(*********************************************************************** Mathematica-Compatible Notebook This notebook can be used on any computer system with Mathematica 3.0, MathReader 3.0, or any compatible application. The data for the notebook starts with the line of stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). 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For each of the following three materials, ", StyleBox["a)", FontSlant->"Italic"], " silver (", StyleBox["a==1.71", FontFamily->"Courier", FontWeight->"Bold"], "), ", StyleBox["b)", FontSlant->"Italic"], " aluminum (", StyleBox["a==0.86", FontFamily->"Courier", FontWeight->"Bold"], "), and ", StyleBox["c)", FontSlant->"Italic"], " iron (", StyleBox["a==0.12", FontFamily->"Courier", FontWeight->"Bold"], "), use the first five non-zero terms of the series to do the following:\n\ 1. Plot ", StyleBox["u[x,t]", FontFamily->"Courier", FontWeight->"Bold"], " as a function of ", StyleBox["x", FontFamily->"Courier", FontWeight->"Bold"], " and ", StyleBox["t", FontFamily->"Courier", FontWeight->"Bold"], "\n2. Plot ", StyleBox["u[x,c]", FontFamily->"Courier", FontWeight->"Bold"], " as a function of ", StyleBox["x", FontFamily->"Courier", FontWeight->"Bold"], " for various choices of ", StyleBox["t==c", FontFamily->"Courier", FontWeight->"Bold"], ". Compare these plots to similar plots using only the first term of the \ series. When do the plots start to coincide?\n3. Plot the temperature \ function of the center of the bar, ", StyleBox["u[10,t]", FontFamily->"Courier", FontWeight->"Bold"], ", as a function of ", StyleBox["t", FontFamily->"Courier", FontWeight->"Bold"], ". Determine how long it takes the center of the bar to cool to ", StyleBox["25", FontFamily->"Courier", FontWeight->"Bold"], ". Compare this answer to that obtained using only one term of the series." }], "Text", Evaluatable->False, AspectRatioFixed->True] }, Open ]], Cell[CellGroupData[{ Cell[TextData["Solution"], "Subsubsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "Let's make the constant ", StyleBox["a", FontFamily->"Courier", FontWeight->"Bold"], " and the number of terms used in the series approximation parameters to \ the function ", StyleBox["u", FontFamily->"Courier", FontWeight->"Bold"], ":" }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell["\<\ u[x_,t_,a_,k_] = Sum[b[n] Sin[n Pi x/20] E^(-(n Pi/20)^2 a t),{n,1,k}]\ \>", "Input", AspectRatioFixed->True], Cell[OutputFormData[ "\<\ Sum[(b[n]*Sin[(n*Pi*x)/20])/E^((a*n^2*Pi^2*t)/400), {n, 1, k}]\ \>", "\<\ n Pi x b[n] Sin[------] 20 Sum[-----------------, {n, 1, k}] 2 2 (a n Pi t)/400 E\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True] }, Open ]], Cell[TextData["The coefficients are:"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData[ "b[n_] = 2/20 Integrate[100 Sin[n Pi x/20],{x,0,20}]//Together"], "Input", AspectRatioFixed->True], Cell[OutputFormData["\<\ (200*(1 - Cos[n*Pi]))/(n*Pi)\ \>", "\<\ 200 (1 - Cos[n Pi]) ------------------- n Pi\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True] }, Open ]], Cell[TextData[{ "Since the even coefficients are ", StyleBox["0", FontFamily->"Courier", FontWeight->"Bold"], ", the first five non-zero terms are given by ", StyleBox["k=9", FontFamily->"Courier", FontWeight->"Bold"], "." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData["u[x,t,a,9]"], "Input", AspectRatioFixed->True], Cell[OutputFormData["\<\ (400*Sin[(Pi*x)/20])/(E^((a*Pi^2*t)/400)*Pi) + (400*Sin[(3*Pi*x)/20])/(3*E^((9*a*Pi^2*t)/400)*Pi) + (80*Sin[(Pi*x)/4])/(E^((a*Pi^2*t)/16)*Pi) + (400*Sin[(7*Pi*x)/20])/(7*E^((49*a*Pi^2*t)/400)*Pi) + (400*Sin[(9*Pi*x)/20])/(9*E^((81*a*Pi^2*t)/400)*Pi)\ \>", "\<\ Pi x 3 Pi x Pi x 400 Sin[----] 400 Sin[------] 80 Sin[----] 20 20 4 ----------------- + --------------------- + ---------------- + 2 2 2 (a Pi t)/400 (9 a Pi t)/400 (a Pi t)/16 E Pi 3 E Pi E Pi 7 Pi x 9 Pi x 400 Sin[------] 400 Sin[------] 20 20 ---------------------- + ---------------------- 2 2 (49 a Pi t)/400 (81 a Pi t)/400 7 E Pi 9 E Pi\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True] }, Open ]], Cell[TextData[{ "Let's set the constants used for ", StyleBox["a", FontFamily->"Courier", FontWeight->"Bold"], ":" }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["silv = 1.71;\nalum = 0.89;\niron = 0.12;"], "Input", AspectRatioFixed->True], Cell[TextData[{ StyleBox["a)", FontSlant->"Italic"], " ", StyleBox["Silver", FontWeight->"Bold"], "" }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData["Plot3D[u[x,t,silv,9],{x,0,20},{t,0,30}]"], "Input", 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.64135 .65117 .62641 Metetra .679 .823 .953 r .65117 .62641 .67013 .64135 .70997 .62718 .6915 .6118 Metetra .741 .836 .923 r .6915 .6118 .70997 .62718 .74992 .60683 .73196 .59071 Metetra .785 .839 .893 r .73196 .59071 .74992 .60683 .78979 .58073 .77235 .56354 Metetra .814 .836 .867 r .77235 .56354 .78979 .58073 .82943 .54962 .81251 .53106 Metetra .831 .829 .845 r .81251 .53106 .82943 .54962 .86877 .51452 .85233 .49436 Metetra .84 .82 .829 r .85233 .49436 .86877 .51452 .9078 .47669 .8918 .45475 Metetra .842 .81 .819 r .8918 .45475 .9078 .47669 .94659 .43752 .931 .41373 Metetra .003 .414 .856 r .36752 .55912 .38895 .57908 .42336 .59521 .40199 .57653 Metetra .017 .439 .867 r .40199 .57653 .42336 .59521 .45886 .61037 .43762 .59289 Metetra .085 .5 .9 r .43762 .59289 .45886 .61037 .49548 .62311 .47443 .60666 Metetra .199 .585 .943 r .47443 .60666 .49548 .62311 .53316 .63208 .51238 .61639 Metetra .338 .673 .977 r .51238 .61639 .53316 .63208 .57178 .63614 .55135 .62093 Metetra .474 .747 .989 r .55135 .62093 .57178 .63614 .61119 .63444 .59116 .61939 Metetra .591 .798 .978 r .59116 .61939 .61119 .63444 .65117 .62641 .6316 .61121 Metetra .68 .826 .954 r .6316 .61121 .65117 .62641 .6915 .6118 .67245 .59616 Metetra .744 .839 .923 r .67245 .59616 .6915 .6118 .73196 .59071 .71345 .5743 Metetra .789 .842 .892 r .71345 .5743 .73196 .59071 .77235 .56354 .75439 .54605 Metetra .819 .838 .864 r .75439 .54605 .77235 .56354 .81251 .53106 .79508 .51217 Metetra .837 .832 .841 r .79508 .51217 .81251 .53106 .85233 .49436 .83541 .47376 Metetra .846 .822 .825 r .83541 .47376 .85233 .49436 .8918 .45475 .87535 .43226 Metetra .848 .811 .814 r .87535 .43226 .8918 .45475 .931 .41373 .91497 .38926 Metetra 0 .376 .83 r .34552 .53864 .36752 .55912 .40199 .57653 .38003 .55745 Metetra 0 .404 .842 r .38003 .55745 .40199 .57653 .43762 .59289 .41576 .57513 Metetra .039 .472 .881 r .41576 .57513 .43762 .59289 .47443 .60666 .45275 .58998 Metetra .165 .567 .933 r .45275 .58998 .47443 .60666 .51238 .61639 .49096 .6005 Metetra .317 .665 .974 r .49096 .6005 .51238 .61639 .55135 .62093 .53027 .60547 Metetra .465 .746 .99 r .53027 .60547 .55135 .62093 .59116 .61939 .57049 .60405 Metetra .588 .799 .98 r .57049 .60405 .59116 .61939 .6316 .61121 .61141 .59571 Metetra .68 .828 .954 r .61141 .59571 .6316 .61121 .67245 .59616 .65278 .5802 Metetra .747 .841 .922 r .65278 .5802 .67245 .59616 .71345 .5743 .69435 .5576 Metetra .793 .844 .89 r .69435 .5576 .71345 .5743 .75439 .54605 .73586 .52826 Metetra .824 .841 .861 r .73586 .52826 .75439 .54605 .79508 .51217 .77713 .49294 Metetra .843 .834 .838 r .77713 .49294 .79508 .51217 .83541 .47376 .81799 .45275 Metetra .853 .824 .82 r .81799 .45275 .83541 .47376 .87535 .43226 .85842 .40922 Metetra .854 .813 .809 r .85842 .40922 .87535 .43226 .91497 .38926 .89847 .36408 Metetra 0 .332 .797 r .32295 .51762 .34552 .53864 .38003 .55745 .35745 .53799 Metetra 0 .363 .811 r .35745 .53799 .38003 .55745 .41576 .57513 .39325 .5571 Metetra 0 .44 .858 r .39325 .5571 .41576 .57513 .45275 .58998 .4304 .57311 Metetra .128 .548 .921 r .4304 .57311 .45275 .58998 .49096 .6005 .46886 .5844 Metetra .296 .658 .971 r .46886 .5844 .49096 .6005 .53027 .60547 .50851 .58976 Metetra .456 .745 .991 r .50851 .58976 .53027 .60547 .57049 .60405 .54915 .5884 Metetra .586 .8 .981 r .54915 .5884 .57049 .60405 .61141 .59571 .59056 .57985 Metetra .681 .83 .955 r .59056 .57985 .61141 .59571 .65278 .5802 .63248 .5639 Metetra .749 .844 .922 r .63248 .5639 .65278 .5802 .69435 .5576 .67463 .54059 Metetra .797 .847 .889 r .67463 .54059 .69435 .5576 .73586 .52826 .71675 .5102 Metetra .829 .844 .859 r .71675 .5102 .73586 .52826 .77713 .49294 .75861 .47341 Metetra .849 .837 .834 r .75861 .47341 .77713 .49294 .81799 .45275 .80005 .43136 Metetra .859 .827 .815 r .80005 .43136 .81799 .45275 .85842 .40922 .84099 .38563 Metetra .861 .815 .803 r .84099 .38563 .85842 .40922 .89847 .36408 .88149 .33817 Metetra 0 .282 .757 r .29977 .49604 .32295 .51762 .35745 .53799 .33422 .51817 Metetra 0 .316 .772 r .33422 .51817 .35745 .53799 .39325 .5571 .37005 .53889 Metetra 0 .404 .83 r .37005 .53889 .39325 .5571 .4304 .57311 .40734 .55611 Metetra .091 .528 .907 r .40734 .55611 .4304 .57311 .46886 .5844 .44605 .56813 Metetra .278 .651 .968 r .44605 .56813 .46886 .5844 .50851 .58976 .48604 .57377 Metetra .45 .744 .991 r .48604 .57377 .50851 .58976 .54915 .5884 .5271 .57236 Metetra .584 .801 .982 r .5271 .57236 .54915 .5884 .59056 .57985 .56901 .56355 Metetra .681 .831 .955 r .56901 .56355 .59056 .57985 .63248 .5639 .61148 .54719 Metetra .75 .845 .922 r .61148 .54719 .63248 .5639 .67463 .54059 .65425 .52324 Metetra .8 .85 .888 r .65425 .52324 .67463 .54059 .71675 .5102 .69702 .49188 Metetra .834 .848 .856 r .69702 .49188 .71675 .5102 .75861 .47341 .73952 .45365 Metetra .855 .841 .83 r .73952 .45365 .75861 .47341 .80005 .43136 .78156 .40963 Metetra .866 .83 .809 r .78156 .40963 .80005 .43136 .84099 .38563 .82304 .36153 Metetra .868 .817 .797 r .82304 .36153 .84099 .38563 .88149 .33817 .86401 .31148 Metetra 0 .223 .706 r .27596 .47387 .29977 .49604 .33422 .51817 .3103 .49806 Metetra 0 .261 .724 r .3103 .49806 .33422 .51817 .37005 .53889 .34613 .52059 Metetra 0 .364 .796 r .34613 .52059 .37005 .53889 .40734 .55611 .38353 .53908 Metetra .053 .508 .891 r .38353 .53908 .40734 .55611 .44605 .56813 .42248 .55173 Metetra .263 .647 .965 r .42248 .55173 .44605 .56813 .48604 .57377 .46281 .55747 Metetra .447 .746 .992 r .46281 .55747 .48604 .57377 .5271 .57236 .50431 .55583 Metetra .584 .802 .982 r .50431 .55583 .5271 .57236 .56901 .56355 .54672 .54668 Metetra .681 .831 .955 r .54672 .54668 .56901 .56355 .61148 .54719 .58977 .52995 Metetra .75 .846 .922 r .58977 .52995 .61148 .54719 .65425 .52324 .63317 .50553 Metetra .801 .852 .888 r .63317 .50553 .65425 .52324 .69702 .49188 .67662 .47336 Metetra .838 .851 .855 r .67662 .47336 .69702 .49188 .73952 .45365 .71982 .43374 Metetra .862 .845 .826 r .71982 .43374 .73952 .45365 .78156 .40963 .7625 .38767 Metetra .874 .833 .804 r .7625 .38767 .78156 .40963 .82304 .36153 .80455 .33694 Metetra .875 .819 .79 r .80455 .33694 .82304 .36153 .86401 .31148 .846 .28399 Metetra 0 .153 .642 r .2515 .4511 .27596 .47387 .3103 .49806 .28565 .47775 Metetra 0 .198 .664 r .28565 .47775 .3103 .49806 .34613 .52059 .32141 .50236 Metetra 0 .32 .753 r .32141 .50236 .34613 .52059 .38353 .53908 .35891 .52217 Metetra .018 .49 .875 r .35891 .52217 .38353 .53908 .42248 .55173 .3981 .53528 Metetra .255 .648 .964 r .3981 .53528 .42248 .55173 .46281 .55747 .43878 .54081 Metetra .451 .75 .993 r .43878 .54081 .46281 .55747 .50431 .55583 .48072 .53866 Metetra .587 .802 .982 r .48072 .53866 .50431 .55583 .54672 .54668 .52365 .52905 Metetra .68 .829 .955 r .52365 .52905 .54672 .54668 .58977 .52995 .56729 .51204 Metetra .748 .845 .923 r .56729 .51204 .58977 .52995 .63317 .50553 .61136 .48739 Metetra .801 .854 .889 r .61136 .48739 .63317 .50553 .67662 .47336 .65553 .45468 Metetra .842 .856 .854 r .65553 .45468 .67662 .47336 .71982 .43374 .69946 .41384 Metetra .869 .85 .822 r .69946 .41384 .71982 .43374 .7625 .38767 .74284 .36564 Metetra .883 .838 .797 r .74284 .36564 .7625 .38767 .80455 .33694 .78549 .31196 Metetra .884 .822 .781 r .78549 .31196 .80455 .33694 .846 .28399 .82743 .25565 Metetra 0 .071 .561 r .22636 .42769 .2515 .4511 .28565 .47775 .26021 .45738 Metetra 0 .124 .587 r .26021 .45738 .28565 .47775 .32141 .50236 .29584 .48446 Metetra 0 .271 .702 r .29584 .48446 .32141 .50236 .35891 .52217 .33341 .50563 Metetra 0 .477 .858 r .33341 .50563 .35891 .52217 .3981 .53528 .37284 .51889 Metetra .259 .657 .966 r .37284 .51889 .3981 .53528 .43878 .54081 .41391 .52372 Metetra .464 .757 .993 r .41391 .52372 .43878 .54081 .48072 .53866 .45631 .52063 Metetra .593 .801 .979 r .45631 .52063 .48072 .53866 .52365 .52905 .49976 .51038 Metetra .677 .824 .954 r .49976 .51038 .52365 .52905 .56729 .51204 .544 .49324 Metetra .742 .841 .925 r .544 .49324 .56729 .51204 .61136 .48739 .58876 .46874 Metetra .798 .855 .892 r .58876 .46874 .61136 .48739 .65553 .45468 .6337 .43596 Metetra .844 .861 .855 r .6337 .43596 .65553 .45468 .69946 .41384 .67843 .39419 Metetra .876 .857 .819 r .67843 .39419 .69946 .41384 .74284 .36564 .72257 .34379 Metetra .893 .844 .79 r .72257 .34379 .74284 .36564 .78549 .31196 .76585 .28673 Metetra .894 .826 .772 r .76585 .28673 .78549 .31196 .82743 .25565 .80829 .22644 Metetra .539 .025 0 r .20051 .40362 .22636 .42769 .26021 .45738 .2339 .43724 Metetra .501 0 0 r .2339 .43724 .26021 .45738 .29584 .48446 .26931 .46733 Metetra 0 .217 .638 r .26931 .46733 .29584 .48446 .33341 .50563 .30693 .48982 Metetra 0 .473 .844 r .30693 .48982 .33341 .50563 .37284 .51889 .34665 .50267 Metetra .282 .677 .972 r .34665 .50267 .37284 .51889 .41391 .52372 .38815 .50605 Metetra .49 .768 .993 r .38815 .50605 .41391 .52372 .45631 .52063 .43105 .50141 Metetra .603 .799 .975 r .43105 .50141 .45631 .52063 .49976 .51038 .47504 .49027 Metetra .673 .816 .952 r .47504 .49027 .49976 .51038 .544 .49324 .51988 .47319 Metetra .732 .835 .928 r .51988 .47319 .544 .49324 .58876 .46874 .56533 .44945 Metetra .79 .855 .898 r .56533 .44945 .58876 .46874 .6337 .43596 .61108 .41733 Metetra .844 .867 .859 r .61108 .41733 .6337 .43596 .67843 .39419 .65667 .37515 Metetra .884 .866 .817 r .65667 .37515 .67843 .39419 .72257 .34379 .70164 .32256 Metetra .904 .852 .781 r .70164 .32256 .72257 .34379 .76585 .28673 .74561 .26153 Metetra .907 .83 .759 r .74561 .26153 .76585 .28673 .80829 .22644 .78854 .19629 Metetra .646 .135 0 r .17391 .37886 .20051 .40362 .2339 .43724 .20661 .41782 Metetra .601 .058 0 r .20661 .41782 .2339 .43724 .26931 .46733 .24166 .45171 Metetra .408 0 0 r .24166 .45171 .26931 .46733 .30693 .48982 .27934 .47528 Metetra 0 .488 .837 r .27934 .47528 .30693 .48982 .34665 .50267 .31944 .48675 Metetra .333 .714 .983 r .31944 .48675 .34665 .50267 .38815 .50605 .36147 .48755 Metetra .53 .783 .99 r .36147 .48755 .38815 .50605 .43105 .50141 .40492 .48054 Metetra .616 .795 .967 r .40492 .48054 .43105 .50141 .47504 .49027 .44946 .4682 Metetra .666 .803 .948 r .44946 .4682 .47504 .49027 .51988 .47319 .49488 .45145 Metetra .714 .822 .932 r .49488 .45145 .51988 .47319 .56533 .44945 .54102 .42923 Metetra .775 .851 .908 r .54102 .42923 .56533 .44945 .61108 .41733 .5876 .39889 Metetra .84 .875 .868 r .5876 .39889 .61108 .41733 .65667 .37515 .63414 .35727 Metetra .892 .879 .816 r .63414 .35727 .65667 .37515 .70164 .32256 .68004 .3027 Metetra .919 .863 .77 r .68004 .3027 .70164 .32256 .74561 .26153 .72475 .23687 Metetra .922 .836 .743 r .72475 .23687 .74561 .26153 .78854 .19629 .76816 .16518 Metetra .751 .257 0 r .14655 .35337 .17391 .37886 .20661 .41782 .17814 .40016 Metetra .696 .162 0 r .17814 .40016 .20661 .41782 .24166 .45171 .21266 .43892 Metetra .47 0 0 r .21266 .43892 .24166 .45171 .27934 .47528 .25048 .46277 Metetra .011 .542 .851 r .25048 .46277 .27934 .47528 .31944 .48675 .29114 .47105 Metetra .422 .766 .996 r .29114 .47105 .31944 .48675 .36147 .48755 .33384 .46767 Metetra .58 .795 .981 r .33384 .46767 .36147 .48755 .40492 .48054 .37792 .45744 Metetra .632 .787 .956 r .37792 .45744 .40492 .48054 .44946 .4682 .42302 .44365 Metetra .657 .786 .943 r .42302 .44365 .44946 .4682 .49488 .45145 .469 .42741 Metetra .689 .803 .936 r .469 .42741 .49488 .45145 .54102 .42923 .51578 .40754 Metetra .746 .841 .923 r .51578 .40754 .54102 .42923 .5876 .39889 .56319 .38055 Metetra .826 .882 .885 r .56319 .38055 .5876 .39889 .63414 .35727 .61077 .34133 Metetra .897 .897 .819 r .61077 .34133 .63414 .35727 .68004 .3027 .65773 .28559 Metetra .936 .88 .755 r .65773 .28559 .68004 .3027 .72475 .23687 .70328 .21379 Metetra .941 .844 .72 r .70328 .21379 .72475 .23687 .76816 .16518 .74711 .13306 Metetra .844 .385 0 r .11838 .32714 .14655 .35337 .17814 .40016 .14803 .38686 Metetra .77 .26 0 r .14803 .38686 .17814 .40016 .21266 .43892 .18182 .43175 Metetra .487 0 0 r .18182 .43175 .21266 .43892 .25048 .46277 .22012 .45301 Metetra .165 .67 .907 r .22012 .45301 .25048 .46277 .29114 .47105 .26175 .45458 Metetra .541 .815 .997 r .26175 .45458 .29114 .47105 .33384 .46767 .30532 .44528 Metetra .626 .794 .963 r .30532 .44528 .33384 .46767 .37792 .45744 .35007 .43162 Metetra .644 .774 .943 r .35007 .43162 .37792 .45744 .42302 .44365 .39572 .4165 Metetra .65 .769 .936 r .39572 .4165 .42302 .44365 .469 .42741 .44223 .4006 Metetra .661 .78 .937 r .44223 .4006 .469 .42741 .51578 .40754 .48959 .38321 Metetra .699 .816 .937 r .48959 .38321 .51578 .40754 .56319 .38055 .53777 .3613 Metetra .786 .879 .915 r .53777 .3613 .56319 .38055 .61077 .34133 .58644 .32805 Metetra .893 .923 .834 r .58644 .32805 .61077 .34133 .65773 .28559 .63469 .27408 Metetra .955 .905 .732 r .63469 .27408 .65773 .28559 .70328 .21379 .68123 .19499 Metetra .966 .857 .681 r .68123 .19499 .70328 .21379 .74711 .13306 .72536 .09986 Metetra .913 .511 .193 r .08936 .30012 .11838 .32714 .14803 .38686 .11483 .38715 Metetra .782 .312 .12 r .11483 .38715 .14803 .38686 .18182 .43175 .14815 .43619 Metetra .306 0 0 r .14815 .43619 .18182 .43175 .22012 .45301 .18832 .44395 Metetra .483 .852 .988 r .18832 .44395 .22012 .45301 .26175 .45458 .23153 .43374 Metetra .634 .812 .968 r .23153 .43374 .26175 .45458 .30532 .44528 .27601 .41895 Metetra .647 .774 .941 r .27601 .41895 .30532 .44528 .35007 .43162 .32135 .4033 Metetra .646 .762 .934 r .32135 .4033 .35007 .43162 .39572 .4165 .36752 .38757 Metetra .649 .763 .932 r .36752 .38757 .39572 .4165 .44223 .4006 .41456 .37122 Metetra .647 .762 .933 r .41456 .37122 .44223 .4006 .48959 .38321 .46248 .35478 Metetra .652 .776 .939 r .46248 .35478 .48959 .38321 .53777 .3613 .5113 .33743 Metetra .692 .83 .949 r .5113 .33743 .53777 .3613 .58644 .32805 .56098 .3153 Metetra .827 .942 .9 r .56098 .3153 .58644 .32805 .63469 .27408 .6109 .27438 Metetra .958 .947 .696 r .6109 .27438 .63469 .27408 .68123 .19499 .65892 .19004 Metetra .993 .871 .595 r .65892 .19004 .68123 .19499 .72536 .09986 .70288 .06555 Metetra .944 .623 .398 r .05946 .27229 .08936 .30012 .11483 .38715 .06794 .46977 Metetra .283 0 .151 r .06794 .46977 .11483 .38715 .14815 .43619 .1117 .44809 Metetra .834 .998 .834 r .1117 .44809 .14815 .43619 .18832 .44395 .15896 .40182 Metetra .152 .41 .854 r .15896 .40182 .18832 .44395 .23153 .43374 .19928 .42043 Metetra .793 .897 .917 r .19928 .42043 .23153 .43374 .27601 .41895 .24606 .38733 Metetra .699 .756 .896 r .24606 .38733 .27601 .41895 .32135 .4033 .29234 .36505 Metetra .38 .577 .917 r .29234 .36505 .32135 .4033 .36752 .38757 .33753 .37084 Metetra .82 .889 .893 r .33753 .37084 .36752 .38757 .41456 .37122 .38635 .33179 Metetra .591 .677 .9 r .38635 .33179 .41456 .37122 .46248 .35478 .4345 .32089 Metetra .442 .649 .947 r .4345 .32089 .46248 .35478 .5113 .33743 .48364 .32112 Metetra .878 .907 .845 r .48364 .32112 .5113 .33743 .56098 .3153 .53415 .26827 Metetra .324 .494 .872 r .53415 .26827 .56098 .3153 .6109 .27438 .58584 .28222 Metetra .58584 .28222 .6109 .27438 .65892 .19004 .63872 .27086 Metetra .844 .754 .239 r .63872 .27086 .65892 .19004 .70288 .06555 .67962 .03005 Metetra P p .002 w .67932 .02494 m .94594 .43277 L s .94594 .43277 m .97506 .64585 L s .97506 .64585 m .69286 .25814 L s .69286 .25814 m .67932 .02494 L s .06024 .26735 m .02494 .49015 L s .02494 .49015 m .69286 .25814 L s .69286 .25814 m .67932 .02494 L s .67932 .02494 m .06024 .26735 L s P p p .002 w .06024 .26735 m .67932 .02494 L s P p .002 w .06024 .26735 m .0648 .2716 L s P [(0)] .05113 .25884 1 .93395 Mshowa p .002 w .20126 .21213 m .2056 .21661 L s P [(5)] .1926 .20316 .96648 1 Mshowa p .002 w .35089 .15354 m .35496 .15826 L s P [(10)] .34275 .1441 .86223 1 Mshowa p .002 w .50994 .09126 m .5137 .09623 L s P [(15)] .50241 .08133 .75799 1 Mshowa p .002 w .67932 .02494 m .68274 .03015 L s P [(20)] .6725 .0145 .65374 1 Mshowa p .001 w .08779 .25656 m .0905 .25914 L s P p .001 w .11567 .24565 m .11835 .24825 L s P p .001 w .14386 .2346 m .14652 .23724 L s P p .001 w .17239 .22343 m .17502 .22609 L s P p .001 w .23048 .20069 m .23305 .20341 L s P p .001 w .26004 .18911 m 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s P p .001 w .1172 .8501 m .11773 .846 L s P p .001 w .14528 .86986 m .14576 .86575 L s P p .001 w .19894 .90762 m .19935 .90351 L s P p .001 w .2246 .92568 m .22498 .92156 L s P p .001 w .24953 .94322 m .24987 .9391 L s P p .001 w .27375 .96027 m .27406 .95615 L s P p .001 w .32021 .99296 m .32046 .98884 L s P p .001 w .34249 1.00864 m .34271 1.00452 L s P p .001 w .36418 1.0239 m .36438 1.01978 L s P p .001 w .3853 1.03877 m .38547 1.03464 L s P P p p .002 w .11113 .25957 m .02755 .78701 L s P p .002 w .10947 .27006 m .11582 .2674 L s P [(0)] .09676 .27537 1 -0.41829 Mshowa p .002 w .09372 .36943 m .10012 .36689 L s P [(25)] .08092 .37452 1 -0.39759 Mshowa p .002 w .07673 .47663 m .08318 .47421 L s P [(50)] .06384 .48147 1 -0.37509 Mshowa p .002 w .05835 .59263 m .06485 .59035 L s P [(75)] .04535 .59718 1 -0.35056 Mshowa p .002 w .03839 .71855 m .04495 .71643 L s P [(100)] .02529 .72279 1 -0.32369 Mshowa p .001 w .10641 .28935 m .11023 .28777 L s P p .001 w .10331 .30892 m .10713 .30736 L s P p .001 w .10016 .32879 m .10399 .32724 L s P p .001 w .09696 .34896 m .1008 .34742 L s P p .001 w .09043 .39021 m .09427 .3887 L s P p .001 w .08708 .41132 m .09093 .40982 L s P p .001 w .08368 .43275 m .08754 .43127 L s P p .001 w .08024 .45452 m .0841 .45305 L s P p .001 w .07317 .49909 m .07705 .49766 L s P p .001 w .06956 .52191 m .07344 .5205 L s P p .001 w .06588 .54511 m .06977 .5437 L s P p .001 w .06215 .56867 m .06604 .56729 L s P p .001 w .05449 .61698 m .0584 .61563 L s P p .001 w .05057 .64173 m .05448 .6404 L s P p .001 w .04658 .66691 m .0505 .6656 L s P p .001 w .04252 .69251 m .04645 .69122 L s P p .001 w .0342 .74503 m .03813 .74378 L s P p .001 w .02993 .77198 m .03387 .77075 L s P P 0 0 m 1 0 L 1 1.08079 L 0 1.08079 L closepath clip newpath p .002 w .11113 .25957 m .02755 .78701 L s .02755 .78701 m .40586 1.05324 L s .40586 1.05324 m .42251 .57709 L s .42251 .57709 m .11113 .25957 L s .66295 .02755 m .90359 .41965 L s .90359 .41965 m .97245 .92345 L s .97245 .92345 m .69507 .58075 L s .69507 .58075 m .66295 .02755 L s .11113 .25957 m .02755 .78701 L s .02755 .78701 m .69507 .58075 L s .69507 .58075 m .66295 .02755 L s .66295 .02755 m .11113 .25957 L s .42251 .57709 m .90359 .41965 L s .90359 .41965 m .97245 .92345 L s .97245 .92345 m .40586 1.05324 L s .40586 1.05324 m .42251 .57709 L s P p .074 .447 .88 r .0015 w .40185 .60417 .4209 .62301 .45223 .63772 .43326 .61931 Metetra .081 .458 .886 r .43326 .61931 .45223 .63772 .48452 .65237 .46567 .63427 Metetra .11 .482 .898 r .46567 .63427 .48452 .65237 .5178 .66655 .49911 .64851 Metetra .153 .511 .913 r .49911 .64851 .5178 .66655 .5521 .68012 .53358 .6618 Metetra .195 .534 .924 r .53358 .6618 .5521 .68012 .58742 .69324 .56909 .67429 Metetra .225 .546 .929 r .56909 .67429 .58742 .69324 .62379 .7064 .60566 .68655 Metetra .234 .541 .926 r .60566 .68655 .62379 .7064 .66129 .72044 .64336 .69953 Metetra .218 .52 .916 r .64336 .69953 .66129 .72044 .70005 .73633 .68229 .7144 Metetra .176 .484 .899 r .68229 .7144 .70005 .73633 .74023 .7551 .72266 .73236 Metetra .114 .436 .875 r .72266 .73236 .74023 .7551 .78207 .77762 .76471 .75447 Metetra .036 .38 .844 r .76471 .75447 .78207 .77762 .8258 .80444 .80872 .78141 Metetra 0 .323 .809 r .80872 .78141 .8258 .80444 .87168 .83568 .85494 .81332 Metetra 0 .273 .773 r .85494 .81332 .87168 .83568 .91988 .87094 .9036 .84979 Metetra 0 .237 .742 r .9036 .84979 .91988 .87094 .97052 .90934 .95483 .88976 Metetra .047 .43 .87 r .38233 .58486 .40185 .60417 .43326 .61931 .4138 .60057 Metetra .057 .445 .877 r .4138 .60057 .43326 .61931 .46567 .63427 .44632 .61595 Metetra .095 .476 .894 r .44632 .61595 .46567 .63427 .49911 .64851 .4799 .63031 Metetra .15 .513 .914 r .4799 .63031 .49911 .64851 .53358 .6618 .51456 .64331 Metetra .205 .544 .929 r .51456 .64331 .53358 .6618 .56909 .67429 .55027 .65509 Metetra .245 .561 .935 r .55027 .65509 .56909 .67429 .60566 .68655 .58705 .66629 Metetra .26 .558 .932 r .58705 .66629 .60566 .68655 .64336 .69953 .62493 .67801 Metetra .245 .536 .922 r .62493 .67801 .64336 .69953 .68229 .7144 .66405 .69164 Metetra .201 .497 .904 r .66405 .69164 .68229 .7144 .72266 .73236 .70461 .70862 Metetra .131 .443 .878 r .70461 .70862 .72266 .73236 .76471 .75447 .74686 .7302 Metetra .044 .38 .844 r .74686 .7302 .76471 .75447 .80872 .78141 .79111 .75722 Metetra 0 .316 .804 r .79111 .75722 .80872 .78141 .85494 .81332 .83768 .7899 Metetra 0 .259 .763 r .83768 .7899 .85494 .81332 .9036 .84979 .8868 .82774 Metetra 0 .217 .728 r .8868 .82774 .9036 .84979 .95483 .88976 .93863 .86955 Metetra .014 .408 .856 r .36231 .56506 .38233 .58486 .4138 .60057 .39383 .58152 Metetra .026 .426 .865 r .39383 .58152 .4138 .60057 .44632 .61595 .42643 .59746 Metetra .074 .467 .888 r .42643 .59746 .44632 .61595 .4799 .63031 .46016 .61202 Metetra .145 .515 .914 r .46016 .61202 .4799 .63031 .51456 .64331 .495 .62472 Metetra .216 .556 .934 r .495 .62472 .51456 .64331 .55027 .65509 .53093 .63568 Metetra .268 .578 .942 r .53093 .63568 .55027 .65509 .58705 .66629 .56792 .64564 Metetra .29 .578 .939 r .56792 .64564 .58705 .66629 .62493 .67801 .60601 .65587 Metetra .276 .554 .928 r .60601 .65587 .62493 .67801 .66405 .69164 .64531 .66801 Metetra .229 .511 .909 r .64531 .66801 .66405 .69164 .70461 .70862 .68604 .68377 Metetra .152 .452 .882 r .68604 .68377 .70461 .70862 .74686 .7302 .72847 .70467 Metetra .054 .381 .845 r .72847 .70467 .74686 .7302 .79111 .75722 .77297 .73174 Metetra 0 .307 .799 r .77297 .73174 .79111 .75722 .83768 .7899 .81986 .76527 Metetra 0 .242 .752 r .81986 .76527 .83768 .7899 .8868 .82774 .86945 .80472 Metetra 0 .194 .711 r .86945 .80472 .8868 .82774 .93863 .86955 .9219 .84869 Metetra 0 .379 .837 r .34178 .54476 .36231 .56506 .39383 .58152 .37331 .56217 Metetra 0 .403 .849 r .37331 .56217 .39383 .58152 .42643 .59746 .40599 .57886 Metetra .048 .454 .879 r .40599 .57886 .42643 .59746 .46016 .61202 .43985 .5937 Metetra .138 .517 .914 r .43985 .5937 .46016 .61202 .495 .62472 .47488 .6061 Metetra .229 .57 .94 r .47488 .6061 .495 .62472 .53093 .63568 .51103 .61613 Metetra .295 .598 .949 r .51103 .61613 .53093 .63568 .56792 .64564 .54826 .62462 Metetra .325 .599 .946 r .54826 .62462 .56792 .64564 .60601 .65587 .58656 .63308 Metetra .313 .575 .934 r .58656 .63308 .60601 .65587 .64531 .66801 .62606 .64343 Metetra .261 .528 .914 r .62606 .64343 .64531 .66801 .68604 .68377 .66695 .65771 Metetra .176 .462 .885 r .66695 .65771 .68604 .68377 .72847 .70467 .70955 .67775 Metetra .065 .382 .845 r .70955 .67775 .72847 .70467 .77297 .73174 .75426 .7048 Metetra 0 .298 .794 r .75426 .7048 .77297 .73174 .81986 .76527 .80148 .7393 Metetra 0 .223 .739 r .80148 .7393 .81986 .76527 .86945 .80472 .85153 .78062 Metetra 0 .168 .691 r .85153 .78062 .86945 .80472 .9219 .84869 .90462 .82714 Metetra 0 .342 .81 r .32071 .52392 .34178 .54476 .37331 .56217 .35222 .54256 Metetra 0 .372 .826 r .35222 .54256 .37331 .56217 .40599 .57886 .38494 .56021 Metetra .016 .438 .867 r .38494 .56021 .40599 .57886 .43985 .5937 .41893 .57546 Metetra .129 .519 .914 r .41893 .57546 .43985 .5937 .47488 .6061 .45416 .58753 Metetra .244 .586 .946 r .45416 .58753 .47488 .6061 .51103 .61613 .49055 .59648 Metetra .327 .621 .957 r .49055 .59648 .51103 .61613 .54826 .62462 .52803 .60327 Metetra .364 .624 .952 r .52803 .60327 .54826 .62462 .58656 .63308 .56657 .60963 Metetra .354 .598 .939 r .56657 .60963 .58656 .63308 .62606 .64343 .60626 .61784 Metetra .299 .547 .919 r .60626 .61784 .62606 .64343 .66695 .65771 .64731 .63032 Metetra .204 .473 .889 r .64731 .63032 .66695 .65771 .70955 .67775 .69007 .64926 Metetra .079 .383 .845 r .69007 .64926 .70955 .67775 .75426 .7048 .73498 .67622 Metetra 0 .288 .787 r .73498 .67622 .75426 .7048 .80148 .7393 .78249 .71178 Metetra 0 .201 .724 r .78249 .71178 .80148 .7393 .85153 .78062 .833 .75531 Metetra 0 .138 .667 r .833 .75531 .85153 .78062 .90462 .82714 .88676 .80486 Metetra 0 .294 .774 r .29909 .50255 .32071 .52392 .35222 .54256 .33052 .52274 Metetra 0 .331 .793 r .33052 .52274 .35222 .54256 .38494 .56021 .36325 .5416 Metetra 0 .416 .849 r .36325 .5416 .38494 .56021 .41893 .57546 .39735 .5574 Metetra .118 .521 .913 r .39735 .5574 .41893 .57546 .45416 .58753 .43279 .56913 Metetra .261 .605 .954 r .43279 .56913 .45416 .58753 .49055 .59648 .46945 .57683 Metetra .363 .647 .964 r .46945 .57683 .49055 .59648 .52803 .60327 .50721 .58162 Metetra .408 .65 .957 r .50721 .58162 .52803 .60327 .56657 .60963 .54601 .58552 Metetra .399 .622 .942 r .54601 .58552 .56657 .60963 .60626 .61784 .58592 .59119 Metetra .341 .567 .922 r .58592 .59119 .60626 .61784 .64731 .63032 .62713 .60148 Metetra .237 .487 .892 r .62713 .60148 .64731 .63032 .69007 .64926 .67003 .61901 Metetra .096 .386 .845 r .67003 .61901 .69007 .64926 .73498 .67622 .71509 .64574 Metetra 0 .276 .781 r .71509 .64574 .73498 .67622 .78249 .71178 .76287 .68249 Metetra 0 .176 .707 r .76287 .68249 .78249 .71178 .833 .75531 .81384 .72861 Metetra 0 .104 .64 r .81384 .72861 .833 .75531 .88676 .80486 .86828 .78182 Metetra 0 .233 .724 r .2769 .4806 .29909 .50255 .33052 .52274 .30818 .50276 Metetra 0 .279 .748 r .30818 .50276 .33052 .52274 .36325 .5416 .34087 .52318 Metetra 0 .388 .824 r .34087 .52318 .36325 .5416 .39735 .5574 .37506 .53966 Metetra .105 .522 .91 r .37506 .53966 .39735 .5574 .43279 .56913 .41072 .551 Metetra .283 .627 .962 r .41072 .551 .43279 .56913 .46945 .57683 .44768 .55724 Metetra .404 .676 .971 r .44768 .55724 .46945 .57683 .50721 .58162 .48576 .55971 Metetra .456 .678 .96 r .48576 .55971 .50721 .58162 .54601 .58552 .52487 .56076 Metetra .448 .647 .943 r .52487 .56076 .54601 .58552 .58592 .59119 .56501 .56344 Metetra .387 .589 .924 r .56501 .56344 .58592 .59119 .62713 .60148 .60639 .57106 Metetra .274 .503 .895 r .60639 .57106 .62713 .60148 .67003 .61901 .6494 .58678 Metetra .118 .391 .846 r .6494 .58678 .67003 .61901 .71509 .64574 .69459 .61306 Metetra 0 .265 .773 r .69459 .61306 .71509 .64574 .76287 .68249 .7426 .6511 Metetra 0 .149 .687 r .7426 .6511 .76287 .68249 .81384 .72861 .79401 .7003 Metetra 0 .064 .608 r .79401 .7003 .81384 .72861 .86828 .78182 .84917 .75797 Metetra 0 .156 .655 r .25411 .45806 .2769 .4806 .30818 .50276 .28514 .48275 Metetra 0 .212 .685 r .28514 .48275 .30818 .50276 .34087 .52318 .31773 .50514 Metetra 0 .352 .788 r .31773 .50514 .34087 .52318 .37506 .53966 .352 .52246 Metetra .091 .526 .907 r .352 .52246 .37506 .53966 .41072 .551 .38789 .53328 Metetra .311 .653 .971 r .38789 .53328 .41072 .551 .44768 .55724 .42519 .53777 Metetra .45 .706 .976 r .42519 .53777 .44768 .55724 .48576 .55971 .46365 .53756 Metetra .507 .706 .96 r .46365 .53756 .48576 .55971 .52487 .56076 .5031 .53536 Metetra .5 .673 .942 r .5031 .53536 .52487 .56076 .56501 .56344 .54352 .53459 Metetra .437 .613 .924 r .54352 .53459 .56501 .56344 .60639 .57106 .58508 .53899 Metetra .318 .522 .896 r .58508 .53899 .60639 .57106 .6494 .58678 .62819 .55234 Metetra .146 .398 .846 r .62819 .55234 .6494 .58678 .69459 .61306 .67346 .5778 Metetra 0 .254 .765 r .67346 .5778 .69459 .61306 .7426 .6511 .72165 .61717 Metetra 0 .118 .665 r .72165 .61717 .7426 .6511 .79401 .7003 .77347 .67005 Metetra 0 .019 .57 r .77347 .67005 .79401 .7003 .84917 .75797 .82937 .73329 Metetra 0 .059 .56 r .23069 .4349 .25411 .45806 .28514 .48275 .26134 .46293 Metetra 0 .128 .598 r .26134 .46293 .28514 .48275 .31773 .50514 .29374 .48781 Metetra 0 .308 .739 r .29374 .48781 .31773 .50514 .352 .52246 .32808 .50605 Metetra .082 .534 .904 r .32808 .50605 .352 .52246 .38789 .53328 .36424 .51606 Metetra .348 .686 .98 r .36424 .51606 .38789 .53328 .42519 .53777 .40194 .51838 Metetra .5 .736 .978 r .40194 .51838 .42519 .53777 .46365 .53756 .44083 .51513 Metetra .557 .731 .956 r .44083 .51513 .46365 .53756 .5031 .53536 .48068 .50938 Metetra .551 .698 .937 r .48068 .50938 .5031 .53536 .54352 .53459 .52143 .50475 Metetra .49 .639 .921 r .52143 .50475 .54352 .53459 .58508 .53899 .5632 .50532 Metetra .369 .546 .897 r .5632 .50532 .58508 .53899 .62819 .55234 .6064 .51549 Metetra .183 .411 .848 r .6064 .51549 .62819 .55234 .67346 .5778 .65169 .53944 Metetra 0 .245 .759 r .65169 .53944 .67346 .5778 .72165 .61717 .69998 .58002 Metetra 0 .085 .641 r .69998 .58002 .72165 .61717 .77347 .67005 .75217 .63733 Metetra 0 0 .527 r .75217 .63733 .77347 .67005 .82937 .73329 .80887 .70771 Metetra .564 .059 0 r .20663 .4111 .23069 .4349 .26134 .46293 .23668 .44371 Metetra .512 0 0 r .23668 .44371 .26134 .46293 .29374 .48781 .26879 .47178 Metetra 0 .258 .675 r .26879 .47178 .29374 .48781 .32808 .50605 .30319 .49082 Metetra .087 .556 .905 r .30319 .49082 .32808 .50605 .36424 .51606 .3397 .49938 Metetra .402 .725 .989 r .3397 .49938 .36424 .51606 .40194 .51838 .37787 .49887 Metetra .552 .763 .975 r .37787 .49887 .40194 .51838 .44083 .51513 .41726 .49224 Metetra .602 .751 .948 r .41726 .49224 .44083 .51513 .48068 .50938 .45759 .48288 Metetra .597 .719 .929 r .45759 .48288 .48068 .50938 .52143 .50475 .49872 .47423 Metetra .544 .666 .916 r .49872 .47423 .52143 .50475 .5632 .50532 .54075 .4703 Metetra .43 .576 .897 r .54075 .4703 .5632 .50532 .6064 .51549 .58404 .47611 Metetra .236 .433 .851 r .58404 .47611 .6064 .51549 .65169 .53944 .62929 .49729 Metetra 0 .243 .755 r .62929 .49729 .65169 .53944 .69998 .58002 .67757 .53855 Metetra 0 .051 .616 r .67757 .53855 .69998 .58002 .75217 .63733 .73005 .60125 Metetra 0 0 .477 r .73005 .60125 .75217 .63733 .80887 .70771 .7876 .68119 Metetra .695 .197 0 r .18189 .38664 .20663 .4111 .23668 .44371 .21099 .42595 Metetra .632 .091 0 r .21099 .42595 .23668 .44371 .26879 .47178 .24265 .45816 Metetra .362 0 0 r .24265 .45816 .26879 .47178 .30319 .49082 .27719 .47734 Metetra .133 .608 .92 r .27719 .47734 .30319 .49082 .3397 .49938 .31419 .48301 Metetra .479 .77 .995 r .31419 .48301 .3397 .49938 .37787 .49887 .35297 .47862 Metetra .603 .78 .965 r .35297 .47862 .37787 .49887 .41726 .49224 .39293 .46845 Metetra .637 .761 .938 r .39293 .46845 .41726 .49224 .45759 .48288 .43376 .45592 Metetra .633 .736 .922 r .43376 .45592 .45759 .48288 .49872 .47423 .47534 .44363 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