00020002010016Linear Programming.ACT0002020018Linear Programming_1.EAC010000001958É Linear Programming_1.EAC†¤.ACT†6†9†CˆF†JŠ †R† kE Ž'*[†)¤w†ŠZŽ ŽWhen the constraints are:Ž: 8x+2yî20ŽKx+†6ŽY 2x+8˜xî+†)0[ŒƒandŠi objective fun† on Ž#is f=10x+5y, fiŽ1ŒÃ ‡F imum of f.–Ø‰q1st:j Transform–ÕŽFequaˆts †! ‡žphingîyî-4î’x+10[ˆx+6[Œ†#Ž-Œ%x-10Œ04\Œ9 CalculatorËˆK†OßˆU Rˆsolve(8x+2yî20,y)RŒ!yŒ)î-’ 8 ’ŒRx&Œb20Œ 2–x simplify(ans´s”g-’4°q+ŒÓ10–iŒáx+†ß6¸Þ”ÒŽ¹ X6¥82x+8ï82Ó88¡8lify(ans)RyŒî-’xŽ&-Œ.10Ž74Rˆ>ŒDˆHÀSGraph2D,† †Š3Š@†ˆLISTSYSˆL4N†Modify†;€†($ˆ<STATCALC¤†Uˆ(ˆŒ<¬\ˆ<Sequence†<,ˆxSheet†Š4†|’ŠŒ°’olveEq†´ˆ´Lwr0†Í(Upˆ<’tupFLG1H†<(Š<†Lis†{p†dD‰ˆPic†´ÜViewWindØ†‰,äŠ±‡@ôxäŠ ŠŒä( ä<´äP$Èäd’ÜäŽx <†ä† † H’’T’’`’’l’’x’’„’’’’œ’’¨’ ’´’!’À’"’Ì’#’Ø’$’ä’%’ð’&’ü’'‘@’(‘T’)‘hˆ¾Œ*‘|,’+‘8’ˆ‹§D¸†ä-†P† †.’\’0’h’1’t’2’€’3’Œ’4’˜’5’¤’E’°’F’¼’H’È’I’Ô’J’à’K’ì’L’ø’M‘,’N‘@’O‘T’‰^‹k(’Q‘|‰ŒR‘@’S‘¤ˆ–ŒT‹¸X††ä]††d† ˆ^’h’_’l’`’p’a’t’b’x’”’|’Œ•’ˆ’Í’ˆ2ŒÎ’ ’Ð’¬Ü††ˆŒ!†” system‰list’‡]=’‡‰’†ü‰’†ø‰’†ô~[LÁŽŽž¾ ¤@LÁŽŽž¾ þ@þ€þÀÿÿ@ÿxLÁLŽŽž¾ þ@þ€þÀÿÿ@ÿx‡¸LÁŽŽž¾ þ@þ€þÀÿÿ@ÿx‡¸LÁŠŽž° ††a†F seq_histb†NewFoldeŠ]ˆ3 ’ systemä]l†0’^]=” –˜††î<ˆxö<þxìðÒ´ emLÁŠŒsystˆä]list’^]= ö<þxþ´<† ††††’’,† †0†@†@PSheet1úê¬^Š2ž3ž 4ž 5ž ‡›ŠX3D´|PSheet1èê¬^Š2ž3ž 4ž 5ž ŒH3D Šˆi™”™††Š ˆŽ¸Dkäûx†††$’S’\¢œ”D²<‡Žh†(žŽ `–PÝÆ$´$“$–p”x”,ˆˆ ™€ `”˜$’.ˆ>(1…0qy`ˆH#Y‡uYƒ†U ˜–0–šH®Æ$ÊHš©š¶´¨5°´þaþ ‰¼†Œ˜ °˜’®`–$ ™†OŠq[Œ;†ˆD2nd:[ŒGraph each of the Œ.constraint equati†.\Œ6Œ6EditorÐˆ_†cÊŒâ SGraph2D† †Š3Š¤†ˆLISTSYSˆ°4N†Modify†;ä†($ˆ<STATCALC †Uˆ(ˆŒ< \ˆ<Sequencel†<,ˆxSheet†Š˜†|’ŠŒ’olveEq†´ˆ´Lwr”†Í(Upˆ ’tupFLG1¬†<(Š<†Lis†{Ô†dD‰ˆPic†ÜViewWind<†‰,x5Š±‡@X‡EH†y†^ŽˆÜŠ2’ˆ†Š3’ˆ Š4’ÀPä‡6Žˆ ŠÜäŽx à†ä† † ì’’ø’(’<’P’d(’x4’Œ@’ L’´X’ˆnŠËd’Üp’ð|’‘ˆ’ ‘”’!‘, ’"‘@¬’#‘T¸’$‘hÄ’%‘|Ð’&‘Ü’'‘¤è¸†ä(†ô† †)ˆŒ*(’+<’,P$’-d0’.x<’ˆŠH’1 T’2´`’3Èl’4Üx’5ð„’E‘’F‘œ’ˆ–‹/¨’I‘@´’J‘TÀ’K‘hÌ’L‘|Ø’M‘ä’N‘¤ð’O‹¸ü††äP††’Q’’R’ ’S’,’T’8’]’D†} ˆ^’H’_’L’`’ˆªŒa’ˆnŒb’X’”’\’Œ•’h’Í’t’Î’€’Ð’Œ‘@††ˆŒ!†” system‰list’‡]=’‡‰’†ü‰’†ø‰systemäb~LÁŽŽž¾ þ@þ€þÀÿÿ@éxLÁ†ŽŽž¾ þ@þ€þÀÿÿ@ÿx‡¾LÁŽŽž¾ þ@þ€þÀÿÿ@ÿxLÁŽŽž¾ þ@¶€††a†§ seq_histb†NewFoldeŒ¾† ’ systemä]l†0’^]=” –ø††î<ˆxö<ôxsystemä]list’^]= †,LÁŠŒö<þxþ´þðß,† ††††RX’’,† †0†@†@PSheet1úê¬^Š2úê¬^Sheet3ž4ž 5ž Ž8D†† †0†@†PŠ,1è”<†|š†|š†|š†|š˜| Šˆi™”™††Š ˆŽ ¸Dkäûxˆ½ŽÀ††$’S’\¢œ“²<‡Žh†(¦d‰G<y0 ˆ¬;x-4*x+1Œ†;x-x+6Šˆ”(-(x-10))/4’”80’`–PPÆ$´$’—ˆ±Ž`2Fu2FwŠxˆu2Š$–„)‡)‡ˆ ™hBRcŠ¨cx”sh¢0‘Xú(1…0qy`Xú#Y‡uYƒ† ˜–˜œ˜H®Æ$ÐH†Ž Žš´Ž ”=–øTøøÌ°Ø¬ð `cx”sg‡H ™†T‹Q[P†‰Y3rd:[ŒFind the corner points on Œ6ˆgraph w†+ow.[‡fŒW0Œ_10,Š1.33Ž4.67Œž&ˆ–1ª%10ŒT0[Œ]†ˆt4th:[ŽZRewrite the objective Œ fun†on’/ f=10x+5y=˜¸xŒÀy†Ž¶”ƒ5o Substitu†reach corner Žè point fo†¬`(.\1 CalculatorËˆæ†êßŠ‡^@RŠ 0Œ10†’Ž 5RŠ350ˆHšH1.33ŒS4.67ÐM36.6†d¢P†E6–[Ð53.2ŠPœè”ÑÒè10Šé*‰;ÀSGraph2D,† †Š3Š@†ˆLISTSYSˆL4N†Modify†;€†($ˆ<STATCALC¤†Uˆ(ˆŒ<¬\ˆ<Sequence†<,ˆxSheet†Š4†|’ŠŒ°’olveEq†´ˆ´Lwr0†Í(Upˆ<’tupFLG1H†<(Š<†Lis†{p†dD‰ˆPic†´ÜViewWindØ†‰,äŠ±‡@ôxäŠ ŠŒä( ä<´äP$Èäd’ÜäŽx <†ä† † H’’T’’`’’l’’x’’„’’’’œ’’¨’ ’´’!’À’"’Ì’#’Ø’$’ä’%’ð’&’ü’'‘@’(‘T’)‘hˆ¾Œ*‘|,’+‘8’ˆ‹§D¸†ä-†P† †.’\’0’h’1’t’2’€’3’Œ’4’˜’5’¤’E’°’F’¼’H’È’I’Ô’J’à’K’ì’L’ø’M‘,’N‘@’O‘T’‰^‹k(’Q‘|‰ŒR‘@’S‘¤ˆ–ŒT‹¸X††ä]††d† ˆ^’h’_’l’`’p’a’t’b’x’”’|’Œ•’ˆ’Í’ˆ2ŒÎ’ ’Ð’¬Ü††ˆŒ!†” system‰list’‡]=’‡‰’†ü‰’†ø‰’†ô~[·ÁŽŽž¾ ¤@·ÁŽŽž¾ þ@þ€þÀÿÿ@ÿx·Á·ŽŽž¾ þ@þ€þÀÿÿ@ÿx‡¸·ÁŽŽž¾ þ@þ€þÀÿÿ@ÿx‡¸·ÁŠŽž° ††a†F seq_histb†RFlashBooŠ]ˆ3 ’ systemä]l†0’^]=” –˜††î<ˆxö<þxìðÒ´ em·ÁŠŒsystˆä]list’^]= ö<þxþ´<† ††††’’,† †0†@†@PSheet1úJë^Š2ž3ž 4ž 5ž ‡›ŠX3D´|PSheet1èJë^Š2ž3ž 4ž 5ž ŒH3D Šˆi™”™††Š ˆŽ¸Dkäûx†††$’S’\¢œ”D²<‡Žh†(žŽ `–PÝÆ$´$“$–p”x”,ˆˆ ™€ `”˜$’.ˆ>(1…0qy`ˆH#Y‡uYƒ†U ˜–0–šH®Æ$ÊHš©š¶´¨5°´þaþ ‰¼†Œ˜ °˜’®`–$ ™†OŠ"[Š<Our function f has its Žminimum value at †vŒ>1.33ŒI4.67ŒT.[Œ]†ˆ¢5th:rTo†fve the exactŒ],twe findŠ! corner point Œ³ algebrically:RˆþŽÒ y=-4î’x+10Œ†x+6Œðx,yR†–x=ŒË43’$,ŒKy ,14 3[Œ The minimum value of f Œ(is:RŠ1Ž?4GŽ1’†)Ž]10Žp5RŠEŽ41’3ŠdŒ›[Œ£eAct020018Linear Programming_2.EAC01000000186eÉ Linear Programming_2.EAC†¤.ACT†6†9†CˆF†JŠ †R†]kE Ž'0[†)¤w[ˆ‘After you work through Ž¦±1, try Œ@ this problem.†pŠ°ŽxGiven:ŽConstraining conditi†: Ž7 4x+5yî12ŽH 5x+3ˆˆÉ‡? 10x+4ˆ8Žkxî†<*ˆƒ‡qObject†ˆ fun† o’f=8x+6yžµFind:‘ ,a)Corner point where minimum occurs.[Œb)MŽvalue of f †ˆ>1st:[ŒJTransform constraˆlŽequati† †" graphingŽ…and drag them ˆŸ:–kˆ¨\Œ° Calculator (†Ct)Ëˆ!†%ßˆÓ†×ˆRˆßŒåsolve(4x+5yî12,y)R†ÂŽ¼yîŽÏ-’ 4 ’7x&G12P5–xŒŽÀSGraph2D,† †Š3Š@†ˆLISTSYSˆL4N†Modify†;€†($ˆ<STATCALC¤†Uˆ(ˆŒ<¬\ˆ<Sequence†<,ˆxSheet†Š4†|’ŠŒ°’olveEq†´ˆ´Lwr0†Í(Upˆ<’tupFLG1H†<(Š<†Lis†{p†dD‰ˆPic†´ÜViewWindØ†‰,äŠ±‡@ôxäŠ ŠŒä( ä<´äP$Èäd’ÜäŽx <†ä† † H’’T’’`’’l’’x’’„’’’’œ’’¨’ ’´’!’À’"’Ì’#’Ø’$’ä’%’ð’&’ü’'‘@’(‘T’)‘hˆ¾Œ*‘|,’+‘8’ˆ‹§D¸†ä-†P† †.’\’0’h’1’t’2’€’3’Œ’4’˜’5’¤’E’°’F’¼’H’È’I’Ô’J’à’K’ì’L’ø’M‘,’N‘@’O‘T’‰^‹k(’Q‘|‰ŒR‘@’S‘¤ˆ–ŒT‹¸X††ä]††d† ˆ^’h’_’l’`’p’a’t’b’x’”’|’Œ•’ˆ’Í’ˆ2ŒÎ’ ’Ð’¬Ü††ˆŒ!†” system‰list’‡]=’‡‰’†ü‰’†ø‰’†ô~[LÁŽŽž¾ ¤@LÁŽŽž¾ þ@þ€þÀÿÿ@ÿxLÁLŽŽž¾ þ@þ€þÀÿÿ@ÿx‡¸LÁŽŽž¾ þ@þ€þÀÿÿ@ÿx‡¸LÁŠŽž° ††a†F seq_histb†NewFoldeŠ]ˆ3 ’ systemä]l†0’^]=” –˜††î<ˆxö<þxìðÒ´ emLÁŠŒsystˆä]list’^]= ö<þxþ´<† ††††’’,† †0†@†@PSheet1úê¬^Š2ž3ž 4ž 5ž ‡›ŠX3D´|PSheet1èê¬^Š2ž3ž 4ž 5ž ŒH3D Šˆi™”™††Š ˆŽ¸Dkäûx†††$’S’\¢œ”D²<‡Žh†(žŽ `–PÝÆ$´$“$–p”x”,ˆˆ ™€ `”˜$’.ˆ>(1…0qy`ˆH#Y‡uYƒ†U ˜–0–šH®Æ$ÊHš©š¶´¨5°´þaþ ‰¼†Œ˜ °˜’®`–$ ™†OŠq[Œ;†ˆD2nd:[ŒGraph each of the Œ.constraint equati†.\Œ6Œ6EditorÐˆ_†cÊŒâÀSGraph2D,† †Š3Š@†ˆLISTSYSˆL4N†Modify†;€†($ˆ<STATCALC¤†Uˆ(ˆŒ<¬\ˆ<Sequence†<,ˆxSheet†Š4†|’ŠŒ°’olveEq†´ˆ´Lwr0†Í(Upˆ<’tupFLG1H†<(Š<†Lis†{p†dD‰ˆPic†´ÜViewWindØ†‰,äŠ±‡@ôxäŠ ŠŒä( ä<´äP$Èäd’ÜäŽx <†ä† † H’’T’’`’’l’’x’’„’’’’œ’’¨’ ’´’!’À’"’Ì’#’Ø’$’ä’%’ð’&’ü’'‘@’(‘T’)‘hˆ¾Œ*‘|,’+‘8’ˆ‹§D¸†ä-†P† †.’\’0’h’1’t’2’€’3’Œ’4’˜’5’¤’E’°’F’¼’H’È’I’Ô’J’à’K’ì’L’ø’M‘,’N‘@’O‘T’‰^‹k(’Q‘|‰ŒR‘@’S‘¤ˆ–ŒT‹¸X††ä]††d† ˆ^’h’_’l’`’p’a’t’b’x’”’|’Œ•’ˆ’Í’ˆ2ŒÎ’ ’Ð’¬Ü††ˆŒ!†” system‰list’‡]=’‡‰’†ü‰’†ø‰’†ô~[LÁŽŽž¾ ¤@LÁŽŽž¾ þ@þ€þÀÿÿ@ÿxLÁLŽŽž¾ þ@þ€þÀÿÿ@ÿx‡¸LÁŽŽž¾ þ@þ€þÀÿÿ@ÿx‡¸LÁŠŽž° ††a†F seq_histb†NewFoldeŠ]ˆ3 ’ systemä]l†0’^]=” –˜††î<ˆxö<þxìðÒ´ emLÁŠŒsystˆä]list’^]= ö<þxþ´<† ††††’’,† †0†@†@ PSheet1ˆL€ dŠ2ž3ž 4ž 5ž ‡›ŠX3D´|PSheet1èê¬^Š2ž3ž 4ž 5ž ŒH3D Šˆi™”™††Š ˆŽ¸Dkäûx†††$’S’\¢œ”D²<‡Žh†(žŽ `–PÝÆ$´$“$–G$8!‰F™RH72GŠ0, fsˆ!5I† ˜qIqp† Y™ 1“gacŒ$ˆ+ˆ/†Xú(1…0qy`Xú#Y‡uYƒ† ˜–0–šH®Æ$ÌH@¬x’’¤˜°´âIâzÂ«†Œ˜ °˜’® `4 @g†h ™†OŠq[Œ;†ˆD3rd:[ŒFind the corner points on Ž"ˆgraph w†,ow a†2ŒSre†7Š@m below.[†ŽvŽ}Œ„,¸,æ;ŽÈ,¨uú\ Answersíˆ†íŠ‰—D;[[0,4.5]],[[1.4,1Ž 235,†12Ž 077,1.538Š*3,0]]†@[ˆG4th:[ŒSRewrite the objective Œp fun†on’/f=8x+6y=Œ˜xŒ y†Ž[Œµk Substitu†neach corner ŽŠ point fo†¬].\ CalculatorËˆk†oß‰*‡.R–‡‰AÀSGraph2D,† †Š3Š@†ˆLISTSYSˆL4N†Modify†;€†($ˆ<STATCALC¤†Uˆ(ˆŒ<¬\ˆ<Sequence†<,ˆxSheet†Š4†|’ŠŒ°’olveEq†´ˆ´Lwr0†Í(Upˆ<’tupFLG1H†<(Š<†Lis†{p†dD‰ˆPic†´ÜViewWindØ†‰,äŠ±‡@ôxäŠ ŠŒä( ä<´äP$Èäd’ÜäŽx <†ä† † H’’T’’`’’l’’x’’„’’’’œ’’¨’ ’´’!’À’"’Ì’#’Ø’$’ä’%’ð’&’ü’'‘@’(‘T’)‘hˆ¾Œ*‘|,’+‘8’ˆ‹§D¸†ä-†P† †.’\’0’h’1’t’2’€’3’Œ’4’˜’5’¤’E’°’F’¼’H’È’I’Ô’J’à’K’ì’L’ø’M‘,’N‘@’O‘T’‰^‹k(’Q‘|‰ŒR‘@’S‘¤ˆ–ŒT‹¸X††ä]††d† ˆ^’h’_’l’`’p’a’t’b’x’”’|’Œ•’ˆ’Í’ˆ2ŒÎ’ ’Ð’¬Ü††ˆŒ!†” system‰list’‡]=’‡‰’†ü‰’†ø‰’†ô~[·ÁŽŽž¾ ¤@·ÁŽŽž¾ þ@þ€þÀÿÿ@ÿx·Á·ŽŽž¾ þ@þ€þÀÿÿ@ÿx‡¸·ÁŽŽž¾ þ@þ€þÀÿÿ@ÿx‡¸·ÁŠŽž° ††a†F seq_histb†RFlashBooŠ]ˆ3 ’ systemä]l†0’^]=” –˜††î<ˆxö<þxìðÒ´ em·ÁŠŒsystˆä]list’^]= ö<þxþ´<† ††††’’,† †0†@†@PSheet1úJë^Š2ž3ž 4ž 5ž ‡›ŠX3D´|PSheet1èJë^Š2ž3ž 4ž 5ž ŒH3D Šˆi™”™††Š ˆŽ¸Dkäûx†††$’S’\¢œ”D²<‡Žh†(žŽ `–PÝÆ$´$“$–p”x”,ˆˆ ™€ `”˜$’.ˆ>(1…0qy`ˆH#Y‡uYƒ†U ˜–0–šH®Æ$ÊHš©š¶´¨5°´þaþ ‰¼†Œ˜ °˜’®`–$ ™†OŠ"[Š<Our function f has its Žminimum value at †vŒ>ŒEŒ.[ŒU†ˆš5th:jFind the exactŽRbykfindingŠ# corner point Œª algebrically:R‰7ŽÊœŒx,yŒâ ìT’}œ×of f isŽ\¨è†¢pRˆ\Œ Answeríˆ†íˆ&†tŒ-76Our function f has its minimum value at (x=1.4, y=1). Ž> Œs %The exactž9of f is 17.2!a[Ž¢eAct010008main.ACT0001020012eActivity Save.EAC010000001924É Linear Programming_1.EAC†¤.ACT†6†9†CˆF†JŠ †R† kE Ž'*[†)¤w†ŠZŽ ŽWhen the constraints are:Ž: 8x+2yî20ŽKx+†6ŽY 2x+8˜xî+†)7>andŠh objective fun† on is f=10x+5y, fiŽ)‡6 imum of f.–È‰a1st¢ ,Transform–Ä equaˆ]s † ‡†phingÖ†Ò-4î’x+1†å-x+6[yîŽ-Œx-10Œ 4\Œ) CalculatorËˆ;†?ßˆE Rˆsolve(8x+2yî20,y)RŒ!yŒ)î-’ 8 ’ŒRx&Œb20Œ 2–x simplify(ans´s”g-’4°q+ŒÓ10–iŒáx+†ß6¸Þ”ÒŽ¹ X6¥82x+8ï82Ó88¡8lify(ans)RyŒî-’xŽ&-Œ.10Ž74Rˆ>ŒDˆHÀSGraph2D,† †Š3Š@†ˆLISTSYSˆL4N†Modify†;€†($ˆ<STATCALC¤†Uˆ(ˆŒ<¬\ˆ<Sequence†<,ˆxSheet†Š4†|’ŠŒ°’olveEq†´ˆ´Lwr0†Í(Upˆ<’tupFLG1H†<(Š<†Lis†{p†dD‰ˆPic†´ÜViewWindØ†‰,äŠ±‡@ôxäŠ ŠŒä( ä<´äP$Èäd’ÜäŽx <†ä† † H’’T’’`’’l’’x’’„’’’’œ’’¨’ ’´’!’À’"’Ì’#’Ø’$’ä’%’ð’&’ü’'‘@’(‘T’)‘hˆ¾Œ*‘|,’+‘8’ˆ‹§D¸†ä-†P† †.’\’0’h’1’t’2’€’3’Œ’4’˜’5’¤’E’°’F’¼’H’È’I’Ô’J’à’K’ì’L’ø’M‘,’N‘@’O‘T’‰^‹k(’Q‘|‰ŒR‘@’S‘¤ˆ–ŒT‹¸X††ä]††d† ˆ^’h’_’l’`’p’a’t’b’x’”’|’Œ•’ˆ’Í’ˆ2ŒÎ’ ’Ð’¬Ü††ˆŒ!†” system‰list’‡]=’‡‰’†ü‰’†ø‰’†ô~[LÁŽŽž¾ ¤@LÁŽŽž¾ þ@þ€þÀÿÿ@ÿxLÁLŽŽž¾ þ@þ€þÀÿÿ@ÿx‡¸LÁŽŽž¾ þ@þ€þÀÿÿ@ÿx‡¸LÁŠŽž° ††a†F seq_histb†NewFoldeŠ]ˆ3 ’ systemä]l†0’^]=” –˜††î<ˆxö<þxìðÒ´ emLÁŠŒsystˆä]list’^]= ö<þxþ´<† ††††’’,† †0†@†@PSheet1úê¬^Š2ž3ž 4ž 5ž ‡›ŠX3D´|PSheet1èê¬^Š2ž3ž 4ž 5ž ŒH3D Šˆi™”™††Š ˆŽ¸Dkäûx†††$’S’\¢œ”D²<‡Žh†(žŽ `–PÝÆ$´$“$–p”x”,ˆˆ ™€ `”˜$’.ˆ>(1…0qy`ˆH#Y‡uYƒ†U ˜–0–šH®Æ$ÊHš©š¶´¨5°´þaþ ‰¼†Œ˜ °˜’®`–$ ™†OŠq[Œ;†ˆD2nd:Ž$'Graph each of the constraint equati†.\Œ/Œ/EditorÐˆW†[ÊŒÚ SGraph2D† †Š3Š¤†ˆLISTSYSˆ°4N†Modify†;ä†($ˆ<STATCALC †Uˆ(ˆŒ< \ˆ<Sequencel†<,ˆxSheet†Š˜†|’ŠŒ’olveEq†´ˆ´Lwr”†Í(Upˆ ’tupFLG1¬†<(Š<†Lis†{Ô†dD‰ˆPic†ÜViewWind<†‰,x5Š±‡@X‡EH†y†^ŽˆÜŠ2’ˆ†Š3’ˆ Š4’ÀPä‡6Žˆ ŠÜäŽx à†ä† † ì’’ø’(’<’P’d(’x4’Œ@’ L’´X’ˆnŠËd’Üp’ð|’‘ˆ’ ‘”’!‘, ’"‘@¬’#‘T¸’$‘hÄ’%‘|Ð’&‘Ü’'‘¤è¸†ä(†ô† †)ˆŒ*(’+<’,P$’-d0’.x<’ˆŠH’1 T’2´`’3Èl’4Üx’5ð„’E‘’F‘œ’ˆ–‹/¨’I‘@´’J‘TÀ’K‘hÌ’L‘|Ø’M‘ä’N‘¤ð’O‹¸ü††äP††’Q’’R’ ’S’,’T’8’]’D†} ˆ^’H’_’L’`’ˆªŒa’ˆnŒb’X’”’\’Œ•’h’Í’t’Î’€’Ð’Œ‘@††ˆŒ!†” system‰list’‡]=’‡‰’†ü‰’†ø‰systemäb~LÁŽŽž¾ þ@þ€þÀÿÿ@éxLÁ†ŽŽž¾ þ@þ€þÀÿÿ@ÿx‡¾LÁŽŽž¾ þ@þ€þÀÿÿ@ÿxLÁŽŽž¾ þ@¶€††a†§ seq_histb†NewFoldeŒ¾† ’ systemä]l†0’^]=” –ø††î<ˆxö<ôxsystemä]list’^]= †,LÁŠŒö<þxþ´þðß,† ††††RX’’,† †0†@†@PSheet1úê¬^Š2úê¬^Sheet3ž4ž 5ž Ž8D†† †0†@†PŠ,1è”<†|š†|š†|š†|š˜| Šˆi™”™††Š ˆŽ ¸Dkäûxˆ½ŽÀ††$’S’\¢œ“²<‡Žh†(¦d‰G<y0 ˆ¬;x-4*x+1Œ†;x-x+6Šˆ”(-(x-10))/4’”80’`–PPÆ$´$’—ˆ±Ž`2Fu2FwŠxˆu2Š$–„)‡)‡ˆ ™hBRcŠ¨cx”sh¢0‘Xú(1…0qy`Xú#Y‡uYƒ† ˜–˜œ˜H®Æ$ÐH†Ž Žš´Ž ”=–øTøøÌ°Ø¬ð `cx”sg‡H ™†T‹Q[P†‰Y3rd:Ž+Find the corner points onŠgraph w†$ow.[‡^ŒO0ŒW10Œa,1.33Ž4.67Œ,˜&ˆ–1ª%10ŒT0[Œ]†ˆf4th:ŽRewrite the objective fun† on:[Ž f=10x+5y=˜ŠxŒ¸y†Ž®”{5@ !Substitu†keach corner point for ®X.\! CalculatorËˆÖ†ÚßŠ‡@@RŠ 0Œ10†’Ž 5RŠ350ˆHšH1.33ŒS4.67ÐM36.6†d¢P†E6–[Ð53.2ŠPœè”ÑÒè10Šé*‰;ÀSGraph2D,† †Š3Š@†ˆLISTSYSˆL4N†Modify†;€†($ˆ<STATCALC¤†Uˆ(ˆŒ<¬\ˆ<Sequence†<,ˆxSheet†Š4†|’ŠŒ°’olveEq†´ˆ´Lwr0†Í(Upˆ<’tupFLG1H†<(Š<†Lis†{p†dD‰ˆPic†´ÜViewWindØ†‰,äŠ±‡@ôxäŠ ŠŒä( ä<´äP$Èäd’ÜäŽx <†ä† † H’’T’’`’’l’’x’’„’’’’œ’’¨’ ’´’!’À’"’Ì’#’Ø’$’ä’%’ð’&’ü’'‘@’(‘T’)‘hˆ¾Œ*‘|,’+‘8’ˆ‹§D¸†ä-†P† †.’\’0’h’1’t’2’€’3’Œ’4’˜’5’¤’E’°’F’¼’H’È’I’Ô’J’à’K’ì’L’ø’M‘,’N‘@’O‘T’‰^‹k(’Q‘|‰ŒR‘@’S‘¤ˆ–ŒT‹¸X††ä]††d† ˆ^’h’_’l’`’p’a’t’b’x’”’|’Œ•’ˆ’Í’ˆ2ŒÎ’ ’Ð’¬Ü††ˆŒ!†” system‰list’‡]=’‡‰’†ü‰’†ø‰’†ô~[·ÁŽŽž¾ ¤@·ÁŽŽž¾ þ@þ€þÀÿÿ@ÿx·Á·ŽŽž¾ þ@þ€þÀÿÿ@ÿx‡¸·ÁŽŽž¾ þ@þ€þÀÿÿ@ÿx‡¸·ÁŠŽž° ††a†F seq_histb†RFlashBooŠ]ˆ3 ’ systemä]l†0’^]=” –˜††î<ˆxö<þxìðÒ´ em·ÁŠŒsystˆä]list’^]= ö<þxþ´<† ††††’’,† †0†@†@PSheet1úJë^Š2ž3ž 4ž 5ž ‡›ŠX3D´|PSheet1èJë^Š2ž3ž 4ž 5ž ŒH3D Šˆi™”™††Š ˆŽ¸Dkäûx†††$’S’\¢œ”D²<‡Žh†(žŽ `–PÝÆ$´$“$–p”x”,ˆˆ ™€ `”˜$’.ˆ>(1…0qy`ˆH#Y‡uYƒ†U ˜–0–šH®Æ$ÊHš©š¶´¨5°´þaþ ‰¼†Œ˜ °˜’®`–$ ™†OŠ"[Š<,(Our function f has its minimum value at †nŒ61.33ŒA4.67ŒL.[ŒU†ˆš5th:Ž?To†]ve the exactŒ\ , we findŠcorner point algebrically:RˆæŽº y=-4î’x+10Œ†x+6ŒØx,yR††ŽêxŽó=Œ»43’$,ŒKy ,1’-Žå)The minimum value of f is:RŠ4Ž3Ž1’†)0Ž15RŠEŽ41’3ŠdŒ\[ŒdeAct0301d70020000800000000134fxœíÝ]Œ\Õað¹ãµ½kÖa¼;$œdN²„!ñ~8!Š[™ú;¬±M»Ø‹¼ª¿°×`R¹*gã¬·Žk"Y´²"TYiêF4B-IyH$¨áaT¡*ªx@J#è¹3wÎÞX ´ ýý¤³÷œÿÜ;÷Ü™»;wÎ=˜ä¹J{ò\Çpò³BOZ ™7Z> Þp|tO¡^ƒ£…[ …Ýcw9ph÷lI_½uÝW …F{Q£Ý_o_ýJóñ¾|{èÆ5× ®˜ÝøbKûÚ–öZÚŸoi¯li¶´ZÚý-í¾–vKÿú[ú×ßÒ¿þ–þõ·ô¯¿¥ý-ýëoé_Kÿú[ú×ßÒ¿¾–þõµô¯¯¥}-ýëké__KÿúZú××Ò¿¾–þõµôï‹-ïÿµ-í/´´?ßÒ^Ùzþ´´ZÚý-í¾ØnÒÖÖÖÖ~Û{FÆöæÛ«¯Û2tË[o¨·“BaýØÞ‘õû„+ŒÏ„‡—o.ª×Ù¿«mh.H·)/jfi¶°-üèL³¡Û¶o»u[š•Âºƒé¶›öí»ûþtO‹Ó•—§ëmÛ~Ýöð!°&d»BÖ¾9ËÛ.Þ_(o«g£÷Ý»c´P¸8íK¥¾í®ÑÑñú]üZøq$fõ¾”Æc¶o÷½£ëîI³´Ïmó›ÙÐ}BöjÚç¤™Ý¼?dK 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