.!\"\"F./F-;F.%)infinityG/F,;F.FR" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 73 "We verify that this sum is a solution. For simplicity, take only 9 terms." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 123 "u:=(t,r,z)->sum(sum(A[n,m]* BesselJ(0,mu[m]*r)*sin(lambda[n]*z)*\n exp(-c*(mu[m]^2+lambda[n]^2)* t),m = 1 .. 3),n = 1 .. 3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"uGf *6%%\"tG%\"rG%\"zG6\"6$%)operatorG%&arrowGF*-%$sumG6$-F/6$**&%\"AG6$% \"nG%\"mG\"\"\"-%(BesselJG6$\"\"!*&&%#muG6#F8F99%F9F9-%$sinG6#*&&%'lam bdaG6#F7F99&F9F9-%$expG6#,$*(%\"cGF9,&*$)F?\"\"#F9F9*$)FGFTF9F9F99$F9! \"\"F9/F8;F9\"\"$/F7FZF*F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "1/r*diff(r*diff(u(t,r,z),r),r)+diff(u(t,r,z),z,z)-1/c*diff(u(t,r ,z),t):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 257 9 "Computing" }{TEXT -1 1 " " }{XPPEDIT 18 0 "mu;" "6#%#muG " }{TEXT -1 4 " 's " }{TEXT 258 3 "and" }{TEXT -1 1 " " }{XPPEDIT 18 0 "lambda;" "6#%'lambdaG" }{TEXT -1 4 " 's." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "We identify the eigenvalues " } {XPPEDIT 18 0 "mu[m];" "6#&%#muG6#%\"mG" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "lambda[n];" "6#&%'lambdaG6#%\"nG" }{TEXT -1 3 " , " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "For " } {XPPEDIT 18 0 "lambda[n];" "6#&%'lambdaG6#%\"nG" }{TEXT -1 21 " , we h ave that sin( " }{XPPEDIT 18 0 "lambda[n];" "6#&%'lambdaG6#%\"nG" } {TEXT -1 18 " b) = 0, so that " }{XPPEDIT 18 0 "lambda[n];" "6#&%'lam bdaG6#%\"nG" }{TEXT -1 7 " b = n " }{XPPEDIT 18 0 "pi;" "6#%#piG" } {TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "for n from 1 to 40 do\n lambda[n]:=n*Pi/(3/2):\nend do:\nn:='n':" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 4 "For " }{XPPEDIT 18 0 "mu[m];" "6#&%#muG6# %\"mG" }{TEXT -1 27 " , we have that BesselJ(0, " }{XPPEDIT 18 0 "mu[m ];" "6#&%#muG6#%\"mG" }{TEXT -1 17 " a) = 0, so that " }{XPPEDIT 18 0 "mu[m];" "6#&%#muG6#%\"mG" }{TEXT -1 8 " is the " }{XPPEDIT 18 0 "m^th ;" "6#)%\"mG%#thG" }{TEXT -1 36 " zero of BesselJ(0,x) divided by a. \+ " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "for m from 1 to 40 do\n \+ mu[m]:=evalf(BesselJZeros(0,m))/5:\nend do:\nm:='m':" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 22 "Computing coeffi cients" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "We choose the coefficients " }{XPPEDIT 18 0 "A[m,n];" "6#&%\"AG6$%\"m G%\"nG" }{TEXT -1 21 " so that when t = 0," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 " 46 - 550 = " } {XPPEDIT 18 0 "sum(sum(A[n,m]*BesselY(0,mu[m]*r)*sin(lambda[n]*z),m = \+ 1 .. infinity),n = 1 .. infinity);" "6#-%$sumG6$-F$6$*(&%\"AG6$%\"nG% \"mG\"\"\"-%(BesselYG6$\"\"!*&&%#muG6#F-F.%\"rGF.F.-%$sinG6#*&&%'lambd aG6#F,F.%\"zGF.F./F-;F.%)infinityG/F,;F.FB" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 59 "These coefficie nts will come from the Fourier coefficients:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 " " }{XPPEDIT 18 0 "A[m,n];" "6#&%\"AG6$%\"mG%\"nG" }{TEXT -1 5 " = " }{XPPEDIT 18 0 "(46-550)*in t(sin(lambda[n]*z),z = 0 .. b)*int(R(mu[m]*r)*r,r = 0 .. a)/(int(sin(l ambda[n]*z)^2,z = 0 .. b)*int(R(mu[m]*r)^2*r,r = 0 .. a));" "6#**,&\"# Y\"\"\"\"$]&!\"\"F&-%$intG6$-%$sinG6#*&&%'lambdaG6#%\"nGF&%\"zGF&/F4; \"\"!%\"bGF&-F*6$*&-%\"RG6#*&&%#muG6#%\"mGF&%\"rGF&F&FDF&/FD;F7%\"aGF& *&-F*6$*$-F-6#*&&F16#F3F&F4F&\"\"#/F4;F7F8F&-F*6$*&-F=6#*&&FA6#FCF&FDF &FQFDF&/FD;F7FGF&F(" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 188 "To speed the calculations, we do all t he computations for n and then all the computations for m and multiply these. Here are the computations for the sine terms, for the terms in volving n." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "a:=5; b:=1.5; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"bG$\"#:!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 122 "for n from 1 to 40 do\nT[n]:=int(sin(lambda[n]*z),z = 0 .. 3/ 2)/\n int(sin(lambda[n]*z)^2,z = 0 .. 3/2):\nend do:\nn:='n':" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 73 "Here are the computation for the Bessel terms, for the terms in volving m." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 129 "for m from 1 \+ to 30 do\nB[m]:=int(BesselJ(0,mu[m]*r)*r,r = 0 .. 5)/\n int(Bes selJ(0,mu[m]*r)^2*r,r = 0 .. 5):\nend do:\nm:='m':" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 42 "Here is mu ltiplying these together to get " }{XPPEDIT 18 0 "A[n,m];" "6#&%\"AG6$ %\"nG%\"mG" }{TEXT -1 2 " ." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 119 "for n from 1 to 40 do\n for m from 1 to 30 do\n A[n,m]:=eval f((46-550)*T[n]*B[m]):\n end do: \nend do:\nn:='n':m:='m':" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 299 "From here on, the computations take some time and space. Some how, this is not a surprise: if you ea t much of this cheese cake, you will take a lot of space! On my little home computer, the total time to execute the entire worksheet with Ma ple 8 was about 100 seconds. The space used was 15.6 Bytes." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 3 "" 0 "" {TEXT 266 49 "A grap h for an approximation of the initial value" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "u0:=(r,z)->sum(sum (A[n,m]*BesselJ(0,mu[m]*r)*sin(lambda[n]*z),n=1..40),m=1..30);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#u0Gf*6$%\"rG%\"zG6\"6$%)operatorG%& arrowGF)-%$sumG6$-F.6$*(&%\"AG6$%\"nG%\"mG\"\"\"-%(BesselJG6$\"\"!*&&% #muG6#F7F89$F8F8-%$sinG6#*&&%'lambdaG6#F6F89%F8F8/F6;F8\"#S/F7;F8\"#IF )F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "plot(u0(r,3/4)+550 ,r=-5..5, numpoints=200);" }}{PARA 13 "" 1 "" {GLPLOT2D 486 486 486 {PLOTDATA 2 "6%-%'CURVESG6$7\\jl7$$!\"&\"\"!$\"+?+++b!\"(7$$!+%*zU$*\\ !\"*$\"+e<$*3^F-7$$!+))f&o)\\F1$\"+8tR>ZF-7$$!+#)RG!)\\F1$\"+3/VLVF-7$ $!+v>rt\\F1$\"+t>.`RF-7$$!+q*Rr'\\F1$\"+%y^,e$F-7$$!+jzcg\\F1$\"+6(pm@ $F-7$$!+df*R&\\F1$\"+=#zV'GF-7$$!+]RUZ\\F1$\"+/.(\\_#F-7$$!+:u)f$\\F1$ \"+-lkp>F-7$$!+!)3bC\\F1$\"+]hJl9F-7$$!+XV68\\F1$\"+:F(y,\"F-7$$!+5yn, \\F1$\"+`]m;j!\")7$$!+Ii\")))[F1$\"+m]2QFFeo7$$!+]Y&f([F1$!)_5*)=Feo7$ $!+qI4j[F1$!+!e>u'>Feo7$$!+!\\J-&[F1$!+O_zXJFeo7$$!+t!eP%[F1$!+@H)fY$F eo7$$!+bYGP[F1$!+\")f'fh$Feo7$$!+Q7\"3$[F1$!+CQi1OFeo7$$!+@yLC[F1$!+d1 %)\\MFeo7$$!+')4R6[F1$!+9rlXFFeo7$$!+_TW)z%F1$!+9b18;Feo7$$!+]NnsZF1$ \"+k!Q^X\"Feo7$$!+\\H!pu%F1$\"+pB\"Q&[Feo7$$!+-.,BZF1$\"++7(>h(Feo7$$! 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This is needed in what follows." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 135 "u:=(t,r,z)->sum(sum(A[n,m]*BesselJ(0,mu[m]*r)*sin(lambda[n]*z)* \+ exp(-c*(mu[m]^2+lambda[n]^2)*t),n=1..40),m=1..30); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"uGf*6%%\"tG%\"rG%\"zG6\"6$%)op eratorG%&arrowGF*-%$sumG6$-F/6$**&%\"AG6$%\"nG%\"mG\"\"\"-%(BesselJG6$ \"\"!*&&%#muG6#F8F99%F9F9-%$sinG6#*&&%'lambdaG6#F7F99&F9F9-%$expG6#,$* (%\"cGF9,&*$)F?\"\"#F9F9*$)FGFTF9F9F99$F9!\"\"F9/F7;F9\"#S/F8;F9\"#IF* F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 123 "We are ready to find w hen the middle of the cake achieves 140. To do this, you need to know \+ c for this cheese cake. It is " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "c:=0.0077;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"cG$\"#x!\"% " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 3 "" 0 "" {TEXT 267 62 "A graphical method for finding when the center is 140 degrees. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 107 "To f ind the time for the center -- u(t, 0, 3/4) -- to be 140, we could dra w make a graphical approximation." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "plot([140,u(t,0,.75)+550],t=12..13);" }}{PARA 13 "" 1 "" {GLPLOT2D 486 486 486 {PLOTDATA 2 "6&-%'CURVESG6$7S7$$\"#7\"\"!$ \"$S\"F*7$$\"3kmmT:(z@?\"!#;F+7$$\"3QLe9ui2/7F0F+7$$\"3um;z_\"4i?\"F0F +7$$\"3gm;aphN37F0F+7$$\"3DLe*=)H\\57F0F+7$$\"3sm\"z/3uC@\"F0F+7$$\"3) **\\7LRDX@\"F0F+7$$\"3om\"zR'ok;7F0F+7$$\"30+D1J:w=7F0F+7$$\"3RLL3En$4 A\"F0F+7$$\"3mm;/RE&GA\"F0F+7$$\"3\"****\\K]4]A\"F0F+7$$\"3&****\\PAvr A\"F0F+7$$\"3*****\\nHi#H7F0F+7$$\"3vm\"z*ev:J7F0F+7$$\"3JLL347TL7F0F+ 7$$\"3KLLLY.KN7F0F+7$$\"3$**\\7o7TvB\"F0F+7$$\"3ILL$Q*o]R7F0F+7$$\"3%* *\\7=lj;C\"F0F+7$$\"3++vV&RY2a7 F0F+7$$\"3om;zXu9c7F0F+7$$\"3)*****\\y))Ge7F0F+7$$\"3)***\\i_QQg7F0F+7 $$\"31+D\"y%3Ti7F0F+7$$\"3'***\\P![hYE\"F0F+7$$\"3MLL$Qx$om7F0F+7$$\"3 +++v.I%)o7F0F+7$$\"3mm\"zpe*zq7F0F+7$$\"3)****\\#\\'QHF\"F0F+7$$\"3OLe 9S8&\\F\"F0F+7$$\"3-+D1#=bqF\"F0F+7$$\"3QLL3s?6z7F0F+7$$\"3#**\\7`Wl7G \"F0F+7$$\"3mmmm*RRLG\"F0F+7$$\"3qm;a<.Y&G\"F0F+7$$\"3SLe9tOc(G\"F0F+7 $$\"3)*****\\Qk\\*G\"F0F+7$$\"3OL$3dg6$\"3)>>lIcZGH\"Ffu7$FA$\"3][#eQ#fY&H\"Ffu7$ FD$\"3!pz#[_Q<)H\"Ffu7$FG$\"3ZEe'35t3I\"Ffu7$FJ$\"3$pH45g\\OI\"Ffu7$FM $\"3Q#R:i7&418Ffu7$FP$\"3w\"e^p<[)38Ffu7$FS$\"3M,D;'[7;J\"Ffu7$FV$\"3? :b$\\NwUJ\"Ffu7$FY$\"3h!=ViM&p;8Ffu7$Ffn$\"3J*4G&y;d>8Ffu7$Fin$\"3av&z $H#3?K\"Ffu7$F\\o$\"3O&=EJTU[K\"Ffu7$F_o$\"3q&etB/^tK\"Ffu7$Fbo$\"3l#o vHG.,L\"Ffu7$Feo$\"31Z\"4j$RsK8Ffu7$Fho$\"3&o!\\,m\"eaL\"Ffu7$F[p$\"3D BAD+*ozL\"Ffu7$F^p$\"3nmR8.pnS8Ffu7$Fap$\"3k$)f1m&*[V8Ffu7$Fdp$\"3u')[ UHy$fM\"Ffu7$Fgp$\"3+&pha$=e[8Ffu7$Fjp$\"3O`)3'RJJ^8Ffu7$F]q$\"3'\\h&) z'\\)RN\"Ffu7$F`q$\"3YFf]#))plN\"Ffu7$Fcq$\"3y!4'4G(R%f8Ffu7$Ffq$\"3#e Q\\qg5lsO\"Ffu7$F_r$\"3_& G,_h\"**p8Ffu7$Fbr$\"3/p>Qtnbs8Ffu7$Fer$\"3'3*eo3yBv8Ffu7$Fhr$\"3)=wF2 reyP\"Ffu7$F[s$\"3/L6NCAg!Q\"Ffu7$F^s$\"3#\\*3HFUC$Q\"Ffu7$Fas$\"3omN[ :d%fQ\"Ffu7$Fds$\"3?E(yL[C')Q\"Ffu7$Fgs$\"35c+\"on&3\"R\"Ffu7$Fjs$\"3] =%y,51RR\"Ffu7$F]t$\"3WLH9m$GkR\"Ffu7$F`t$\"3m^\"[+D<\"*R\"Ffu7$Fct$\" 3`-$Q9[!p,9Ffu7$Fft$\"3X`@8``X/9Ffu-Fit6&F[uF_uF\\uF_u-%+AXESLABELSG6$ Q\"t6\"Q!F]_l-%%VIEWG6$;F(Fft%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{PARA 0 "" 0 "" {TEXT -1 72 "It seems clear that 140 degrees is achieved between 12 .9 and 13 minutes." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 38 "plot([140,u(t,0,.75)+550],t=12.9..13);" }} {PARA 13 "" 1 "" {GLPLOT2D 486 486 486 {PLOTDATA 2 "6&-%'CURVESG6$7S7$ $\"3/++++++!H\"!#;$\"$S\"\"\"!7$$\"3mm;arz@!H\"F*F+7$$\"3I$e9ui2/H\"F* F+7$$\"3ym\"z_\"4i!H\"F*F+7$$\"3umT&phN3H\"F*F+7$$\"3O$e*=)H\\5H\"F*F+ 7$$\"3y;z/3uC\"H\"F*F+7$$\"3-]7LRDX\"H\"F*F+7$$\"3u;zR'ok;H\"F*F+7$$\" 31]i5`h(=H\"F*F+7$$\"3KL$3En$4#H\"F*F+7$$\"3kmT!RE&G#H\"F*F+7$$\"35+]K ]4]#H\"F*F+7$$\"3++]PAvr#H\"F*F+7$$\"3-+]nHi#HH\"F*F+7$$\"3w;z*ev:JH\" F*F+7$$\"3ML$347TLH\"F*F+7$$\"3WLLjM?`$H\"F*F+7$$\"3'*\\7o7Tv$H\"F*F+7 $$\"3ULLQ*o]RH\"F*F+7$$\"35]7=lj;%H\"F*F+7$$\"35]PaRY2aH\"F*F+7$$\"3km\"zXu9cH\"F*F+7$$\"3%****\\y)) GeH\"F*F+7$$\"3-+DE&QQgH\"F*F+7$$\"3'*\\7y%3TiH\"F*F+7$$\"3)**\\P![hY' H\"F*F+7$$\"3KLLQx$omH\"F*F+7$$\"33+]P+V)oH\"F*F+7$$\"3q;zpe*zqH\"F*F+ 7$$\"31+]#\\'QH(H\"F*F+7$$\"3S$e9S8&\\(H\"F*F+7$$\"3-]i?=bq(H\"F*F+7$$ \"3QL$3s?6zH\"F*F+7$$\"3-]7`Wl7)H\"F*F+7$$\"3omm'*RRL)H\"F*F+7$$\"3umT vJga)H\"F*F+7$$\"3S$e9tOc()H\"F*F+7$$\"3)****\\Qk\\*)H\"F*F+7$$\"3QL3d g6<*H\"F*F+7$$\"3emmw(Gp$*H\"F*F+7$$\"3'*\\7oK0e*H\"F*F+7$$\"3/](=5s#y *H\"F*F+7$$\"#8F-F+-%'COLOURG6&%$RGBG$\"#5!\"\"$F-F-F_u-F$6$7S7$F($\"3 wb:)R'os\"R\"!#:7$F/$\"3!3Q#e,W+#R\"Ffu7$F2$\"3QggK#)eC#R\"Ffu7$F5$\"3 Y`%o!\\u^#R\"Ffu7$F8$\"3!3ci?\"3z#R\"Ffu7$F;$\"31vg%>(G1$R\"Ffu7$F>$\" 3%4)=$G5:LR\"Ffu7$FA$\"3U%H&zpid$R\"Ffu7$FD$\"3W%)yFlj%QR\"Ffu7$FG$\"3 #y'=u!f:TR\"Ffu7$FJ$\"3)HTN#=DR%R\"Ffu7$FM$\"3UaUwKkj%R\"Ffu7$FP$\"37) ))z*=5\"\\R\"Ffu7$FS$\"3w'RU'Gn=&R\"Ffu7$FV$\"3s0!f@U_aR\"Ffu7$FY$\"3! )Q01%p$p&R\"Ffu7$Ffn$\"3!RmKce!)fR\"Ffu7$Fin$\"3'47rphBiR\"Ffu7$F\\o$ \"3/rl>;j]'R\"Ffu7$F_o$\"3O)*yJ\\lv'R\"Ffu7$Fbo$\"3C+(=34JqR\"Ffu7$Feo $\"3g\"oba^#H(R\"Ffu7$Fho$\"3GcFo!GlvR\"Ffu7$F[p$\"3)**>U2w:yR\"Ffu7$F ^p$\"3Krt[Lf3)R\"Ffu7$Fap$\"3yM=\"Gcm$)R\"Ffu7$Fdp$\"3k/([z%3h)R\"Ffu7 $Fgp$\"3q*\\0*yY())R\"Ffu7$Fjp$\"39#Rt!Qs9*R\"Ffu7$F]q$\"3q*ev+)QT*R\" Ffu7$F`q$\"33G!yh'=n*R\"Ffu7$Fcq$\"3C&yL9Je**R\"Ffu7$Ffq$\"3KsZM\"p:-S \"Ffu7$Fiq$\"3[w1E&\\!\\+9Ffu7$F\\r$\"3AI!4[]R2S\"Ffu7$F_r$\"32?Z/M<,, 9Ffu7$Fbr$\"3%)\\DuwyE,9Ffu7$Fer$\"3yY5s:c`,9Ffu7$Fhr$\"32pPgvtz,9Ffu7 $F[s$\"3SSRz492-9Ffu7$F^s$\"36U\"oKLNBS\"Ffu7$Fas$\"3xs.0H_g-9Ffu7$Fds $\"3APg'e)G(GS\"Ffu7$Fgs$\"3Cg-3J)=JS\"Ffu7$Fjs$\"3T4!*z22S.9Ffu7$F]t$ \"3nl4%y\"Gl.9Ffu7$F`t$\"3\\n'4dh@RS\"Ffu7$Fct$\"3\\ZWn&))yTS\"Ffu7$Ff t$\"3X`@8``X/9Ffu-Fit6&F[uF_uF\\uF_u-%+AXESLABELSG6$Q\"t6\"Q!F]_l-%%VI EWG6$;$\"$H\"F^uFft%(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 70 "Now, I surmise that the desired t solution is between 12.96 and 12.98." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "plot([140,u(t,0,.75)+550],t=12.96..12.98);" }}{PARA 13 "" 1 "" {GLPLOT2D 486 486 486 {PLOTDATA 2 "6&-%'CURVESG6$7S7$$\"33+ +++++'H\"!#;$\"$S\"\"\"!7$$\"3WL$3VfVgH\"F*F+7$$\"3y;H[D:3'H\"F*F+7$$ \"3WLe0$=ChH\"F*F+7$$\"3OL3RBr;'H\"F*F+7$$\"3w;zjf)4iH\"F*F+7$$\"3S$e4 ;[\\iH\"F*F+7$$\"3;]i'y]!H'H\"F*F+7$$\"3]$ezs$HL'H\"F*F+7$$\"3-]7iI_P' H\"F*F+7$$\"3om;_M(=kH\"F*F+7$$\"3SL3y_qX'H\"F*F+7$$\"3++]1!>+lH\"F*F+ 7$$\"3/+]Z/Na'H\"F*F+7$$\"3;+]$fC&e'H\"F*F+7$$\"3Y$ez6:BmH\"F*F+7$$\"3 om;=C#omH\"F*F+7$$\"3wmm#pS1nH\"F*F+7$$\"33]i`A3v'H\"F*F+7$$\"3smm(y8! 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The value is about 13 minutes." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 3 "" 0 "" {TEXT 268 36 "A numerical procedure for finding t." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 102 "To find the time for the center to be 140, we co uld ask Maple to solve the following equation for t. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "Digits:=4;\nfsolve(u(t,0,3/4)+550=140,t,1 2.9..13);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'DigitsG\"\"%" }}{PARA 7 "" 1 "" {TEXT -1 33 "Warning, computation interrupted\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "Digits:=10;" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "EMAIL: herod@math.gatech. edu or jherod@tds.net" }}{PARA 0 "" 0 "" {TEXT -1 38 "URL: http:// www.math.gatech.edu/~herod" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 111 "(The receipe for the cheese cake is not included in the copyright. I have no idea where my grandmother got it.)" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 259 "" 0 "" {TEXT -1 36 "Copyrig ht \251 2003 by James V. Herod" }}{PARA 259 "" 0 "" {TEXT -1 19 "All rights reserved" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{MARK "0 0 0" 0 } {VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 1 1 2 33 1 1 }