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Bottom side:" }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell["bot[x_] = f[x,-1/2]", "Input", AspectRatioFixed->True], Cell[BoxData[ \(1\/4 + x\^2\/2 + x\^3 + x\^4\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["CriticalPoints[bot[x]] //N", "Input", AspectRatioFixed->True], Cell[BoxData[ \({{x \[Rule] 0}, {x \[Rule] \(-0.375`\) - 0.330718913883073817`\ I}, { x \[Rule] \(-0.375`\) + 0.330718913883073817`\ I}}\)], "Output"] }, Open ]], Cell[TextData[ "We drop the compex numbers and get one critical point on the bottom side:"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell["bb = bot[0]", "Input", AspectRatioFixed->True], Cell[BoxData[ \(1\/4\)], "Output"] }, Open ]], Cell[TextData["Top side:"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell["top[x_] = f[x,1]", "Input", AspectRatioFixed->True], Cell[BoxData[ \(1 - x\^2 + x\^3 + x\^4\)], "Output"] }, Open ]], Cell[CellGroupData[{ 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Metetra .478 .783 .998 r .47007 .22041 .5143 .21508 .48641 .19795 .44171 .20416 Metetra .427 .806 .992 r .44171 .20416 .48641 .19795 .45745 .18547 .41223 .19309 Metetra .214 .739 .812 r .41223 .19309 .45745 .18547 .42719 .1849 .38133 .19442 Metetra .185 0 0 r .38133 .19442 .42719 .1849 .3952 .20635 .34852 .21821 Metetra .375 .016 .222 r .34852 .21821 .3952 .20635 .36074 .26355 .31295 .27815 Metetra .647 .586 .772 r .72897 .4096 .76994 .3915 .74789 .32851 .70666 .34348 Metetra .654 .659 .843 r .70666 .34348 .74789 .32851 .72628 .29185 .68464 .304 Metetra .638 .757 .934 r .68464 .304 .72628 .29185 .7046 .27244 .66247 .28207 Metetra .562 .833 .996 r .66247 .28207 .7046 .27244 .68248 .26289 .63982 .27035 Metetra .451 .833 .988 r .63982 .27035 .68248 .26289 .65969 .25733 .61647 .26303 Metetra .396 .809 .977 r .61647 .26303 .65969 .25733 .63606 .25137 .59229 .25575 Metetra .405 .794 .989 r .59229 .25575 .63606 .25137 .6115 .24217 .56722 .24575 Metetra .434 .777 .997 r .56722 .24575 .6115 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.3611 .58097 L p .75 .436 .454 r F P 0 g s .4449 .56291 m .44785 .60718 L .53307 .6689 L p .756 .432 .437 r F P 0 g s .53307 .6689 m .53413 .63334 L .4449 .56291 L p .756 .432 .437 r F P 0 g s .3611 .58097 m .35461 .53239 L .27642 .648 L p .758 .45 .463 r F P 0 g s .53413 .63334 m .53307 .6689 L .61929 .79939 L p .785 .437 .398 r F P 0 g s .61929 .79939 m .62484 .77905 L .53413 .63334 L p .785 .437 .398 r F P 0 g s .3611 .58097 m .36784 .62636 L .44785 .60718 L p .75 .436 .454 r F P 0 g s .45091 .64882 m .44785 .60718 L .36784 .62636 L p .745 .427 .446 r F P 0 g s .44785 .60718 m .45091 .64882 L .53198 .70245 L p .753 .424 .428 r F P 0 g s .53198 .70245 m .53307 .6689 L .44785 .60718 L p .753 .424 .428 r F P 0 g s .27642 .648 m .2867 .68834 L .3611 .58097 L p .758 .45 .463 r F P 0 g s .36784 .62636 m .3611 .58097 L .2867 .68834 L p .752 .44 .456 r F P 0 g s .53307 .6689 m .53198 .70245 L .61364 .81835 L p .783 .429 .386 r F P 0 g s .61364 .81835 m .61929 .79939 L .53307 .6689 L p .783 .429 .386 r F P 0 g s .36784 .62636 m .37484 .66848 L .45091 .64882 L p .745 .427 .446 r F P 0 g s .45407 .68773 m .45091 .64882 L .37484 .66848 L p .739 .417 .436 r F P 0 g s .45091 .64882 m .45407 .68773 L .53085 .7339 L p .749 .414 .417 r F P 0 g s .53085 .7339 m .53198 .70245 L .45091 .64882 L p .749 .414 .417 r F P 0 g s .2867 .68834 m .29732 .72512 L .36784 .62636 L p .752 .44 .456 r F P 0 g s .37484 .66848 m .36784 .62636 L .29732 .72512 L p .746 .43 .447 r F P 0 g s .53198 .70245 m .53085 .7339 L .60788 .83588 L p .78 .419 .372 r F P 0 g s .60788 .83588 m .61364 .81835 L .53198 .70245 L p .78 .419 .372 r F P 0 g s .63028 .75736 m .62484 .77905 L .71141 .97834 L p .855 .472 .321 r F P 0 g s .71141 .97834 m .72192 .97971 L .63028 .75736 L p .855 .472 .321 r F P 0 g s .37484 .66848 m .38208 .70728 L .45407 .68773 L p .739 .417 .436 r F P 0 g s .45734 .72382 m .45407 .68773 L .38208 .70728 L p .733 .405 .425 r F P 0 g s .45407 .68773 m .45734 .72382 L .5297 .76314 L p .744 .403 .405 r F P 0 g s .5297 .76314 m .53085 .7339 L .45407 .68773 L p .744 .403 .405 r F P 0 g s .29732 .72512 m .30826 .75835 L .37484 .66848 L p .746 .43 .447 r F P 0 g s .38208 .70728 m .37484 .66848 L .30826 .75835 L p .739 .418 .438 r F P 0 g s .53085 .7339 m .5297 .76314 L .60202 .85194 L p .777 .408 .356 r F P 0 g s .60202 .85194 m .60788 .83588 L .53085 .7339 L p .777 .408 .356 r F P 0 g s .2867 .68834 m .27642 .648 L .21402 .83215 L p .764 .446 .448 r F P 0 g s .62484 .77905 m .61929 .79939 L .70084 .9767 L p .852 .462 .306 r F P 0 g s .70084 .9767 m .71141 .97834 L .62484 .77905 L p .852 .462 .306 r F P 0 g s .38208 .70728 m .38954 .74273 L .45734 .72382 L p .733 .405 .425 r F P 0 g s .4607 .75699 m .45734 .72382 L .38954 .74273 L p .727 .392 .413 r F P 0 g s .45734 .72382 m .4607 .75699 L .52852 .79011 L p .739 .391 .39 r F P 0 g s .52852 .79011 m .5297 .76314 L .45734 .72382 L p .739 .391 .39 r F P 0 g s .30826 .75835 m .31951 .78809 L .38208 .70728 L p .739 .418 .438 r F P 0 g s .38954 .74273 m .38208 .70728 L .31951 .78809 L p .732 .404 .426 r F P 0 g s .61929 .79939 m .61364 .81835 L .6902 .97478 L p .849 .451 .289 r F P 0 g s .6902 .97478 m .70084 .9767 L .61929 .79939 L p .849 .451 .289 r F P 0 g s .5297 .76314 m .52852 .79011 L .59606 .86649 L p .773 .394 .337 r F P 0 g s .59606 .86649 m .60202 .85194 L .5297 .76314 L p .773 .394 .337 r F P 0 g s .21402 .83215 m .22828 .85365 L .2867 .68834 L p .764 .446 .448 r F P 0 g s .29732 .72512 m .2867 .68834 L .22828 .85365 L p .757 .434 .439 r F P 0 g s .38954 .74273 m .39722 .77481 L .4607 .75699 L p .727 .392 .413 r F P 0 g s .46415 .78719 m .4607 .75699 L .39722 .77481 L p .719 .376 .397 r F P 0 g s .4607 .75699 m .46415 .78719 L .52731 .8147 L p .734 .376 .373 r F P 0 g s .52731 .8147 m .52852 .79011 L .4607 .75699 L p .734 .376 .373 r F P 0 g s .61364 .81835 m .60788 .83588 L .67949 .97258 L p .845 .438 .269 r F P 0 g s .67949 .97258 m .6902 .97478 L .61364 .81835 L p .845 .438 .269 r F P 0 g s .31951 .78809 m .33104 .81441 L .38954 .74273 L p .732 .404 .426 r F P 0 g s .39722 .77481 m .38954 .74273 L .33104 .81441 L p .723 .389 .413 r F P 0 g s .52852 .79011 m .52731 .8147 L .59 .87948 L p .768 .378 .314 r F P 0 g s .59 .87948 m .59606 .86649 L .52852 .79011 L p .768 .378 .314 r F P 0 g s .22828 .85365 m .24282 .87212 L .29732 .72512 L p .757 .434 .439 r F P 0 g s .30826 .75835 m .29732 .72512 L .24282 .87212 L p .748 .42 .428 r F P 0 g s .60788 .83588 m .60202 .85194 L .66872 .97009 L p .84 .423 .246 r F P 0 g s .66872 .97009 m .67949 .97258 L .60788 .83588 L p .84 .423 .246 r F P 0 g s .39722 .77481 m .40511 .8035 L .46415 .78719 L p .719 .376 .397 r F P 0 g s .46769 .81432 m .46415 .78719 L .40511 .8035 L p .71 .358 .378 r F P 0 g s .46415 .78719 m .46769 .81432 L .52608 .83686 L p .727 .358 .351 r F P 0 g s .52608 .83686 m .52731 .8147 L .46415 .78719 L p .727 .358 .351 r F P 0 g s .33104 .81441 m .34283 .83741 L .39722 .77481 L p .723 .389 .413 r F P 0 g s .40511 .8035 m .39722 .77481 L .34283 .83741 L p .713 .371 .396 r F P 0 g s .52731 .8147 m .52608 .83686 L .58385 .89088 L p .762 .357 .286 r F P 0 g s .58385 .89088 m .59 .87948 L .52731 .8147 L p .762 .357 .286 r F P 0 g s .24282 .87212 m .25761 .8877 L .30826 .75835 L p .748 .42 .428 r F P 0 g s .31951 .78809 m .30826 .75835 L .25761 .8877 L p .738 .404 .415 r F P 0 g s .60202 .85194 m .59606 .86649 L .65787 .96731 L p .833 .404 .218 r F P 0 g s .65787 .96731 m .66872 .97009 L .60202 .85194 L p .833 .404 .218 r F P 0 g s .40511 .8035 m .41319 .82883 L .46769 .81432 L p .71 .358 .378 r F P 0 g s .47132 .83835 m .46769 .81432 L .41319 .82883 L p .699 .335 .354 r F P 0 g s .46769 .81432 m .47132 .83835 L .52482 .85649 L p .718 .335 .324 r F P 0 g s .52482 .85649 m .52608 .83686 L .46769 .81432 L p .718 .335 .324 r F P 0 g s .34283 .83741 m .35486 .85719 L .40511 .8035 L p .713 .371 .396 r F P 0 g s .41319 .82883 m .40511 .8035 L .35486 .85719 L p .701 .349 .376 r F P 0 g s .25761 .8877 m .27263 .90056 L .31951 .78809 L p .738 .404 .415 r F P 0 g s .33104 .81441 m .31951 .78809 L .27263 .90056 L p .728 .385 .4 r F P 0 g s .52608 .83686 m .52482 .85649 L .57761 .90064 L p .753 .332 .25 r F P 0 g s .57761 .90064 m .58385 .89088 L .52608 .83686 L p .753 .332 .25 r F P 0 g s .59606 .86649 m .59 .87948 L .64696 .96422 L p .824 .381 .184 r F P 0 g s .64696 .96422 m .65787 .96731 L .59606 .86649 L p .824 .381 .184 r F P 0 g s .41319 .82883 m .42145 .8508 L .47132 .83835 L p .699 .335 .354 r F P 0 g s .47503 .85922 m .47132 .83835 L .42145 .8508 L p .685 .306 .323 r F P 0 g s .47132 .83835 m .47503 .85922 L .52354 .87354 L p .706 .306 .289 r F P 0 g s .52354 .87354 m .52482 .85649 L .47132 .83835 L p .706 .306 .289 r F P 0 g s .35486 .85719 m .36711 .87388 L .41319 .82883 L p .701 .349 .376 r F P 0 g s .42145 .8508 m .41319 .82883 L .36711 .87388 L p .687 .323 .351 r F P 0 g s .59 .87948 m .58385 .89088 L .63598 .96083 L p .811 .352 .142 r F P 0 g s .63598 .96083 m .64696 .96422 L .59 .87948 L p .811 .352 .142 r F P 0 g s .27263 .90056 m .28784 .91087 L .33104 .81441 L p .728 .385 .4 r F P 0 g s .34283 .83741 m .33104 .81441 L .28784 .91087 L p .715 .364 .381 r F P 0 g s .52482 .85649 m .52354 .87354 L .57128 .90874 L p .74 .298 .203 r F P 0 g s .57128 .90874 m .57761 .90064 L .52482 .85649 L p .74 .298 .203 r F P 0 g s .72192 .97971 m .71141 .97834 L .78496 1.1868 L p .905 .495 .124 r F P 0 g s .78496 1.1868 m .8009 1.21745 L .72192 .97971 L p .905 .495 .124 r F P 0 g s .71141 .97834 m .70084 .9767 L .76918 1.15807 L p .896 .478 .102 r F P 0 g s .76918 1.15807 m .78496 1.1868 L .71141 .97834 L p .896 .478 .102 r F P 0 g s .58385 .89088 m .57761 .90064 L .62494 .95714 L p .794 .316 .089 r F P 0 g s .62494 .95714 m .63598 .96083 L .58385 .89088 L p .794 .316 .089 r F P 0 g s .28784 .91087 m .30323 .91881 L .34283 .83741 L p .715 .364 .381 r F P 0 g s .35486 .85719 m .34283 .83741 L .30323 .91881 L p .7 .339 .359 r F P 0 g s .42145 .8508 m .42988 .86947 L .47503 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.88793 L p .688 .19 .055 r F P 0 g s .57128 .90874 m .56487 .91514 L .60266 .94878 L p .729 .205 0 r F P 0 g s .60266 .94878 m .61383 .95312 L .57128 .90874 L p .729 .205 0 r F P 0 g s .66872 .97009 m .65787 .96731 L .70723 1.0608 L p .836 .378 0 r F P 0 g s .67678 1.02185 m .69197 1.04057 L .68784 1.03356 L p .775 .29 0 r F P 0 g s .68784 1.03356 m .64696 .96422 L p .67582 1.01999 L .775 .29 0 r F P 0 g s .67678 1.02185 m .68784 1.03356 L .67582 1.01999 L .775 .29 0 r F .64656 .98876 m .66165 1.0046 L .65995 1.00241 L p .667 .156 0 r F P 0 g s .61383 .95312 m .60266 .94878 L .6315 .97428 L p .583 .064 0 r F P 0 g s .61647 .9611 m .6315 .97428 L .62976 .97274 L p .472 0 0 r F P 0 g s .62976 .97274 m .60266 .94878 L p .61647 .9611 L .472 0 0 r F P 0 g s .60266 .94878 m .59143 .94412 L .61647 .9611 L p .472 0 0 r F P 0 g s .61647 .9611 m .60146 .94919 L .60243 .94881 L p .078 .376 .84 r F P 0 g s .59143 .94412 m .60266 .94878 L .56487 .91514 L p .67 .118 0 r F P 0 g s .62494 .95714 m .61383 .95312 L .64656 .98876 L p .667 .156 0 r F P 0 g s .6315 .97428 m .64656 .98876 L .64242 .98425 L p .583 .064 0 r F P 0 g s .64242 .98425 m .61383 .95312 L p .6307 .97331 L .583 .064 0 r F P 0 g s .6315 .97428 m .64242 .98425 L .6307 .97331 L .583 .064 0 r F .60243 .94881 m .60386 .94825 L p .61647 .9611 L .078 .376 .84 r F P 0 g s .60386 .94825 m .61319 .94461 L p .61621 .95979 L .078 .376 .84 r F P 0 g s .61621 .95979 m .61647 .9611 L .60386 .94825 L .078 .376 .84 r F .6315 .97428 m .61647 .9611 L .61885 .96077 L p 0 .281 .784 r F P 0 g s .65754 .99929 m .62494 .95714 L p .64569 .9875 L .667 .156 0 r F P 0 g s .63598 .96083 m .62494 .95714 L .66165 1.0046 L p .729 .23 0 r F P 0 g s .61885 .96077 m .63037 .95915 L p .63143 .97329 L 0 .281 .784 r F P 0 g s .63143 .97329 m .6315 .97428 L .61885 .96077 L 0 .281 .784 r F .65995 1.00241 m .65754 .99929 L p .64656 .98876 L .667 .156 0 r F P 0 g s .64656 .98876 m .65754 .99929 L .64569 .9875 L .667 .156 0 r F .64656 .98876 m .6315 .97428 L .63246 .97439 L p 0 .185 .718 r F P 0 g s .64696 .96422 m .63598 .96083 L .67678 1.02185 L p .775 .29 0 r F P 0 g s .66165 1.0046 m .67678 1.02185 L .67268 1.01571 L p .729 .23 0 r F P 0 g s .67268 1.01571 m .63598 .96083 L p .66073 1.00304 L .729 .23 0 r F P 0 g s .66165 1.0046 m .67268 1.01571 L .66073 1.00304 L .729 .23 0 r F .63246 .97439 m .63388 .97456 L p .64656 .98876 L 0 .185 .718 r F P 0 g s .63388 .97456 m .64765 .97618 L p .64662 .98807 L 0 .185 .718 r F P 0 g s .64662 .98807 m .64656 .98876 L .63388 .97456 L 0 .185 .718 r F .66165 1.0046 m .64656 .98876 L .64895 .98968 L p 0 .091 .647 r F P 0 g s .64895 .98968 m .66507 .99585 L p .66181 1.00419 L 0 .091 .647 r F P 0 g s .66181 1.00419 m .66165 1.0046 L .64895 .98968 L 0 .091 .647 r F .69197 1.04057 m .67678 1.02185 L .67926 1.02415 L p 0 0 .503 r F P 0 g s .7183 1.07369 m .66872 .97009 L p .70617 1.05829 L .836 .378 0 r F P 0 g s .65787 .96731 m .64696 .96422 L .69197 1.04057 L p .809 .338 0 r F P 0 g s .69197 1.04057 m .70723 1.0608 L .70304 1.05287 L p .809 .338 0 r F P 0 g s .70304 1.05287 m .65787 .96731 L p .69096 1.03839 L .809 .338 0 r F P 0 g s .69197 1.04057 m .70304 1.05287 L .69096 1.03839 L .809 .338 0 r F .67678 1.02185 m .66165 1.0046 L .66407 1.00619 L p 0 .003 .574 r F P 0 g s .66407 1.00619 m .68263 1.01832 L p .67703 1.02171 L 0 .003 .574 r F P 0 g s .67703 1.02171 m .67678 1.02185 L .66407 1.00619 L 0 .003 .574 r F .67926 1.02415 m .70037 1.04375 L p .69229 1.04069 L 0 0 .503 r F P 0 g s .69229 1.04069 m .69197 1.04057 L .67926 1.02415 L 0 0 .503 r F .70723 1.0608 m .72256 1.0826 L .7183 1.07369 L p .836 .378 0 r F P 0 g s .70723 1.0608 m .7183 1.07369 L .70617 1.05829 L .836 .378 0 r F .31877 .92458 m .33444 .92838 L .36711 .87388 L p .683 .308 .331 r F P 0 g s .37956 .88763 m .36711 .87388 L .33444 .92838 L p .661 .27 .295 r F P 0 g s .17951 .99669 m .19701 .99569 L .22828 .85365 L p .753 .399 .382 r F P 0 g s .24282 .87212 m .22828 .85365 L .19701 .99569 L p .739 .376 .364 r F P 0 g s .39219 .89858 m .40498 .9069 L .43846 .88489 L p .647 .248 .274 r F P 0 g s .44719 .89712 m .43846 .88489 L .40498 .9069 L p .615 .191 .213 r F P 0 g s .43846 .88489 m .44719 .89712 L .48267 .89132 L p .642 .219 .225 r F P 0 g s .48659 .90251 m .48267 .89132 L .44719 .89712 L p .603 .147 .142 r F P 0 g s .48267 .89132 m .48659 .90251 L .51957 .90854 L p .624 .139 .081 r F P 0 g s .51957 .90854 m .52091 .89962 L .48267 .89132 L p .624 .139 .081 r F P 0 g s .52091 .89962 m .51957 .90854 L .55182 .92272 L p .634 .097 0 r F P 0 g s .55182 .92272 m .55838 .91981 L .52091 .89962 L p .634 .097 0 r F P 0 g s .56487 .91514 m .55838 .91981 L .59143 .94412 L p .67 .118 0 r F P 0 g s .33444 .92838 m .35022 .93043 L .37956 .88763 L p .661 .27 .295 r F P 0 g s .39219 .89858 m .37956 .88763 L .35022 .93043 L p .633 .222 .248 r F P 0 g s .19701 .99569 m .21457 .99331 L .24282 .87212 L p .739 .376 .364 r F P 0 g s .25761 .8877 m .24282 .87212 L .21457 .99331 L p .722 .349 .342 r F P 0 g s .40498 .9069 m .41791 .91277 L .44719 .89712 L p .615 .191 .213 r F P 0 g s .45604 .90625 m .44719 .89712 L .41791 .91277 L p .568 .109 .124 r F P 0 g s .44719 .89712 m .45604 .90625 L .48659 .90251 L p .603 .147 .142 r F P 0 g s .49057 .91046 m .48659 .90251 L .45604 .90625 L p .537 .036 .012 r F P 0 g s .48659 .90251 m .49057 .91046 L .51821 .91465 L p .553 .02 0 r F P 0 g s .51821 .91465 m .51957 .90854 L .48659 .90251 L p .553 .02 0 r F P 0 g s .55838 .91981 m .55182 .92272 L .58014 .93913 L p .575 0 0 r F P 0 g s .58014 .93913 m .59143 .94412 L .55838 .91981 L p .575 0 0 r F P 0 g s .51957 .90854 m .51821 .91465 L .54519 .92384 L p .536 0 0 r F P 0 g s .54519 .92384 m .55182 .92272 L .51957 .90854 L p .536 0 0 r F P 0 g s .21457 .99331 m .23217 .98971 L .25761 .8877 L p .722 .349 .342 r F P 0 g s .27263 .90056 m .25761 .8877 L .23217 .98971 L p .703 .317 .315 r F P 0 g s .35022 .93043 m .36609 .93092 L .39219 .89858 L p .633 .222 .248 r F P 0 g s .40498 .9069 m .39219 .89858 L .36609 .93092 L p .595 .159 .183 r F P 0 g s .5362 .27524 m .53747 .21945 L .64184 .37347 L p .648 .462 .618 r F P 0 g s .53747 .21945 m .5362 .27524 L .43578 .34098 L p .737 .538 .624 r F P 0 g s .43578 .34098 m .43192 .28965 L .53747 .21945 L p .737 .538 .624 r F P 0 g s .23217 .98971 m .2498 .98507 L .27263 .90056 L p .703 .317 .315 r F P 0 g s .28784 .91087 m .27263 .90056 L .2498 .98507 L p .678 .279 .281 r F P 0 g s .41791 .91277 m .43097 .91638 L .45604 .90625 L p .568 .109 .124 r F P 0 g s .46501 .91237 m .45604 .90625 L .43097 .91638 L p .489 0 0 r F P 0 g s .45604 .90625 m .46501 .91237 L .49057 .91046 L p .537 .036 .012 r F P 0 g s .55182 .92272 m .54519 .92384 L .56879 .9338 L p .427 0 0 r F P 0 g s .56879 .9338 m .58014 .93913 L .55182 .92272 L p .427 0 0 r F P 0 g s .36609 .93092 m .38204 .93007 L .40498 .9069 L p .595 .159 .183 r F P 0 g s .41791 .91277 m .40498 .9069 L .38204 .93007 L p .54 .073 .093 r F P 0 g s .4946 .91515 m .49057 .91046 L .46501 .91237 L p .412 0 0 r F P 0 g s .49057 .91046 m .4946 .91515 L .51683 .91791 L p .415 0 0 r F P 0 g s .51683 .91791 m .51821 .91465 L .49057 .91046 L p .415 0 0 r F P 0 g s .51821 .91465 m .51683 .91791 L .5385 .92315 L p .359 0 0 r F P 0 g s .5385 .92315 m .54519 .92384 L .51821 .91465 L p .359 0 0 r F P 0 g s .2498 .98507 m .26743 .97956 L .28784 .91087 L p .678 .279 .281 r F P 0 g s .30323 .91881 m .28784 .91087 L .26743 .97956 L p .649 .232 .24 r F P 0 g s .64184 .37347 m .63521 .41191 L .5362 .27524 L p .648 .462 .618 r F P 0 g s .53491 .32944 m .5362 .27524 L .63521 .41191 L p .653 .468 .622 r F P 0 g s .5362 .27524 m .53491 .32944 L .4397 .39054 L p .738 .542 .628 r F P 0 g s .4397 .39054 m .43578 .34098 L .5362 .27524 L p .738 .542 .628 r F P 0 g s .38204 .93007 m .39805 .92811 L .41791 .91277 L p .54 .073 .093 r F P 0 g s .43097 .91638 m .41791 .91277 L .39805 .92811 L p .457 0 0 r F P 0 g s .43097 .91638 m .44413 .91792 L .46501 .91237 L p .489 0 0 r F P 0 g s .47409 .91559 m .46501 .91237 L .44413 .91792 L p .347 0 0 r F P 0 g s .46501 .91237 m .47409 .91559 L .4946 .91515 L p .412 0 0 r F P 0 g s .55738 .92813 m .56879 .9338 L .54519 .92384 L p 0 .345 .717 r F P 0 g s .54519 .92384 m .5385 .92315 L .55738 .92813 L p 0 .345 .717 r F P 0 g s .60146 .94919 m .61647 .9611 L .59143 .94412 L p 0 .179 .652 r F P 0 g s .59143 .94412 m .58014 .93913 L .60146 .94919 L p 0 .179 .652 r F P 0 g s .26743 .97956 m .28506 .97333 L .30323 .91881 L p .649 .232 .24 r F P 0 g s .31877 .92458 m .30323 .91881 L .28506 .97333 L p .611 .174 .187 r F P 0 g s .51543 .91828 m .51683 .91791 L .4946 .91515 L p .149 0 0 r F P 0 g s .53174 .92063 m .5385 .92315 L .53161 .92149 L p 0 .488 .776 r F P 0 g s .53161 .92149 m .51683 .91791 L p .52645 .91966 L 0 .488 .776 r F P 0 g s .53174 .92063 m .53161 .92149 L .52645 .91966 L 0 .488 .776 r F .51543 .91828 m .51402 .91575 L .51557 .91583 L p .232 .674 .955 r F P 0 g s .51557 .91583 m .52493 .91626 L p .51986 .91734 L .232 .674 .955 r F P 0 g s .51683 .91791 m .51543 .91828 L .53174 .92063 L p 0 .488 .776 r F P 0 g s .51986 .91734 m .51543 .91828 L .51557 .91583 L .232 .674 .955 r F .49869 .91662 m .4946 .91515 L .47409 .91559 L p .165 0 0 r F P 0 g s .48325 .91603 m .49869 .91662 L p .49244 .91636 L .165 0 0 r F P 0 g s .47409 .91559 m .48325 .91603 L p .49244 .91636 L .165 0 0 r F P 0 g s .44413 .91792 m .45739 .9176 L .47409 .91559 L p .347 0 0 r F P 0 g s .48325 .91603 m .47409 .91559 L .45739 .9176 L p .087 0 0 r F P 0 g s .48325 .91603 m .47074 .91562 L p .47105 .9156 L .269 .767 .89 r F P 0 g s .49249 .9138 m .48325 .91603 L p .47105 .9156 L .269 .767 .89 r F P 0 g s .49869 .91662 m .48325 .91603 L p .48635 .91585 L .212 .725 .883 r F P 0 g s .4946 .91515 m .49869 .91662 L p .50834 .91722 L .149 0 0 r F P 0 g s .50282 .91487 m .49869 .91662 L p .48635 .91585 L .212 .725 .883 r F P 0 g s .49869 .91662 m .51543 .91828 L p .50834 .91722 L .149 0 0 r F P 0 g s .39805 .92811 m .41411 .92526 L .43097 .91638 L p .457 0 0 r F P 0 g s .44413 .91792 m .43097 .91638 L .41411 .92526 L p .324 0 0 r F P 0 g s .58014 .93913 m .56879 .9338 L .58645 .9385 L p 0 .32 .774 r F P 0 g s .58645 .9385 m .60146 .94919 L .58014 .93913 L p 0 .32 .774 r F P 0 g s .28506 .97333 m .30268 .96656 L .31877 .92458 L p .611 .174 .187 r F P 0 g s .33444 .92838 m .31877 .92458 L .30268 .96656 L p .562 .102 .118 r F P 0 g s .63521 .41191 m .62854 .4499 L .53491 .32944 L p .653 .468 .622 r F P 0 g s .5336 .38192 m .53491 .32944 L .62854 .4499 L p .657 .475 .626 r F P 0 g s .53491 .32944 m .5336 .38192 L .44367 .43826 L p .74 .546 .632 r F P 0 g s .44367 .43826 m .4397 .39054 L .53491 .32944 L p .74 .546 .632 r F P 0 g s .5385 .92315 m .53174 .92063 L .54591 .9221 L p .036 .524 .881 r F P 0 g s .54591 .9221 m .55738 .92813 L .5385 .92315 L p .036 .524 .881 r F P 0 g s .43192 .28965 m .43578 .34098 L .33873 .56787 L p .817 .615 .629 r F P 0 g s .33873 .56787 m .32928 .53887 L .43192 .28965 L p .817 .615 .629 r F P 0 g s .30268 .96656 m .32026 .95942 L .33444 .92838 L p .562 .102 .118 r F P 0 g s .35022 .93043 m .33444 .92838 L .32026 .95942 L p .497 .009 .027 r F P 0 g s .56879 .9338 m .55738 .92813 L .57144 .92899 L p .026 .455 .874 r F P 0 g s .57144 .92899 m .58645 .9385 L .56879 .9338 L p .026 .455 .874 r F P 0 g s .41411 .92526 m .43021 .92174 L .44413 .91792 L p .324 0 0 r F P 0 g s .45739 .9176 m .44413 .91792 L .43021 .92174 L p .109 0 0 r F P 0 g s .51402 .91575 m .51543 .91828 L .49869 .91662 L p .214 .704 .924 r F P 0 g s .52493 .91626 m .53174 .92063 L .51543 .91828 L p .232 .674 .955 r F P 0 g s .49869 .91662 m .50282 .91487 L .51402 .91575 L p .214 .704 .924 r F P 0 g s .32026 .95942 m .33782 .95206 L .35022 .93043 L p .497 .009 .027 r F P 0 g s .36609 .93092 m .35022 .93043 L .33782 .95206 L p .407 0 0 r F P 0 g s .53174 .92063 m .52493 .91626 L .53439 .91573 L p .273 .652 .967 r F P 0 g s .53439 .91573 m .54591 .9221 L .53174 .92063 L p .273 .652 .967 r F P 0 g s .62854 .4499 m .62184 .48733 L .5336 .38192 L p .657 .475 .626 r F P 0 g s .53227 .43256 m .5336 .38192 L .62184 .48733 L p .662 .482 .631 r F P 0 g s .5336 .38192 m .53227 .43256 L .44769 .48407 L p .741 .551 .637 r F P 0 g s .44769 .48407 m .44367 .43826 L .5336 .38192 L p .741 .551 .637 r F P 0 g s .55738 .92813 m .54591 .9221 L .55641 .92061 L p .206 .571 .939 r F P 0 g s .55641 .92061 m .57144 .92899 L .55738 .92813 L p .206 .571 .939 r F P 0 g s .45739 .9176 m .47074 .91562 L .48325 .91603 L p .087 0 0 r F P 0 g s .48325 .91603 m .49249 .9138 L .50282 .91487 L p .212 .725 .883 r F P 0 g s .43578 .34098 m .4397 .39054 L .34822 .59585 L p .817 .618 .634 r F P 0 g s .34822 .59585 m .33873 .56787 L .43578 .34098 L p .817 .618 .634 r F P 0 g s .35534 .94465 m .36609 .93092 L p .33791 .95199 L .407 0 0 r F P 0 g s .38204 .93007 m .36609 .93092 L .35534 .94465 L p .283 0 0 r F P 0 g s .37281 .93735 m .38204 .93007 L p .35563 .94449 L .283 0 0 r F P 0 g s .35534 .94465 m .37281 .93735 L p .35563 .94449 L .283 0 0 r F P 0 g s .39805 .92811 m .38204 .93007 L .37281 .93735 L p .115 0 0 r F P 0 g s .37281 .93735 m .37915 .91544 L p .39012 .93016 L .769 .927 .938 r F P 0 g s .39024 .93033 m .37281 .93735 L p .39012 .93016 L .769 .927 .938 r F P 0 g s .35534 .94465 m .36228 .92175 L p .37278 .9373 L .761 .943 .941 r F P 0 g s .37281 .93735 m .35534 .94465 L p .37278 .9373 L .761 .943 .941 r F P 0 g s .33782 .95206 m .35534 .94465 L p .33791 .95199 L .407 0 0 r F P 0 g s .43021 .92174 m .44635 .91777 L .45739 .9176 L p .109 0 0 r F P 0 g s .47074 .91562 m .45739 .9176 L .44635 .91777 L p .179 .714 .797 r F P 0 g s .51261 .9103 m .51402 .91575 L .50282 .91487 L p .484 .79 .999 r F P 0 g s .51402 .91575 m .51261 .9103 L .51808 .91003 L p .458 .747 .991 r F P 0 g s .51808 .91003 m .52493 .91626 L .51402 .91575 L p .458 .747 .991 r F P 0 g s .54591 .9221 m .53439 .91573 L .54136 .91334 L p .363 .657 .969 r F P 0 g s .54136 .91334 m .55641 .92061 L .54591 .9221 L p .363 .657 .969 r F P 0 g s .50698 .90995 m .50282 .91487 L .49249 .9138 L p .505 .828 1 r F P 0 g s .50282 .91487 m .50698 .90995 L .51261 .9103 L p .484 .79 .999 r F P 0 g s .44635 .91777 m .46252 .91358 L .47074 .91562 L p .179 .714 .797 r F P 0 g s .48416 .9122 m .47074 .91562 L .46252 .91358 L p .433 .846 .966 r F P 0 g s .47074 .91562 m .48416 .9122 L .49249 .9138 L p .269 .767 .89 r F P 0 g s .37281 .93735 m .39024 .93033 L .39805 .92811 L p .115 0 0 r F P 0 g s .41411 .92526 m .39805 .92811 L .39024 .93033 L p 0 0 0 r F P 0 g s .52493 .91626 m .51808 .91003 L .52281 .909 L p .452 .72 .982 r F P 0 g s .52281 .909 m .53439 .91573 L .52493 .91626 L p .452 .72 .982 r F P 0 g s .63037 .95915 m .64765 .97618 L .6315 .97428 L p 0 .281 .784 r F P 0 g s .61319 .94461 m .63037 .95915 L .61647 .9611 L p .078 .376 .84 r F P 0 g s .64765 .97618 m .66507 .99585 L .64656 .98876 L p 0 .185 .718 r F P 0 g s .62184 .48733 m .61509 .52407 L .53227 .43256 L p .662 .482 .631 r F P 0 g s .53093 .48126 m .53227 .43256 L .61509 .52407 L p .667 .489 .636 r F P 0 g s .53227 .43256 m .53093 .48126 L .45177 .52788 L p .742 .556 .642 r F P 0 g s .45177 .52788 m .44769 .48407 L .53227 .43256 L p .742 .556 .642 r F P 0 g s .50181 .90905 m .49249 .9138 L .48416 .9122 L p .528 .85 .998 r F P 0 g s .49249 .9138 m .50181 .90905 L .50698 .90995 L p .505 .828 1 r F P 0 g s .60146 .94919 m .58645 .9385 L .59609 .9324 L p .193 .465 .885 r F P 0 g s .59609 .9324 m .61319 .94461 L .60146 .94919 L p .193 .465 .885 r F P 0 g s .66507 .99585 m .68263 1.01832 L .66165 1.0046 L p 0 .091 .647 r F P 0 g s .4397 .39054 m .44367 .43826 L .35775 .62277 L p .816 .621 .639 r F P 0 g s .35775 .62277 m .34822 .59585 L .4397 .39054 L p .816 .621 .639 r F P 0 g s .39024 .93033 m .40762 .92373 L .41411 .92526 L p 0 0 0 r F P 0 g s .43021 .92174 m .41411 .92526 L .40762 .92373 L p .296 .793 .836 r F P 0 g s .58645 .9385 m .57144 .92899 L .57905 .9224 L p .301 .544 .916 r F P 0 g s .57905 .9224 m .59609 .9324 L .58645 .9385 L p .301 .544 .916 r F P 0 g s .68263 1.01832 m .70037 1.04375 L .67678 1.02185 L p 0 .003 .574 r F P 0 g s .53439 .91573 m .52281 .909 L .52629 .90711 L p .489 .714 .971 r F P 0 g s .52629 .90711 m .54136 .91334 L .53439 .91573 L p .489 .714 .971 r F P 0 g s .57144 .92899 m .55641 .92061 L .56206 .91448 L p .399 .611 .935 r F P 0 g s .56206 .91448 m .57905 .9224 L .57144 .92899 L p .399 .611 .935 r F P 0 g s .40762 .92373 m .42497 .91771 L .43021 .92174 L p .296 .793 .836 r F P 0 g s .44635 .91777 m .43021 .92174 L .42497 .91771 L p .46 .874 .945 r F P 0 g s .70723 1.0608 m .69197 1.04057 L .71832 1.07233 L p 0 0 .436 r F P 0 g s .70037 1.04375 m .71832 1.07233 L .69197 1.04057 L p 0 0 .503 r F P 0 g s .46252 .91358 m .47871 .90939 L .48416 .9122 L p .433 .846 .966 r F P 0 g s .49764 .90756 m .48416 .9122 L .47871 .90939 L p .577 .854 .996 r F P 0 g s .48416 .9122 m .49764 .90756 L .50181 .90905 L p .528 .85 .998 r F P 0 g s .63521 .41191 m .64184 .37347 L .74028 .70859 L p .556 .395 .617 r F P 0 g s .55641 .92061 m .54136 .91334 L .54509 .9085 L p .484 .667 .944 r F P 0 g s .54509 .9085 m .56206 .91448 L .55641 .92061 L p .484 .667 .944 r F P 0 g s .42497 .91771 m .44227 .91245 L .44635 .91777 L p .46 .874 .945 r F P 0 g s .46252 .91358 m .44635 .91777 L .44227 .91245 L p .565 .891 .987 r F P 0 g s .51118 .90191 m .51261 .9103 L .50698 .90995 L closepath p .614 .773 .956 r F P 0 g s .51118 .90191 m .51808 .91003 L .51261 .9103 L closepath p .588 .753 .957 r F P 0 g s .73799 1.10603 m .72256 1.0826 L .74366 1.1198 L p 0 0 .32 r F P 0 g s .75353 1.13117 m .73799 1.10603 L .76208 1.15522 L p .663 .306 0 r F P 0 g s .76208 1.15522 m .77373 1.179 L p .7694 1.16875 L .663 .306 0 r F P 0 g s .7694 1.16875 m .75353 1.13117 L .76208 1.15522 L .663 .306 0 r F .75496 1.13974 m .77373 1.179 L .7718 1.17505 L p 0 0 .32 r F P 0 g s .7718 1.17505 m .73799 1.10603 L p .75496 1.13974 L 0 0 .32 r F P 0 g s .75496 1.13974 m .75097 1.13182 L .75006 1.13032 L 0 0 .32 r F .75496 1.13974 m .75006 1.13032 L .74366 1.1198 L F .75097 1.13182 m .73799 1.10603 L .74366 1.1198 L F .75496 1.13974 m .7365 1.10426 L .73669 1.10141 L p 0 .083 .62 r F P 0 g s .73669 1.10141 m .73938 1.06071 L p .75496 1.13974 L 0 .083 .62 r F P 0 g s .7365 1.10426 m .75496 1.13974 L .72786 1.09194 L p 0 0 .375 r F P 0 g s .72786 1.09194 m .72256 1.0826 L p .7241 1.08498 L 0 0 .375 r F P 0 g s .7365 1.10426 m .72786 1.09194 L .7241 1.08498 L 0 0 .375 r F .72256 1.0826 m .70723 1.0608 L .7365 1.10426 L p F P 0 g s .71832 1.07233 m .7365 1.10426 L .71248 1.06859 L p 0 0 .436 r F P 0 g s .71248 1.06859 m .70723 1.0608 L p .7086 1.06223 L 0 0 .436 r F P 0 g s .71832 1.07233 m .71248 1.06859 L .7086 1.06223 L 0 0 .436 r F .51118 .90191 m .50698 .90995 L .50181 .90905 L closepath p .634 .797 .961 r F P 0 g s .51118 .90191 m .52281 .909 L .51808 .91003 L closepath p .573 .747 .959 r F P 0 g s .51118 .90191 m .50181 .90905 L .49764 .90756 L closepath p .64 .813 .966 r F P 0 g s .44227 .91245 m .45953 .90807 L .46252 .91358 L p .565 .891 .987 r F P 0 g s .47871 .90939 m .46252 .91358 L .45953 .90807 L p .621 .871 .989 r F P 0 g s .54136 .91334 m .52629 .90711 L .52814 .90435 L p .556 .712 .945 r F P 0 g s .52814 .90435 m .54509 .9085 L .54136 .91334 L p .556 .712 .945 r F P 0 g s .61509 .52407 m .60831 .55999 L .53093 .48126 L p .667 .489 .636 r F P 0 g s .52956 .52789 m .53093 .48126 L .60831 .55999 L p .671 .497 .641 r F P 0 g s .53093 .48126 m .52956 .52789 L .45588 .56963 L p .744 .561 .648 r F P 0 g s .45588 .56963 m .45177 .52788 L .53093 .48126 L p .744 .561 .648 r F P 0 g s .51118 .90191 m .52629 .90711 L .52281 .909 L closepath p .585 .749 .956 r F P 0 g s .44367 .43826 m .44769 .48407 L .36731 .64863 L p .815 .624 .644 r F P 0 g s .36731 .64863 m .35775 .62277 L .44367 .43826 L p .815 .624 .644 r F P 0 g s .47871 .90939 m .49493 .90543 L .49764 .90756 L p .577 .854 .996 r F P 0 g s .51118 .90191 m .49764 .90756 L .49493 .90543 L closepath p .64 .811 .965 r F P 0 g s .45953 .90807 m .47676 .90476 L .47871 .90939 L p .621 .871 .989 r F P 0 g s .49493 .90543 m .47871 .90939 L .47676 .90476 L p .645 .834 .973 r F P 0 g s .74028 .70859 m .72752 .71081 L .63521 .41191 L p .556 .395 .617 r F P 0 g s .62854 .4499 m .63521 .41191 L .72752 .71081 L p .566 .405 .622 r F P 0 g s .51118 .90191 m .52814 .90435 L .52629 .90711 L closepath p .616 .748 .939 r F P 0 g s .47676 .90476 m .49398 .90265 L .49493 .90543 L p .645 .834 .973 r F P 0 g s .51118 .90191 m .49493 .90543 L .49398 .90265 L closepath p .649 .792 .951 r F P 0 g s .56206 .91448 m .54509 .9085 L .54515 .90133 L p .58 .659 .891 r F P 0 g s .54509 .9085 m .52814 .90435 L .52815 .90092 L p .613 .697 .903 r F P 0 g s .52815 .90092 m .54515 .90133 L .54509 .9085 L p .613 .697 .903 r F P 0 g s .51118 .90191 m .52815 .90092 L .52814 .90435 L closepath p .642 .733 .914 r F P 0 g s .57905 .9224 m .56206 .91448 L .56219 .90335 L p .541 .618 .879 r F P 0 g s .54515 .90133 m .56219 .90335 L .56206 .91448 L p .58 .659 .891 r F P 0 g s .51118 .90191 m .52284 .89389 L .52632 .89723 L closepath p .651 .703 .885 r F P 0 g s .51118 .90191 m .51809 .89149 L .52284 .89389 L closepath p .653 .708 .888 r F P 0 g s .51118 .90191 m .51259 .89049 L .51809 .89149 L closepath p .666 .727 .895 r F P 0 g s .51118 .90191 m .50693 .89107 L .51259 .89049 L closepath p .687 .746 .896 r F P 0 g s .51118 .90191 m .50174 .89306 L .50693 .89107 L closepath p .703 .754 .892 r F P 0 g s .51118 .90191 m .49757 .89602 L .50174 .89306 L closepath p .704 .755 .891 r F P 0 g s .51118 .90191 m .49489 .8994 L .49757 .89602 L closepath p .691 .757 .903 r F P 0 g s .51118 .90191 m .49398 .90265 L .49489 .8994 L closepath p .669 .77 .925 r F P 0 g s .51118 .90191 m .52632 .89723 L .52815 .90092 L closepath p .651 .713 .893 r F P 0 g s .47856 .89822 m .49489 .8994 L .49398 .90265 L p .699 .799 .927 r F P 0 g s .49398 .90265 m .47676 .90476 L .47856 .89822 L p .699 .799 .927 r F P 0 g s .72752 .71081 m .71492 .71505 L .62854 .4499 L p .566 .405 .622 r F P 0 g s .62184 .48733 m .62854 .4499 L .71492 .71505 L p .576 .415 .626 r F P 0 g s .59609 .9324 m .57905 .9224 L .57928 .90719 L p .497 .575 .865 r F P 0 g s .56219 .90335 m .57928 .90719 L .57905 .9224 L p .541 .618 .879 r F P 0 g s .46217 .8983 m .47856 .89822 L .47676 .90476 L p .723 .825 .928 r F P 0 g s .47676 .90476 m .45953 .90807 L .46217 .8983 L p .723 .825 .928 r F P 0 g s .60831 .55999 m .60149 .59497 L .52956 .52789 L p .671 .497 .641 r F P 0 g s .52819 .57235 m .52956 .52789 L .60149 .59497 L p .675 .505 .647 r F P 0 g s .52956 .52789 m .52819 .57235 L .46004 .60926 L p .745 .567 .654 r F P 0 g s .46004 .60926 m .45588 .56963 L .52956 .52789 L p .745 .567 .654 r F P 0 g s .54515 .90133 m .52815 .90092 L .52632 .89723 L p .644 .675 .865 r F P 0 g s .44769 .48407 m .45177 .52788 L .3769 .67341 L p .814 .627 .65 r F P 0 g s .3769 .67341 m .36731 .64863 L .44769 .48407 L p .814 .627 .65 r F P 0 g s .44573 .89959 m .46217 .8983 L .45953 .90807 L p .742 .849 .929 r F P 0 g s .45953 .90807 m .44227 .91245 L .44573 .89959 L p .742 .849 .929 r F P 0 g s .61319 .94461 m .59609 .9324 L .59645 .91305 L p .448 .53 .849 r F P 0 g s .57928 .90719 m .59645 .91305 L .59609 .9324 L p .497 .575 .865 r F P 0 g s .48391 .89049 m .49757 .89602 L .49489 .8994 L p .73 .771 .885 r F P 0 g s .49489 .8994 m .47856 .89822 L .48391 .89049 L p .73 .771 .885 r F P 0 g s .76918 1.15807 m .75353 1.13117 L .79284 1.22229 L p .698 .345 0 r F P 0 g s .77373 1.179 m .75496 1.13974 L .75517 1.13709 L p 0 .026 .58 r F P 0 g s .79085 1.21768 m .78989 1.21543 L p .77373 1.179 L .663 .306 0 r F P 0 g s .78989 1.21543 m .75353 1.13117 L p .77345 1.17835 L .663 .306 0 r F P 0 g s .77373 1.179 m .78989 1.21543 L .77345 1.17835 L .663 .306 0 r F .73938 1.06071 m .75844 1.09577 L .75517 1.13709 L p 0 .083 .62 r F P 0 g s .75517 1.13709 m .75496 1.13974 L p .75385 1.13411 L 0 .083 .62 r F P 0 g s .73938 1.06071 m .75517 1.13709 L .75385 1.13411 L 0 .083 .62 r F .78496 1.1868 m .76918 1.15807 L .81235 1.26988 L p .727 .378 0 r F P 0 g s .79284 1.22229 m .81235 1.26988 L .80701 1.25604 L p .698 .345 0 r F P 0 g s .80701 1.25604 m .76918 1.15807 L p .7918 1.21945 L .698 .345 0 r F P 0 g s .79284 1.22229 m .80701 1.25604 L .7918 1.21945 L .698 .345 0 r F .75517 1.13709 m .75555 1.13226 L p .77373 1.179 L 0 .026 .58 r F P 0 g s .75555 1.13226 m .75844 1.09577 L p .77274 1.1736 L 0 .026 .58 r F P 0 g s .77274 1.1736 m .77373 1.179 L .75555 1.13226 L 0 .026 .58 r F .77373 1.179 m .79284 1.22229 L .79085 1.21768 L p .663 .306 0 r F P 0 g s .56219 .90335 m .54515 .90133 L .54151 .89278 L p .631 .639 .84 r F P 0 g s .52632 .89723 m .54151 .89278 L .54515 .90133 L p .644 .675 .865 r F P 0 g s .54151 .89278 m .52632 .89723 L .52284 .89389 L p .657 .661 .842 r F P 0 g s .42921 .90202 m .44573 .89959 L .44227 .91245 L p .756 .87 .931 r F P 0 g s .44227 .91245 m .42497 .91771 L .42921 .90202 L p .756 .87 .931 r F P 0 g s .49225 .88306 m .50174 .89306 L .49757 .89602 L p .742 .753 .861 r F P 0 g s .49757 .89602 m .48391 .89049 L .49225 .88306 L p .742 .753 .861 r F P 0 g s .71492 .71505 m .70244 .72111 L .62184 .48733 L p .576 .415 .626 r F P 0 g s .61509 .52407 m .62184 .48733 L .70244 .72111 L p .586 .426 .631 r F P 0 g s .52284 .89389 m .53455 .8845 L .54151 .89278 L p .657 .661 .842 r F P 0 g s .53455 .8845 m .52284 .89389 L .51809 .89149 L p .669 .665 .837 r F P 0 g s .63037 .95915 m .61319 .94461 L .61371 .92115 L p .395 .481 .83 r F P 0 g s .59645 .91305 m .61371 .92115 L .61319 .94461 L p .448 .53 .849 r F P 0 g s .4702 .8853 m .48391 .89049 L .47856 .89822 L p .761 .779 .868 r F P 0 g s .47856 .89822 m .46217 .8983 L .4702 .8853 L p .761 .779 .868 r F P 0 g s .50267 .87772 m .50693 .89107 L .50174 .89306 L p .736 .737 .852 r F P 0 g s .50174 .89306 m .49225 .88306 L .50267 .87772 L p .736 .737 .852 r F P 0 g s .41261 .90551 m .42921 .90202 L .42497 .91771 L p .766 .891 .933 r F P 0 g s .42497 .91771 m .40762 .92373 L .41261 .90551 L p .766 .891 .933 r F P 0 g s .57928 .90719 m .56219 .90335 L .55674 .88877 L p .614 .606 .819 r F P 0 g s .54151 .89278 m .55674 .88877 L .56219 .90335 L p .631 .639 .84 r F P 0 g s .51809 .89149 m .52504 .87841 L .53455 .8845 L p .669 .665 .837 r F P 0 g s .52504 .87841 m .51809 .89149 L .51259 .89049 L p .689 .686 .842 r F P 0 g s .51401 .87597 m .51259 .89049 L .50693 .89107 L p .715 .714 .848 r F P 0 g s .50693 .89107 m .50267 .87772 L .51401 .87597 L p .715 .714 .848 r F P 0 g s .51259 .89049 m .51401 .87597 L .52504 .87841 L p .689 .686 .842 r F P 0 g s .55674 .88877 m .54151 .89278 L .53455 .8845 L p .655 .626 .809 r F P 0 g s .60149 .59497 m .59464 .62886 L .52819 .57235 L p .675 .505 .647 r F P 0 g s .5268 .61453 m .52819 .57235 L .59464 .62886 L p .68 .514 .654 r F P 0 g s .52819 .57235 m .5268 .61453 L .46422 .6467 L p .746 .574 .661 r F P 0 g s .46422 .6467 m .46004 .60926 L .52819 .57235 L p .746 .574 .661 r F P 0 g s .39593 .91001 m .41261 .90551 L .40762 .92373 L p .77 .91 .935 r F P 0 g s .40762 .92373 m .39024 .93033 L .39593 .91001 L p .77 .91 .935 r F P 0 g s .45177 .52788 m .45588 .56963 L .38651 .6971 L p .813 .631 .656 r F P 0 g s .38651 .6971 m .3769 .67341 L .45177 .52788 L p .813 .631 .656 r F P 0 g s .70244 .72111 m .69007 .72878 L .61509 .52407 L p .586 .426 .631 r F P 0 g s .60831 .55999 m .61509 .52407 L .69007 .72878 L p .595 .436 .637 r F P 0 g s .64765 .97618 m .63037 .95915 L .63109 .93172 L p .337 .43 .809 r F P 0 g s .61371 .92115 m .63109 .93172 L .63037 .95915 L p .395 .481 .83 r F P 0 g s .48273 .8719 m .49225 .88306 L .48391 .89049 L p .77 .749 .834 r F P 0 g s .48391 .89049 m .4702 .8853 L .48273 .8719 L p .77 .749 .834 r F P 0 g s .59645 .91305 m .57928 .90719 L .57203 .88546 L p .595 .574 .8 r F P 0 g s .55674 .88877 m .57203 .88546 L .57928 .90719 L p .614 .606 .819 r F P 0 g s .45642 .88043 m .4702 .8853 L .46217 .8983 L p .786 .785 .851 r F P 0 g s .46217 .8983 m .44573 .89959 L .45642 .88043 L p .786 .785 .851 r F P 0 g s .53455 .8845 m .5463 .87396 L .55674 .88877 L p .655 .626 .809 r F P 0 g s .5463 .87396 m .53455 .8845 L .52504 .87841 L p .673 .629 .798 r F P 0 g s .37915 .91544 m .39593 .91001 L .39024 .93033 L p .769 .927 .938 r F P 0 g s .57203 .88546 m .55674 .88877 L .5463 .87396 L p .649 .596 .781 r F P 0 g s .52504 .87841 m .53201 .86279 L .5463 .87396 L p .673 .629 .798 r F P 0 g s .53201 .86279 m .52504 .87841 L .51401 .87597 L p .698 .652 .801 r F P 0 g s .49839 .86186 m .50267 .87772 L .49225 .88306 L p .758 .721 .818 r F P 0 g s .49225 .88306 m .48273 .8719 L .49839 .86186 L p .758 .721 .818 r F P 0 g s .51401 .87597 m .51543 .85838 L .53201 .86279 L p .698 .652 .801 r F P 0 g s .51543 .85838 m .51401 .87597 L .50267 .87772 L p .73 .687 .809 r F P 0 g s .50267 .87772 m .49839 .86186 L .51543 .85838 L p .73 .687 .809 r F P 0 g s .69007 .72878 m .6778 .73785 L .60831 .55999 L p .595 .436 .637 r F P 0 g s .60149 .59497 m .60831 .55999 L .6778 .73785 L p .604 .447 .643 r F P 0 g s .59464 .62886 m .58775 .66154 L .5268 .61453 L p .68 .514 .654 r F P 0 g s .5254 .65433 m .5268 .61453 L .58775 .66154 L p .684 .523 .662 r F P 0 g s .5268 .61453 m .5254 .65433 L .46844 .6819 L p .747 .581 .669 r F P 0 g s .46844 .6819 m .46422 .6467 L .5268 .61453 L p .747 .581 .669 r F P 0 g s .8009 1.21745 m .78496 1.1868 L .8323 1.32206 L p .751 .407 0 r F P 0 g s .79284 1.22229 m .77373 1.179 L .77439 1.17214 L p 0 0 .54 r F P 0 g s .82675 1.30621 m .78496 1.1868 L p .81121 1.26642 L .727 .378 0 r F P 0 g s .77439 1.17214 m .77791 1.1356 L p .79195 1.21711 L 0 0 .54 r F P 0 g s .79195 1.21711 m .79284 1.22229 L .77439 1.17214 L 0 0 .54 r F .81235 1.26988 m .8323 1.32206 L .82675 1.30621 L p .727 .378 0 r F P 0 g s .81235 1.26988 m .82675 1.30621 L .81121 1.26642 L .727 .378 0 r F .61371 .92115 m .59645 .91305 L .58738 .88308 L p .573 .543 .784 r F P 0 g s .57203 .88546 m .58738 .88308 L .59645 .91305 L p .595 .574 .8 r F P 0 g s .36228 .92175 m .37915 .91544 L .37281 .93735 L p .761 .943 .941 r F P 0 g s .44259 .87588 m .45642 .88043 L .44573 .89959 L p .806 .788 .836 r F P 0 g s .44573 .89959 m .42921 .90202 L .44259 .87588 L p .806 .788 .836 r F P 0 g s .66507 .99585 m .64765 .97618 L .64861 .94498 L p .274 .375 .785 r F P 0 g s .63109 .93172 m .64861 .94498 L .64765 .97618 L p .337 .43 .809 r F P 0 g s .45588 .56963 m .46004 .60926 L .39614 .71968 L p .812 .636 .663 r F P 0 g s .39614 .71968 m .38651 .6971 L .45588 .56963 L p .812 .636 .663 r F P 0 g s .5463 .87396 m .55808 .86245 L .57203 .88546 L p .649 .596 .781 r F P 0 g s .55808 .86245 m .5463 .87396 L .53201 .86279 L p .672 .601 .767 r F P 0 g s .47318 .85958 m .48273 .8719 L .4702 .8853 L p .791 .744 .809 r F P 0 g s .4702 .8853 m .45642 .88043 L .47318 .85958 L p .791 .744 .809 r F P 0 g s .58738 .88308 m .57203 .88546 L .55808 .86245 L p .64 .569 .759 r F P 0 g s .53201 .86279 m .53899 .84475 L .55808 .86245 L p .672 .601 .767 r F P 0 g s .53899 .84475 m .53201 .86279 L .51543 .85838 L p .701 .625 .769 r F P 0 g s .6778 .73785 m .66561 .74812 L .60149 .59497 L p .604 .447 .643 r F P 0 g s .59464 .62886 m .60149 .59497 L .66561 .74812 L p .612 .458 .649 r F P 0 g s .3453 .92887 m .36228 .92175 L .35534 .94465 L p .744 .955 .943 r F P 0 g s .35534 .94465 m .33782 .95206 L .3453 .92887 L p .744 .955 .943 r F P 0 g s .4941 .8435 m .49839 .86186 L .48273 .8719 L p .773 .707 .79 r F P 0 g s .48273 .8719 m .47318 .85958 L .4941 .8435 L p .773 .707 .79 r F P 0 g s .51543 .85838 m .51686 .83777 L .53899 .84475 L p .701 .625 .769 r F P 0 g s .51686 .83777 m .51543 .85838 L .49839 .86186 L p .738 .665 .778 r F P 0 g s .58775 .66154 m .58084 .69287 L .5254 .65433 L p .684 .523 .662 r F P 0 g s .52399 .69164 m .5254 .65433 L .58084 .69287 L p .688 .533 .671 r F P 0 g s .5254 .65433 m .52399 .69164 L .47268 .7148 L p .747 .588 .679 r F P 0 g s .47268 .7148 m .46844 .6819 L .5254 .65433 L p .747 .588 .679 r F P 0 g s .49839 .86186 m .4941 .8435 L .51686 .83777 L p .738 .665 .778 r F P 0 g s .55808 .86245 m .56989 .85017 L .58738 .88308 L p .64 .569 .759 r F P 0 g s .56989 .85017 m .55808 .86245 L .53899 .84475 L p .669 .577 .743 r F P 0 g s .42869 .87163 m .44259 .87588 L .42921 .90202 L p .823 .79 .821 r F P 0 g s .42921 .90202 m .41261 .90551 L .42869 .87163 L p .823 .79 .821 r F P 0 g s .63109 .93172 m .61371 .92115 L .60281 .88187 L p .549 .513 .768 r F P 0 g s .58738 .88308 m .60281 .88187 L .61371 .92115 L p .573 .543 .784 r F P 0 g s .46004 .60926 m .46422 .6467 L .40578 .74115 L p .81 .64 .671 r F P 0 g s .40578 .74115 m .39614 .71968 L .46004 .60926 L p .81 .64 .671 r F P 0 g s .66561 .74812 m .65349 .75941 L .59464 .62886 L p .612 .458 .649 r F P 0 g s .58775 .66154 m .59464 .62886 L .65349 .75941 L p .621 .47 .656 r F P 0 g s .3282 .93673 m .3453 .92887 L .33782 .95206 L p .715 .963 .942 r F P 0 g s .33782 .95206 m .32026 .95942 L .3282 .93673 L p .715 .963 .942 r F P 0 g s .53899 .84475 m .54598 .82441 L .56989 .85017 L p .669 .577 .743 r F P 0 g s .54598 .82441 m .53899 .84475 L .51686 .83777 L p .702 .603 .743 r F P 0 g s .81235 1.26988 m .79284 1.22229 L .79359 1.21612 L p 0 0 .5 r F P 0 g s .84694 1.36109 m .8009 1.21745 L p .83105 1.31793 L .751 .407 0 r F P 0 g s .79359 1.21612 m .79785 1.18058 L p .81154 1.26491 L 0 0 .5 r F P 0 g s .81154 1.26491 m .81235 1.26988 L .79359 1.21612 L 0 0 .5 r F .8323 1.32206 m .85273 1.37915 L .84694 1.36109 L p .751 .407 0 r F P 0 g s .8323 1.32206 m .84694 1.36109 L .83105 1.31793 L .751 .407 0 r F .4636 .84611 m .47318 .85958 L .45642 .88043 L p .806 .738 .788 r F P 0 g s .45642 .88043 m .44259 .87588 L .4636 .84611 L p .806 .738 .788 r F P 0 g s .68263 1.01832 m .66507 .99585 L .66629 .96118 L p .209 .319 .758 r F P 0 g s .64861 .94498 m .66629 .96118 L .66507 .99585 L p .274 .375 .785 r F P 0 g s .60281 .88187 m .58738 .88308 L .56989 .85017 L p .63 .546 .741 r F P 0 g s .51686 .83777 m .51829 .8142 L .54598 .82441 L p .702 .603 .743 r F P 0 g s .51829 .8142 m .51686 .83777 L .4941 .8435 L p .743 .647 .753 r F P 0 g s .58084 .69287 m .5739 .7227 L .52399 .69164 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What radius and height do you recommend for \ the tank?"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["Solution"], "Subsubsection", Evaluatable->False, PageBreakWithin->Automatic, AspectRatioFixed->True], Cell[TextData[ "The cost of material is proportional to the surface area so we must minimize \ (side and two ends):"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(\(Clear[s, r, t]; \)\)], "Input"], Cell[CellGroupData[{ Cell["s[r_,h_] = 2Pi r h + 2 Pi r^2", "Input", AspectRatioFixed->True], Cell[BoxData[ \(2\ h\ \[Pi]\ r + 2\ \[Pi]\ r\^2\)], "Output"] }, Open ]], Cell[TextData[{ "Subject to the constraint on the volume: ", StyleBox["Pi r^2 h == 8000", "Input"] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell["lp = Lagrange[s[r,h], Pi r^2 h == 8000, {r,h}]", "Input", AspectRatioFixed->True], Cell[BoxData[ \({{h \[Rule] \(-20\)\ 2\^\(2/3\)\ \((\(-\(1\/\[Pi]\)\))\)\^\(1/3\), r \[Rule] \(-10\)\ 2\^\(2/3\)\ \((\(-\(1\/\[Pi]\)\))\)\^\(1/3\)}, { h \[Rule] \(20\ \((\(-2\))\)\^\(2/3\)\)\/\[Pi]\^\(1/3\), r \[Rule] \(10\ \((\(-2\))\)\^\(2/3\)\)\/\[Pi]\^\(1/3\)}, { h \[Rule] \(20\ 2\^\(2/3\)\)\/\[Pi]\^\(1/3\), r \[Rule] \(10\ 2\^\(2/3\)\)\/\[Pi]\^\(1/3\)}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["N[lp]", "Input", AspectRatioFixed->True], Cell[BoxData[ \({{h \[Rule] \(-10.838521402785779`\) - 18.77286974854767`\ I, r \[Rule] \(-5.41926070139288906`\) - 9.38643487427383504`\ I}, { h \[Rule] \(-10.838521402785779`\) + 18.7728697485476736`\ I, r \[Rule] \(-5.41926070139288995`\) + 9.38643487427383682`\ I}, { h \[Rule] 21.677042805571558`, r \[Rule] 10.838521402785779`}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["s[r,h] /. lp", "Input", AspectRatioFixed->True], Cell[BoxData[ \({1200\ \((\(-1\))\)\^\(2/3\)\ \((2\ \[Pi])\)\^\(1/3\), \(-1200\)\ \((\(-2\)\ \[Pi])\)\^\(1/3\), 1200\ \((2\ \[Pi])\)\^\(1/3\)} \)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["N[%]", "Input", AspectRatioFixed->True], Cell[BoxData[ \({\(-1107.16208918641734`\) + 1917.66099068497979`\ I, \(-1107.16208918641689`\) - 1917.66099068497837`\ I, 2214.32417837283379`}\)], "Output"] }, Open ]], Cell[TextData[{ "The minimum surface area is ", StyleBox["1200 (2Pi)^(1/3) == 2214.32", FontFamily->"Courier", FontWeight->"Bold"], " and occurs when\n", StyleBox["h == 20(4/Pi)^(1/3) == 21.677", FontFamily->"Courier", FontWeight->"Bold"], ",", StyleBox["\nr == 10(4/Pi)^(1/3) == 10.8385", FontFamily->"Courier", FontWeight->"Bold"] }], "Text", Evaluatable->False, AspectRatioFixed->True] }, Open ]] }, FrontEndVersion->"Macintosh 3.0", ScreenRectangle->{{0, 1024}, {0, 748}}, AutoGeneratedPackage->None, WindowToolbars->{"RulerBar", "EditBar"}, CellGrouping->Manual, WindowSize->{520, 509}, WindowMargins->{{Automatic, 67}, {Automatic, 2}}, PrintingCopies->1, PrintingPageRange->{1, Automatic}, PrivateNotebookOptions->{"ColorPalette"->{RGBColor, 128}}, ShowCellLabel->False, ShowCellTags->False, RenderingOptions->{"ObjectDithering"->True, "RasterDithering"->False}, CharacterEncoding->"MacintoshAutomaticEncoding", MacintoshSystemPageSetup->"\<\ 00<0004/0B`000002n88o?moogl" ] (*********************************************************************** Cached data follows. 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