0002000201000aLadder.ACT000102000aLadder.EAC010000009ad8É Ladder-new.EAC† Œ.ACT† †#†-ˆ0†4† Š†G ~I Ž'Á[ˆ4A lŠc problem†ŠX\Œ Author툈XíŽxŒ.Professor Wei-Chi Yang ŒLRadford University Ž9 e-mail: wy†9@rŒ2.eduŽE!URL: http://www.–#/Š5‡/ŽÖŽ Object†vs:[ŽŽP1.†Ĩ( will use ClassPad to explore a Calculus‘ that can be found in many 1@ctextbooks. And see how we can generate many different scenarios.[Q2. We will use its dynamic†Fometry a†bCAS features to help†2†b ke conjecŒŒX%before solving problems†L alytically.[Œ…Ž[†—ˆ˜ Example 1.ŒĻN A†nce q†‡et ta†ē runs parallelˆ–a buildˆrat a†ûstaˆ6of pŒ9from ŽþWthe’2 . What is ˆlengthˆ8ˆ$shortest ladder tˆ*‰jreachŒTˆLgrou‡\+over the fence toŠ wall ofŠbuilding?[Note. ˆ #1For demonstration purpose, we useŒ^ollow†Fs:[Œ9$#1. We set p=4 and q=6 respectively.Ž+J2. T†E point E inœT Geometry Strip isžcoor†ž ate (0,0).\ŒķExample 1=>play‹animŠÂΈį†ëΆņŠ ††‰ 3X "‡ `63H‚`† ` cv"0#† b$Bf8Š ŽD‰>v’K†r‰PŽ A† 'LŽ'‰rC˜ u˜x††yŽ †Ž ČČ  Ž'†-4fˆ8Q)u`Ž CAbPŽH ('€Ž$SWE€ Y5‡pŽeWv0r1H0yaY“@Ž`‡Vss0Žl–%SWŽ`xrtŽ)#gŽ„'’v‘Žœ@ˆ`ŽU@e•ŽĻqyeŽīA””ŽĻyŠį6UpŠŨ†ĻeYt"Žä&CŽ$A„†T“)“ŽĻX(‡pŽCU!Ž$X˜1  #$ƒ‘Da3‹ˆccccP‰”‹pDƒ‹o9@B‹3™f’(a0pŽ #$€`Ž‘&f Ž$`xf3y8@Ž<yaŽ0–’@Ž0‘y48Œ`†rQ0PŒls0Œxv‚ rh’ g3i„Žb€”•HXF˜d€ŒīT6I1ŽœPUu)`G62ŽØD!5SŽĻA—I@e–ŽĀ@gbEBeb5Gi‡` W'!Āw(—Iü‰7D†‡OÄH ‹3‰\ˆÃH!‹EB‰4Ð6–°‡x%Šf – Š†U†UŽCEC††Š” ž’ †3†8ČČ" †J)gQP†T Y™8Cg0Œ ca„ŠgY™„C$wŠ7Y™r`ŽGŽ7aHF€ŠH˜”’TuŠu ˜H˜Š ™ ™bŠ< ™IBXpŠ$™‰e`ŠĨ ™B$2TŠ ™ ‰aQŽ SUŠĻ))h7 QX0@Œæv8T ŒōŠë5G8Ž$qˆˆŠÁ‡tŽ<Y ŠĀ'%d:y†i5‹AX™ ?V1RŽHx’$xŽœ 8)AŽx2ˆw‹Ŧ‘p`QuaX8w €Ž V) ‘@ŽT# U0Ž$R!P#y5$H1PP`ŽHFD“<DdyT`B‘”†lA'8Ql9r'™l8(xŽ6–)Yœ5y `4x„Iī3˜SBĀ3Aƒ<30—ī3t5Ā3eI’Ø4cPŽü6$(ð8eR@ðB2S’Ø‘!E0SUp‰<bux&‘Due™xT” pP8† ‡gČČ € ‹p†‡y(†'EŠōY™WC &Œ `e"‹RY™—‹ŽY™0rpŒ0#1CXY™%2PŒ ƒ…#`ŒY4E0Œ$U%1Fˆ1 ˜G`2tŠ ™‘4Ž F”'QpŒG58Œ0™b!sŽ BŠT#•8( IB$!Š`w—6gŒ$ 2–ŠĻF‡ŠĻˆ–€ŒH7w8Ž•u'ŽHc"wRŠĖEq7%Ž<GU9IŽTvFV$ Œ D‰vg„Cr‹†† Y`T‹(WPG@Œ UC€vŽ,S@‘„ŒžQB"t0IH"‹GYEXŽ\EvX#‹dD5#`B1dV‹ @qITŽx9!%€7—x@Ž 6Uyh`Ž5DgPŽ$4Q‰ Ž03x130q’€T3'U9ˆ_ˆ`3‡ 5wH6yBv$9C&E0ŽC$ŽœH34vœUEƒ‡e1•‚$yH1œ™VŽx† †âČČ €Šŧˆï†ô(†'EˆþY™WC &ŠnY™`e"ŠþY™—‹Y™0rpŒ0#1CX‰9†<%2‹"Y™ƒ…#Ž<Y4ŒÎY™U%1FŠl ˜G`2tŠ ™‘4Ž F”'QŠ` ™G58Šœ ™b!sŽ B"#•8(0IB$!Ž w—6gŒ 2–ˆ#†$F‡PŒ0ˆ–€Œ<7w8Ž•u'ŽHc"wR`Œ`Eq7%Ž<GU9IŽTvFV$ Œ„ D‰vg„CrŒ<†† Y`TŒ€WPG@Ž UC€vŒhS@‘„ŒžQB"t0IH"pŽ<GYEX0EvX#D5#`B1dVŒø@qITx9!%Œø7—x„6Uyh„5Dg’Ļ4Q‰Œā3x1(30q’€T3'U9x3‡ˆýŒTw(6yBv9C&E0Ž C$ŽH34v@Ž$UEƒ‡PŽ0e1•‚$yH1 ŽH™VŽT† †_ČČ €Šˆl†q(†'Eˆ{Y™WC &ŠnY™`e"`Œ—ŠzY™0rpŒ0#1CXŠķY™%2Š’Y™ƒ…#Ž<Y4ŒÎY™U%1FŠl ˜G`2tŠ ™‘4Ž F”'QŠ` ™G58Šœ ™b!sŽ B"#•8( IB$!ŠĖw—6g: 2–^F‡:ˆ–€ŒH7w8R•u'ŽHc"wR‹Eq7%vGU9IēvFV$  D‰vgŒ Cr`Š†† Y`TPŠ WPG@Ž UC€vŽ,S@‘„Ž$QB"t0IH"pŽ<GYEXŽ\EvX#0ŽTD5#`B1dVˆ—ˆl@qITx9!%€Ž„7—x„6UyhŽ°5Dg’Ļ4Q‰Œā3x1Ā30q’€T3'U9x3‡ˆýŒTwH6yBvœ9C&EĀC$üH34v“ Eƒ‡Øe1•‚‘ yH1œ™VŽð† ‡’ČČ €‹›‰“‹‚(†'E‹ĒY™WC &Y™`e"`Ž —@Œ0rpŒ$#1CXˆ/†0%2PŒ<ƒ…#Ž<Y4E0ŒTU%1Fˆ` ˜G`2tŠ ™‘4Ž F”'QŠ` ™G58Œ0™b!sŽ BŠT#•8( IB$!Š`w—6gŠĖ 2–ˆŨ†0F‡ŠĻˆ–€ŒH7w8Ž•u'ŽHc"wR‹Eq7%Ž<GU9IŽTvFV$ Œ D‰vg„Cr‹D†† Y`T‹(WPG‹XUC€vŽ,S@‘„‹ˆQB"t0IH"‹|GYEXŽ\EvX#0D5#PŽ B1dVˆˆ@qIT9!%€Ž07—x@Ž<6Uyh`ŽH5DgH4Q‰ Ž`3x1`30q’€T3'U9x3‡„5wH6yBvœ9C&EĀC$ŽĖH34vœUEƒ‡Øe1•‚$yH1œ™VŽð† ‡ČČ €Šŧ‰‡$(†'E‰.Y™WC &ŠnY™`e"ŠþY™—‹Y™0rpŒ0#1CX‰i†<%2‹jY™ƒ…#Ž<Y4ŒÎY™U%1FŠl ˜G`2tŠ ™‘4Ž F”'Qp ™G58Œ b!sŽ B0ˆ$#•8( IB$!Š0w—6gŒ$ 2–ˆS†0F‡PŒ<ˆ–€ŒH7w8Ž•u'ŽHc"wR`ŒlEq7%Ž<GU9IŽTvFV$ Œ D‰vg„CrŒ<†† Y`TŒ€WPG@Œ UC€vŒhS@‘„ŒžQB"t0IH"‹GYEX0EvX#D5#`B1dVŒø@qITx9!%Œø7—x„6Uyh„5Dg’Ļ4Q‰Œā3x1(30qŒø3€€3'U9ˆ Š ‡ 5w Ž$6yBv$9C&E0Ž<C$ŽHH34v@ŽTUEƒ‡PŽ`e1•‚$yH1T™VŽx††ŽČČ €Šsˆ›† (†'EˆŠY™WC &ŠnY™`e"`Œ—ŠzY™0rpŒ0#1CXˆå†<%2Š’Y™ƒ…#Ž<Y4ŒÎY™U%1FŠl ˜G`2tŠ ™‘4Ž F”'QŠ` ™G58Šœ ™b!sŽ B"#•8( IB$!ŠĖw—6g: 2–‚F‡:ˆ–Ķ7w8R•u'c"wR`Œ Eq7%PŒGU9Iˆ#†$vFV$ Œ0 D‰vgŽ<CrŒ<†† Y`TŒDWPG@Ž UC€vŒhS@‘„Ž$QB"t0IH"pŽ<GYEX0EvX#0ŽTD5#`B1dVŒĪ@qITx9!%€Ž„7—x„6Uyh„5Dg’Ļ4Q‰Œā3x1Ā30q’€T3'U9x3‡ˆýŒTwH6yBvœ9C&EĀC$üH34v“ Eƒ‡Øe1•‚‘ yH1| ™V†† ČČ € Š††(†'Eˆ'Y™WC &Œ `e"`Œ—@Œ$0rpŒ0#1CXŠbY™%2PŒHƒ…#Ž<Y4E0Œ`U%1FŠl ˜G`2tŠ ™‘4Ž F”'QŠ` ™G58Šœ ™b!sŽ BŠT#•8( IB$!ŠĖw—6gŠĖ 2– F‡ŠĻˆ–€ŒH7w8Ž•u'ŽHc"wR‹Eq7%Ž<GU9I^vFV$‹W D‰vg„Cr‹D‹‚‹pY`T‹(WPG‹XUC€vŽ,S@‘„QB"tPŽ IH"pŽGYEX`Ž$EvX#0Ž0D5#0B1dVˆGˆH@qITH9!%€Ž`7—x@Žl6UyhT5Dgx4Q‰ Ž3x130q’€T3'U9x3‡„5wH6yBvœ9C&EĀC$üH34vœUEƒ‡Øe1•‚‘ yH1œ™VŽð†‡BČČ €Šŧ‰O‡T(†'EˆY™`e"‹FY™0r‹^Y™%2‹vY™Y4ŒžY™G`2tŠ  ™F”'QŠ0 ™b!sŽ #•8(0w—6gŒ F‡PŒ7w8Ž c"wR`Œ0GU9Iˆ;†< D‰vgŽHPQUƒŒ •3p2ŒTT`gsŒ<’'ƒŠ`pxŠ0`ƒ 4 Bp5Gu Œ0'@!Š„` 5‚vŠĀ`VTxˆË`•A7”ŠĖ`FƒT€Œ$0H™Ž tP˜Ž$††QY`TŒøUC€vŒėQB"tGYEXD5#0@qIT<7—x@Žø5Dg’T3x1L3€Š˜ˆ‡L6yBv C$‘@UEƒ‡”yH1 )A43PŒ h‚5@ŒaF„ €Š$'ƒpŽ0•‘‡ˆ;†<$U’pŒHWEBŒ01CRŒ0y")…0Š`eR13ŒTsugŽ•'hŒT%"†r`ŽH1W—ˆĻ ™ˆˆŽ†ū9ČĀ ŠĮ2ˆķÐ6–°†tR˜ˆ°E3•YˆŒ žC Length: ĸĸü†žc†’ŒÄČ ‹2G˜k‰Eˆ `e(sV3‹VžK‡lĀHCH˜Ū“)“(˜ĻĒžôõ 1@†Ð6–°†‡E!9ˆ ™9U˜ˆ ˆ*ˆ.Ž Coord: ĸĸü† †Ž†iÅH Šrˆ†{ÁH Œ„D˜„…‡CFQ4†Yˆ’Tžƒž†ÎÄ@ ’eˆ†ä ČȎa–í”@†”âĪiŽKˆæÄH"”a†čŠaˆ{ˆ‰ˆĒ‹Œð!ŽðF™t‰‡†`–”ĒðœĘb‡· Ð6–°††† U‘ƒ† ‡–†† ††4Æ@  ˆ†Žˆ"ˆ†^ÅH ’*††t ËČ#Œp@†„‹’r†˜—˜Ž(ˆKˆ‚ʋˆˆaĸĸĸĸŽ"ˆq!†‹5†‹œĩ"†ĸ ČĀ˜Š™ ywR#ĐˆÅH"“‹ A–ÍŒ+† `I33334†%”Z”ģ$ĒɈnˆŸ%ŠãH!œã"Ð $y1 ™†† † Y™ˆŒ ™y1D‚vˆ$˜†&†-Į@  0†'†C ËĀ  Š@†RÐ6–°† ”W†’ ˜†€Š((†‰ÄH Š\C–GeF„18† ”F E)†CÄH#”ŸˆYˆŸ*Š˜đˆoˆ†ĸ+ĒˆŨ,‡ČȐ˜ŨrE%A†s;5$’6y‹G4•Bˆu`‡W-ĒvŽX.‡vÆ@"’Ž/ĒŠˆ40†Œ œŒÐ6–°††Š’ –`†'† †1†4Ä@  ˆ0†/dŽ"ˆ2Œ*!š*"†3†xÅ D4ˆŽČČ Œ@†ÞĪ5†r!šœˆn4W–2g”‰f€†6ĒˆŲˆ7ĒA'†8†™ †™Œóq˜š™2 ywR#‡U ‡˜ēŽa†9‹H!œ{ˆa:ĢXˆ0ˆ{ŒĀ•ŠōĀŽ+98†;†ĮH! † ††'†<†"!Ä@ ”"†4'W–2g•†€†*† )Ž3%†$Ž)#†(††4†††,††ŽkŽ-ˆˆ9ˆ5ˆMˆ9ˆŒˆĶA0–E.†-ŽM††††ŠŸ†† † † † † ††Žõˆ…!‘[ŽŠExploration and ApproximŠ:Ž. L-play the†$ŽŠ.collectŠdata forŠ'point HŠSˆ7 distance GHŽ‚R-drag the x-coordinate of H andŠlength GH back toŠ0 curve, we see a mimimum.[Ž Analytically:[ŒFLet BE=p, DE=q, EH=aŠrGF=b,ˆ’nˆ\consider two s†cla† riangles ˆmGFDŽXŠķ† ˆ…DEHŒŽ.†zŽ{WŒģŒĐGFŒē’E=DEŒÎEHš or<bŒíp’:ŽWqaŒ‚. Šý‰Step 1.ŒQ †‘no‡vthat b’CpžDÆ”D2. We define the ˆsquar†istanceŒ-,+ function of GH below by using Pythagorean Œ_The†m.RˆmŒs Clear_a_z R†nŒ‰done–&Ž•f(a,b)=(p+a)^2+(q+b†ķ9[Ž[ŽÍStep 3.ŒÅˆō substitue bˆŋpî’qaŒ/ intoŽ‡ŒĮ†W‰;b:=Ļ8ŠÕĻQ—ŒŨŠĮķ€+qŦ2Žy+‡ū a+p† 2R†Œ [ˆStep 4.Œ$%E Find the critical points by settingŠderivative of f equals to 0.–hsolve(diff(f(a,b),a)=0†R†€†‡Š–a=Ž pî’q’īŽŋ1ŽĮ3ŒÏ,a=-pŠk!ŒãŒcŽ)aø|a=ŽeĪdŠžŸ6î’p“IQ†Ļ”ŽÉ+4ãî’Ō‡ +2î’† pî’qŒ 2Œ4Œ3Ð)3R†[ŒT[†6ˆ`Note.Žl&$ Since p,q>0, it is not hard to see !Œ˜f(a,b)ŒĨaĄˆX†Ä|a=ŽÃ’›1ÄŒę>0 so f indeed†us a Q minimum at â[.[’úŒø Numerically:Œþ Suppos‡ˆø4 and q‰6:—= p:=4RM† “Zq:=6R† †Œa:=Ž pî’qŽ'2Ž11Ž93ŠFŽ92î’18Ļ$†Ž bŠ1Ž#12Žt°8ŒĢf(a,b)Šĩ"Š( 14.046965@F–GXD†5–l[ŒÞ,Conclusion: The above is t† minimum length.†4‰ DiscusŠ4ŽGFWhenŠ3GH reaches itsB, willŠT triangle B†+b†qn†noceles?Ž•ˆą‡(taníü((y+q)/(x+p))/î)*180˜ų49.596534•–SCywR†† x+pR"Š8.545144TQDXˆ/˜/y+q–/ 10.206051ff’_Œ_[†^ Remark:ŒwH Since †~ is not same as †b which implies that† e triangle3an ŒÆ isoceles.[ŒnŽ'Exa†F e 2. Repla†tˆJwall by a circle\ļ$ΈÓ=†Ũ΋<ŠâŠí‰M 2U‰C† `4(iS‡p Y™bV )I‰|PhDvv‰ˆD‰‰ˆ‹8‡Ēr† ‡Ŧ A†† † L† †Œ C˜ u˜x˜yŽ †N† >†WÅH  ŽN?†iĀH ŠJ@†yÐ6–°†2…a€€"–†& 'R)5x†— —Ž;Ž‘ˆ3 ČĀ ŒU–Br%hFTB†ÍY…ˆÆŒ1ŽNw‡†NYˆNˆéˆ…Š—B˜…9y5w˜”…wˆ‘Ē…ˆdĒCC˜Č3 C0#–Č!ĻȉUÄ@ČŽ˜‰DĒˆÓ@mŽˆÕE‹U˜8?F†ČČ  † @†Ð6–°†XW`ˆ”†% ™)E2…Rˆ$ ™ƒE”q!†.`†A?† G†FČČ"ŒF†(†\(‚Y…ˆS phpŽ 2CŠ+†eg”I$’€Ž<!9 "`† Šy†T$(  &€0Žl'ˆ”! )ƒs'l1‰“„4!'65€€ŽĻ8yĻA75"īD˜‚lGa‚0PSQB–ŽðW5Ļ`•w—eif‘iaSPŽðtI`u yĖ…a€€‘,‘™‚Ž`™…`0€89 Ž GuwPŽ% e@Ž$6qH€Ž0HfE2Ž<b™„3<yb`ŽT™ƒDŒ`"4"†ŽHPCpŽT†(†‚(†—saŽt……‘Ž€„wWŽPƒupƒpŒ°‚ytŽŒ1ˆĮŠ ch$€2)Ž€ydV<y€v0xU’cTxw†T)whuŠĮŠcS lwrv’0•DDx5&Q4x“—Lypqf€p&d” LƒD—™d…&”‡AŽü‰”60’•ˆpŽ –6“ Œ9IdPŒ$4ˆ QŒ<ƒa™ˆG†H$!3`ŒT2ƒ”c<B—†”ŽlT–DA`i#‰ql†9€EŽ&84Žœ37dŒ†H†ēČČ"€Š7ˆ(†Ä(‚Y…@ŒÎphŽÎ2CÎegŽæ”æI$’€ !9 "`† Î$( Ú&€:'ˆ”!F)ƒs'‘:1‰“R4!'^65€€‘:8yvA75"‚D˜‚lGa‚‘jP^SQB–W5`Ž `•w—eifpŽ$iaSPŽ0tI`u y0Ž$…a€€ ŽT‘™‚`™…`Ž$€89Ž$GuwŽT% e@Œ6qH€ŒœHfE2ŽĻb™„3Ž`ybŽī™ƒDŒĖ"4"†ŽHPCpŽT†(†î(†—saŽČ……‘ŽÔ„wWƒupƒŽø‚ytŽŒ1‰3Š ch€2)Ž€ydV4y€v0xU’cLx(w†T)@whuŠĮŠcS lwrvXw•DDx5&Q@ “—PŽypqf`Ž €p&” <ƒD—™€ŽH…& ŽT‡ApŽ0‰”60Žl’•ˆ–6“09IdŽx4ˆŽ„QŒĻƒa™ˆģ†ī$!3Žœ2ƒ”cŽīB—†”ŽlT–DAŽi#‰qŽœ†9€EŽ&84Žœ37d†I‡9ĘČ ŠÓ3‰"Ð6–°‡5E‚Dsa?"XG“Su7ˆ;Ž î IJB: ĸĸü†ž  †F† ‰€–J‡œ9ČČ"‹Q2˜~•‚DsaPf4ty4Š ††Œ† Length IJ: ĸĸü†ž$†K†LÄČ ƒ †UI@†\Ð6–°†9y5w˜ˆ&f#5’ˆjĒnL†ÅH ŒCJ˜Cc5—p€fŠąU†TƒC†― ™ ē†M†Ö9ČČ"ŒG1˜ŠE’ø°ø Coord of īųŽķ‡HN†w†w€‹‡Z(†r(‚Y…‰ ‡jphpŽ 2CŠ+eg”I$’€Ž<!9Ž "`† `$(  &€0Ž'ˆ”! )ƒs' Ž01‰“pŽ<4!' 65€€ŽT8y$A75"0D˜‚lGa‚0PSQB–ŽœW5Ļ`•w—eif„iaSPŽĖtI`u yĖ…a€€Ā‘™‚`™…`Žð€89ŽäGuwŽT% e@Œä6qH€8HfE2ŽĻb™„3 yb\™ƒD"4"†ŽHPCpŽT†(‡Š(†—saŽČ……‘ŽÔ„wWŽƒupƒp‚yt€Ž 1ˆˆch$€2)Ž0ydV<y€v0xU’cTx Ž`w†T)0ŽlwhuŠGˆxwcS lwrv’0•DDŽœx5&Q@ŽĻx“—PŽīypqf`Ž €p&” <ƒD—™Ø…&‡Aü‰”6œ’•ˆ‘–6“09IdŽx4ˆŽ„Qƒa™8$!3Žœ2ƒ”cŽīB—†”T–DA i#‰q,†9€E,&84837d†O Å@  † † B†P†ČĀ@†%Ð6–°†,r‡ia€˜†7Y…4 'R)5w‡†OY—†QˆXĘXAŽXˆ\R†v5Å@!’K†S†v ČĀ˜v˜jQ1E˜w†đ ‡9y5w˜†`†ÉT†XÅH ŠÎA–CŒ+Š­)G$wCŠđY–§UĒ―ˆYˆŸV‰/ÄH!œŨSÜ:‰5‡P ™˜Q‘Y ™VAV‰  ‰BWˆhÆ@–NL†FŽXĢ›‰%ˆˆY†<ƆŠ †F†Z† ĀČ Œ@†Ð6–°†#—)5w˜† €'R)5x†: ™uA ™†[†OČČ `ˆUF–C°C‚Ž‰\†’ÄH Š›GēC… —žC]†Õ%ĘH!ŽČŽ˜ˆÞ^†]Ä@âŽoˆ_‡ Æ@ ’ˆÃĀ’‰H—)A X‹r†…3‹ž‰‰U‰YaŠ]ˆwˆbĒ{ˆ•c‡K ‡KĨ‘ĩ‡†Š—)5w˜† `ˆ[† \†d†$Ä@  +ˆc†b˜Z„Ž"ˆe†NĮ *f†d ČĀ Œ@†sÐ6–°† š~’ €'R)5x†Y™†g†ĶÃH#Š‚E–C9yŽĩ˜7 ˆ7˜^ˆ‰hŽÉŽ‰ŽYˆŸˆđˆˆ"ˆ]†^†Z†i‡#ÆH–6‰ˆÆj‡=ŘPˆk†ï†ïĪï”ã‹a§yFl†Ž!šŽ‰8 kW–2g”‰fŠƒm Å@  † † [†k††n†Ęg†0[u- ™†oˆDš&ˆDp†ZČČ Œ@†iÐ6–°†pr%hFTB†{Y…x €'R)5x†XYˆXˆFq† !Æ@ ’\ˆIˆ\°ŠrĒĮˆĐˆ'ˆ+sĒĄˆSŽ&tĒcˆ@W–2g•Š™€†D† CŽoV†UŽCT†A†ˆÏ–ˆˆäˆ1Ž‘]\†Z†f†c†`†F†K†L†P†Q†O†N†M†J†I†H†G†?‡ŧ(†R† ††[†Given conditions:Ž -AB is perpe† cular to BC.Ž# *-The circleˆ+adjacentˆ&both †Band–2[ˆyKey:Œƒ R Let GJ=x>0Š0 IE=y>0, th†šIH=ˆŠI†$&JH=x. So we are minimizing JI=x+y but ŒÜ@subjecŠt†Špoi†H lyˆ/onˆaŽŋ : (x-r)^2+(yŠ=r^2.‹.( Clear_a_z RŽdone–&‘V Note. Conside‡7‡(triang‡*IBJ:‹Š^IB‹2“+BJš =JI† 2Ž.[Œ So (IE+EB) & +(GJ+BG) @ĻT and ’X(y+rĒS+(xšŽ†=†y)’§RŠ5ŽĢsimplify(solve(šZ ëŽZĄ ,y))R y=r2†U -rŠ—K[ŽŽŒģŒLy:ČMžs x-rRˆ[Œ .We define th† istance function IJ as follows:–BŽ7f(x,r)=x+yRŽdone–o simplify(Œ/)Š,ŒŽxŒ—2Ž+r’Œ$œÁdiffŽN,xÄP-ržQ -2î’rî’x”b’ˆ–―solve(œq=0Œttx=r-žĮ†b,x=’Œ†y+ŒÎ!’I‘žý|x= 2Œ î’r+rRžr[Œ/ Note that !ŒAf(x,r)Ž3xVŽ0|x= k8Šk is positive so we n’\the distance function Ž„-achieves its minimum Ž6cr†Rcal†\int Ī~Šé.[ŒĸWe let x0=GJ and y0=IERŠÅŒÔx0:=r+Ą1‡1‹/°#”J y†J simplify(Ž:rî’Œlx0+r’ -Ž )Ķi2ˆî’rRŠŒ taníü(Œ y0+rŒ+x’ )RŽ$îŒM4[ŽH=This shows that† e ladder Œ rtest whenŠ angle IJBˆĪZŒŊ or”4tri”7ŒÐ†van†c oceles.[†Ļ†KŽA Challenging Problem.\ WˆŠabouˆ°is?Ή%u‡1Ή7‰<ˆP‰@ tPb)1† `e6gY€Š ™Dd6†ž wQuw`†D‰|v’K†r† Ž  A† 'LŽ'‰°C‰―†u† † Œ x˜ yŽ †+† v†4Å@  Ž4w†FÁH ŠI@†VÐ6–°†]i“‚“ˆ4 Y™’%!e†2Ž:ŽAˆsČĀ  Š–B ™˜™6c)†B ™HbqXe †ķ —Rycr!bŠ†3ˆĻÄHž—zXŒ–šFWV46ŠX7@—A•WŠdwt&“€†.`F{Š\˜ó|‡9Ā’óH˜ó08†%&Šį…y“#7Š―˜žó}Š›ČĻóHYT†!†įY—ˆĸ8BŒĸyX†gfŠĸ†3~ Å@  † † |†zŽˆĘ€†ÄH Š0G@†@Ð6–°†H7C† %hVq0†^ ˜<ŽC}†ĒY‚œYF˜Y’•f$H™†Ŧ Y™•1Yˆ†2žXxŽGˆaˆ ƒ†ÜČČ `ŠšŦxCQ†ü 00bPŒ žR„ŽBōB…Œ„!„Ķ„€3“y‰"ˆ„Q 2Š§ĒÖ†īB€3“y0Q 2`ˆ ˆˆŒ†‡†"ČČ Š@†1Ð6–°†8U08 ˆBY™p—–€ŒNžBˆīBY—ˆIpŽB—h–Š=ĒB‰ŒB"ô„Š†čÉøÆ‹†„ČČ"Šņˆ†ÂWIfQŽøVƒWƒ‹A‡RU)RPjHx6Šō‡vC)$46‘Ž)y—H!r`2<v"—’ ’Bb0Ž `Ži2Œ#ˆ$V’e—PŽ0D&‰ 0‡p€ŽH9—EpŽTt)ŽH ˆˆk ™†† †’6‡Ž€V(•Œ€9 !ŒPSg…”ŒŒ$ yh “hŒČ hE„Ž€ xg@ŒāQ7uŒ˜(T Œørƒr°YBy(e8E•2ue(cw%Žā ”‘!6G€Œ`yi—@˜ä†Œ‡nČČ ‹_ˆÐ6–°‡„c@‹vY™†uRžB†B9ČĀ"€ 1@†Ð6–°† 1&5e…p† ˆ‚H9C(Š ††Œ†5 î : ĸĸü†ž †z† Ž†a ČČ" Š–j a10h† Y™u‘˜ d† ™u„D†B&Š †3†BČČ  †ŽA˜­ !PŒšŠ+ Ĩ †æÄČ Š…E˜ðV’e—HŒð)$1@cĻð‘†CÄ@ ŒČŽ˜ˆÞ’ŠH"”ˆoŽ[ˆą–Dˆ*ˆHˆ6“†Ö•†‡‹WIfQ€EVƒWƒ`QU)RŠũHx6pC)$€Ž 6‘Ž)y—`Ž$!r`2PŽ0vŽ<—’ Ž0’Bb0ŒTŽ<i2Œ#†lV’e—ŽHD&‰ŽT0‡pŽ„9—EŽœt)Ž„ ˆˆģ ™†† †’6‡Ž€V(•Œž9 !ŒPSg…”ŒŒ$ yh “hŒø hE„ xg@(Q7uŽø(T @rƒr°YBy(e8EL•2ueXcw%Žā ”‘!6G€Xyi|˜ä†ˆŋ5Ä@  ††•† ĀĀ !PŒ ™'ˆ/WIfQƒ8† F5‰1y†F — L ™Ž:ˆ$Ž †–†bĀH ŠxB@†Ð6–°†5 "CĶ?`˜zˆ—œCC˜C°šžC˜†õ ČĀ Œþ–… ™˜™8ˆ— ™HYT†!ˆĀ†Ü X1@9†Y‰ Ė‰ƙHˆˆ`šŽ’ˆ0ˆˆ ›‡w ËŠ‚Œv†vY—šŽvi5–qÐ‰N—œ†Ä@  † ††›†Ē—Ž†7ž†<&Č 8•†Ÿ†V'Į˜–Ž †R'ĒRˆĄ†!ÄH!”˜†œ›W–2g”‰f@ 4!&18†ī ™&†ÁPš ™1W‰G6„†A ˆAĒ†Ĩ.šĨĢ†3ČĀ–ˆ†z†ĪĒ0ˆˆ4ĨĒJˆ4ˆˆdˆ<ˆ"ˆ@ˆˆĶ†Ó1šíˆ2ˆPHŽ §Œ"!’"Ļ†ÄH ‹D@‡ĄÐ6–°‡ĻRWB&ˆY™$"g ˜˜†u”“@ˆ ™†•†H†Ą††œ†›†š†ˆ ˜† †Ÿ†žŽ0†—†–†Ļ†€†YŽˆˆˆ)ˆ†z†“†Ž††Œ†‹†Š†‰†ˆ†‡†††…†„†ƒ†x†}†‚Že|†w††Éˆq”Ž [†ßHere a†given conditions:Ž"-We draw two lines DF and DG.Ž%/’%!an ellipse that is tangent to FD †D.[ŒZ -We construct the tangent line aŒpoi†E, which is lying onŠ3 ellipse. T–@1ŠGinterse†b FD and DGˆ[IŠ H resp† ively.[Œ;1. † ˆG %Equation of˜rˆŠ as followsŒrŽz>6.25î’x^2+2.768î’y† 0.03124† î’y-11.39†$-†2847Š2.11=0RˆūŒÄJdef‰ k(x,y)=øXRŽVdone[&[ô2’ôL‰eųDG:R3 y=4.486îėë3î’x-4.088Š. We define it as f(x,y)Rˆ(ŽŒ=y†>–L+”LRŽ4done[Œ@[ŒI†n† ˆULˆq equation FD:Œl,†Ŧ-222.9ˆ§167.8˜ĒthisˆĪg Ī!ŽĪŒ†Ī(žF†ŅŠ 4Objective:‰ want to †Md†ne shortest distance IH.˜Ų‰#Step 1.ŒÄH‰S”Alope ofŠN†Bge†bŽ ellipse a‡y g†„n poi†‚‹ƒ:‘@RˆBeq:=6.25î’x^2+2.768î’y† 0.03124† î’y-11.39†$-†2847Š2.11=0RŒSŠMŒ 2Œh4Ž+(346î’yž)125’+ŒĶ 781Œ‰’c000ŒÂ-'11ŠĨJ00Ž“-ŽIŽžŒ"†"†H’%H1211”E*ˆę“8 impdiff(eq,x)Ry'=‘ -‡@Œï‡N†ķ‡h+ŠÎy-‰H5ŽÆl+5536ˆ&Œ–r[†Uˆ -Step 2.1 We define the slope of IH to b˜above.Rˆ7Œ>Ž:m(x,y)=Ž-4î’Œa312500î’x+781î’y-284750Œ††,x+5536ˆ&ŒRŒcdoneŠtŒą[Ž[†nˆÄŠŅ3ŽŅQ If E=(a,b) is a point onŠĘ ellipse and†ã, whereˆíŠ+tange†-l‰(‡‰)@Ž:) atŽSE. Now,‹8Š2equation‰=can‰> written asŽļ y-b=‡0-a). ŽĖ:-We define the following function to representŠ$lˆ- IH:RˆŽCh(x,y)=y-b-m(a,b)*(x-a)RŽ*done–:[ŒBONotˆƒat kŒE0”wsŠœellipse. Since E is ˆ ”,†<ˆv=0.ŽW_g[† Step 4.ŒÁ J Now we †Į d H, whichˆb‰interse between DG a†,IH or fŠųŠ  : —''temp1:=solve({Œ5=0,70},{‡@}0ˆŠ‹cx= 6250000†00î’a 2ˆ +27680000†0î’bŒ2Œ!6+3124a†"-569493614544î’a-2919416Š?-5819268Œ^*625007007132î’a+2803724†qŽI 506385821Ž,y=Ž Š/†Bˆša’™Žŧ+†™1’N đ 4+1401426ˆĪŠ·10977††|a-1277†7†ó+23281ˆÁ0â·R‰NKH_x:=getright(temp1[1])R’ÔŠ\“pĒÖő+3124000000î’aî’b-569493614544†-2919416ˆ †819268*625007007132†+ +28037248Ž, 69506385821Rˆ0Œ6 H_y:=getright(temp1[2])RŒ\ ŠL†_Š˜Œ2Ž+†ķ1’k’ Œ 4+1401426ˆÁŠÔ10977†Šˆå-1277† 7î’b+23281ˆÞ0øÔ[ˆļ‰Step 5."7J Now we find I, which is the intersection between IH a†, DF or h(x,y)ŠsgŠ:‘P#temp2:=solve({h(x,y),gŠ},{† })R†1x=  12500000î’aŒ 2Œ( +5536Šb  -6147928†9+bî’Œ]6248ˆN+†912330Œ1 -4777266Œ… ˆ\604†x-ˆ&94316†† b-5044037Ž­,y†ŪŒC-Ä 2786Œŧ†š’đŽâ+ˆy97†IŠ―žŲ2+13926†ū†öˆs441331-1103†šˆ‹1911242‡!ÎĩR‹`‹fI_x:=getright(‹Ķ[1]Œ€ 12500000î’a†2Œ +5536Šb  -6147928†9+bî’Œ=6248ˆN+†912330Œ1 -4777266Œe ˆ\604†x-ˆ&94316†† b-5044037R†žŒ‘I_y:=getright(temp2[2])RŽķ-Ž  2786ŒØ†Ũ’ÖŽ)+ˆ–97†fŠÚžö2+13926†Û‡ˆ441331#-1103†ŨˆĻ1911242‡>äŌ[†‰mStep 6.z6= Set up the distance function and subject to k(a,b)=0:Rˆ 4define Lˆ,L1)=(I_x-H_x)^2+† y† yˆ L1*(ŒB)RŽ@done–P eq1:=diff(’S,a†uŠ4Œw L1î’Œ†25î’aŒ 2Œš+78†.b’+000ŽG-ŽA1139Œ 1’ +2w 28037ˆA0î’aqŽ/ +12417248ˆb’ ++14014264î’aî’b-5109777ŒM-1277† 7î’b+2328116†Đf*5625007007132î’a+2803724800î’b-569506385821+Ž  278Š@ˆ-aŽ 2Ž) +12339744ŠL’ Œ 2+1392679ˆwˆl441331ŒL-1103891ˆ„ 1911242000Œ 5536604Š'ˆb4316ŒŪ0†l37ŒĻ î’Œ6#557ŒŦ†Ųa–|–yÎ`‘ - ’’†pĸ‰c2+13926792î’aî’b-441331000†-1103891ˆ1911242† 5536604Š)23394316†7 b-5044037Œ(2Œ2+Œ;$56075†eŠg+1401426†J b-5109777†Œf*62500†713ˆĪ +28037248ˆĒb-569506385821f-Žpœ<ˆ ŠDŠ€†ö”§ŒĶ+†é1’c’ĮŒ4–Ŋ‹2”ē‹3277† 7î’b+23281‡•4ÖΠ—-ŒÃC+1250000††î’a+3124b-569493614544*6ˆ27007132ˆ128037248’1 506385821Œ2-Ž œ<ŽŠ”…Œk2ŽB+2768”Ķb’"Œ!6š·a š†§-2919416ŽŨ819268âԔ›”ÞŒ‘J‰F6‰‰A‡;7928< 553660Š1233943†’‹h044‡GT+‘2 ’2“- ‘ģ‡Ū‡2Š +5536000î’bŒ-6147928î’a+bî’Œ56248†2a+†912330Œ1 -4777266* ˆ^604î’a-ˆ(94316ˆl-5044037’rŒˆfŽ 125†ˆŸa žþŧ°ŧÎđŽ°-’Ž6ŽŽŽ° Û+2768”!bĄo6+3124000000î’aî’b-569493614544†-2919416ˆ †819268*625007007132†+ +28037248Ž, 69506385821Œ1=0Rˆ9Œ?eq2:=diff(L(a,b,L1),b)ˆ$Œf L1î’Œu78†aŒ‚ˆ0ŒŽ+’#69†bŒĨ125ŽJ-ŽD2847Œ1Šýl+†Č} ŠÓ†æ‹Œ2Ž/+‡=1’ō’ Œ 4+1401426‰H‹[10977‡1‰l-1277† 7î’b+23281‰eŽÍ*5625007007132î’a+2803724800î’b-569506385821+Ž  278Š@ˆ-aŽ 2Ž) +12339744ŠL’ Œ 2+1392679ˆwˆl441331ŒL-1103891ˆ„ 1911242000Œ 5536604Š'ˆb4316ŒŪ0†l37ŒĻ î’Žļ ˜,†þÃîÃÎÅĄN‡w€"14014264î’a+2483449600î’b-1277†7*625007007132Š-8037†2Œ-569506385821Œ3+Œ<!1392679Œ84† 48Ž81103891Œd 553660ˆŒ†~3394316ŠŒ5044†oY-Žc ˜†y Šœ†ŊˆÉaŒ2Ž—+1241’ŧ’ Œ4+—ˆĢ510977†ú†Šˆ˜ î’b+23281†Ē00ã& —+‡W†Ã46‡Ž‡ˆa+‡#72ŠËb+92‡-‡0• *5536604î’a-1233943160î’b-5044037+Ž  28†2480†'Ž 6250000†ˆaŽ72Ž@+2768”!b’"Œ!6+3124’@Š‚ 6949361454Š 29194†žŒŸ819268Œ€*Š€7007132†-+˜ĶŠK 506385821”›Ü-Œå'šƒ+‰”Ób”€âsĨE™l“E 1E‡ŽĄ  ‰<™=-6147928î’a+bî’ˆ62480ˆ†912330Œ -4777266* 5536604†A-ˆ(94316†: b-5044037ŒR2Œ[ˆfŽ 1250000ˆm’,Œ† +ˆ]Šb ôŧÎđŽ°-’Ž6ŽŽŽ° Û+2768”!b Ņ6+3124’@‹&69493‡†54‹D29194‡BC8192689*4625007007132î’a+2803724800î’b-569506385821=0RŠŠeq3:=diff(L(a,b,L1)†)ˆ%Œ7 Šh†TaŒH2Ž +276Œo’Œ&+3124†–ˆ‹1139ŒB-2847ˆž1211000Œ™ˆ  œĶ[ŽŽ–Âsolve({eq1,eq2†3},{ŒË})ŠÆŒųERROR:Overflow†K‰ Remark:[ŽčQ1) It is inevitable to switchˆ an external compu†  algebra system for further wŒ+ation†Cd†Galysis.=[C2) By switching to Maple 10, we get a = 1.784381519,b†2.123030625[ŒM 3) We dra†Hhe point Ž# –HŒ‚ –JŒ”3in††ˆAGeometry Strip above†–hich will see this ŒÎ4ma†Į es closely †Ũ†*hatˆËhad cojectured earlier. Žž‰,eActDGXXFy €ĸHč à Ąx125F_x  €EũL €xzŒIb \{Ta€ €2åH  Ę^7 ôab yTb €XÏ]a  žūaF_y € €=ëq €Øb ˆÔba €ôŒIb €8-Na ÍĶa €EũL €xzŒIb \{Ta€yG_x  €Ï9™„ r§b œR…‘a€ €„ËX €āŽfb u4šba Аr@b €Ðïv˜„a *į„‘aG_y  €Ï9™„ r§b œR…‘a€  ;Ċ €ÍßÂb 8ŨÕba €7Jb €h‘0a pÔ§aH_x%ûH_yPP  oPx€^3a44 (ans1 X((Fx  €Ï9™„ r§b œR…‘a€ €„ËX €āŽfb u4šba Аr@b €Ðïv˜„a *į„‘a$Fy  €Ï9™„ r§b œR…‘a€  ;Ċ €ÍßÂb 8ŨÕba €7Jb €h‘0a pÔ§ab44 qpa€e1$$F   €sd  Ļab aL1 €  €œR…‘ €Ï9™„ r§b œR…‘a€ €„ËX €āŽfb u4šba Аr@b €Ðïv˜„a *į„‘a  €Ï9™„ r§b œR…‘a€ €Ðïv˜„ u4šb TÎ #a € €XÏ] ôb @x}a €EũL €xzŒIb \{Ta€ \{T €EũL €xzŒIb \{Ta€ €2åH  Ę^7 ôab yTb €XÏ]a  žūa  € €Ï9™„ r§b œR…‘a€ €„ËX €āŽfb u4šba Аr@b €Ðïv˜„a *į„‘a  €EũL €xzŒIb \{Ta€ €2åH  Ę^7 ôab yTb €XÏ]a  žūa   €œR…‘ €Ï9™„ r§b œR…‘a€  ;Ċ €ÍßÂb 8ŨÕba €7Jb €h‘0a pÔ§a  €Ï9™„ r§b œR…‘a€ €h‘0 8ŨÕb āĻ;Na €\{T €=ëq €Øb ˆÔba €ôŒIb €8-Na ÍĶa €EũL €xzŒIb \{Ta€  €8-N ˆÔb  š%La €EũL €xzŒIb \{Ta€   €=ëq €Øb ˆÔba €ôŒIb €8-Na ÍĶa €EũL €xzŒIb \{Ta€  €Ï9™„ r§b œR…‘a€  ;Ċ €ÍßÂb 8ŨÕba €7Jb €h‘0a pÔ§a  €  €œR…‘ €Ï9™„ r§b œR…‘a€ €„ËX €āŽfb u4šba Аr@b €Ðïv˜„a *į„‘a  €Ï9™„ r§b œR…‘a€ €Ðïv˜„ u4šb TÎ #a € €XÏ] ôb @x}a €EũL €xzŒIb \{Ta€ \{T €EũL €xzŒIb \{Ta€ €2åH  Ę^7 ôab yTb €XÏ]a  žūa  € €Ï9™„ r§b œR…‘a€ €„ËX €āŽfb u4šba Аr@b €Ðïv˜„a *į„‘a  €EũL €xzŒIb \{Ta€ €2åH  Ę^7 ôab yTb €XÏ]a  žūa   €œR…‘ €Ï9™„ r§b œR…‘a€  ;Ċ €ÍßÂb 8ŨÕba €7Jb €h‘0a pÔ§a  €Ï9™„ r§b œR…‘a€ €h‘0 8ŨÕb āĻ;Na €\{T €=ëq €Øb ˆÔba €ôŒIb €8-Na ÍĶa €EũL €xzŒIb \{Ta€  €8-N ˆÔb  š%La €EũL €xzŒIb \{Ta€   €=ëq €Øb ˆÔba €ôŒIb €8-Na ÍĶa €EũL €xzŒIb \{Ta€  €Ï9™„ r§b œR…‘a€  ;Ċ €ÍßÂb 8ŨÕba €7Jb €h‘0a pÔ§aeqF €ŧd €  †y  Ļayx Z}y €sdx x1eq1 Ø ØF   €sd  Ļab aL1 €  €œR…‘ €Ï9™„ r§b œR…‘a€ €„ËX €āŽfb u4šba Аr@b €Ðïv˜„a *į„‘a  €Ï9™„ r§b œR…‘a€ €Ðïv˜„ u4šb TÎ #a € €XÏ] ôb @x}a €EũL €xzŒIb \{Ta€ \{T €EũL €xzŒIb \{Ta€ €2åH  Ę^7 ôab yTb €XÏ]a  žūa  € €Ï9™„ r§b œR…‘a€ €„ËX €āŽfb u4šba Аr@b €Ðïv˜„a *į„‘a  €EũL €xzŒIb \{Ta€ €2åH  Ę^7 ôab yTb €XÏ]a  žūa   €œR…‘ €Ï9™„ r§b œR…‘a€  ;Ċ €ÍßÂb 8ŨÕba €7Jb €h‘0a pÔ§a  €Ï9™„ r§b œR…‘a€ €h‘0 8ŨÕb āĻ;Na €\{T €=ëq €Øb ˆÔba €ôŒIb €8-Na ÍĶa €EũL €xzŒIb \{Ta€  €8-N ˆÔb  š%La €EũL €xzŒIb \{Ta€   €=ëq €Øb ˆÔba €ôŒIb €8-Na ÍĶa €EũL €xzŒIb \{Ta€  €Ï9™„ r§b œR…‘a€  ;Ċ €ÍßÂb 8ŨÕba €7Jb €h‘0a pÔ§aeq2 Ô ÔF   €  † ī}b  ĻaaL1   €r§ €Ï9™„ r§b œR…‘a€  ;Ċ €ÍßÂb 8ŨÕba €7Jb €h‘0a pÔ§a  €Ï9™„ r§b œR…‘a€ €Íß o”b 8ŨÕa xzŒI €=ëq €Øb ˆÔba €ôŒIb €8-Na ÍĶa €EũL €xzŒIb \{Ta€  €EũL €xzŒIb \{Ta€ €Ø é“b ˆÔa   €=ëq €Øb ˆÔba €ôŒIb €8-Na ÍĶa €EũL €xzŒIb \{Ta€  €Ï9™„ r§b œR…‘a€  ;Ċ €ÍßÂb 8ŨÕba €7Jb €h‘0a pÔ§a   € €Ï9™„ r§b œR…‘a€ €„ËX €āŽfb u4šba Аr@b €Ðïv˜„a *į„‘a  €EũL €xzŒIb \{Ta€ €2åH  Ę^7 ôab yTb €XÏ]a  žūa  r§ €Ï9™„ r§b œR…‘a€ €„ËX €āŽfb u4šba Аr@b €Ðïv˜„a *į„‘a € €Ï9™„ r§b œR…‘a€ €āŽf  !å€b u4ša xzŒI €EũL €xzŒIb \{Ta€ €2åH  Ę^7 ôab yTb €XÏ]a  žūa  €EũL €xzŒIb \{Ta€ Ę^7 ōĻb ôaeq3āāF  † €xz € b 4 ba @9b €8aa h‰ a^fH$a,b(p+a)^2+(q+b)^2f1H$x,y(p+x)^2+(q+y)^2lineFGF  €by €  €8a € €rb 4 a€ 4  € €rb 4 a€b Ð € €rb 4 a€a  €axp q temp1 X((Fx  €Ï9™„ r§b œR…‘a€ €„ËX €āŽfb u4šba Аr@b €Ðïv˜„a *į„‘a$Fy  €Ï9™„ r§b œR…‘a€  ;Ċ €ÍßÂb 8ŨÕba €7Jb €h‘0a pÔ§atemp2 HFx  €EũL €xzŒIb \{Ta€ €2åH  Ę^7 ôab yTb €XÏ]a  žūa(Fy € €=ëq €Øb ˆÔba €ôŒIb €8-Na ÍĶa €EũL €xzŒIb \{Ta€x44 qx0DD  rry0DD  rr010008main.ACT0001020012eActivity Save.EAC010000009acfÉ  Ladder.EAC–ACT†††)ˆ,†0† Š†C ~I Ž'Á[ˆ4A lŠ_ problem†ŠX\Œ Author툈XíŽxŒ.Professor Wei-Chi Yang ŒLRadford University Ž9 e-mail: wy†9@rŒ2.eduŽE!URL: http://www.–#/Š5‡+ŽÖŽ Object†vs:[ŽŽP1.†Ĩ( will use ClassPad to explore a Calculus‘ that can be found in many 1@ctextbooks. And see how we can generate many different scenarios.[Q2. We will use its dynamic†Fometry a†bCAS features to help†2†b ke conjecŒŒX%before solving problems†L alytically.[Œ…Ž[†—ˆ˜ Example 1.ŒĻN A†nce q†‡et ta†ē runs parallelˆ–a buildˆrat a†ûstaˆ6of pŒ9from ŽþWthe’2 . What is ˆlengthˆ8ˆ$shortest ladder tˆ*‰jreachŒTˆLgrou‡\+over the fence toŠ wall ofŠbuilding?[Note. ˆ #1For demonstration purpose, we useŒ^ollow†Fs:[Œ9$#1. We set p=4 and q=6 respectively.Ž+J2. 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L-play the†$ŽŠ.collectŠdata forŠ'point HŠSˆ7 distance GHŽ‚R-drag the x-coordinate of H andŠlength GH back toŠ0 curve, we see a mimimum.[Ž Analytically:[ŒFLet BE=p, DE=q, EH=aŠrGF=b,ˆ’nˆ\consider two s†cla† riangles ˆmGFDŽXŠķ† ˆ…DEHŒŽ.†zŽ{WŒģŒĐGFŒē’E=DEŒÎEHš or<bŒíp’:ŽWqaŒ‚. Šý‰Step 1.ŒQ †‘no‡vthat b’CpžDÆ”D2. We define the ˆsquar†istanceŒ-,+ function of GH below by using Pythagorean Œ_The†m.RˆmŒs Clear_a_z R†nŒ‰done–&Ž•f(a,b)=(p+a)^2+(q+b†ķ9[Ž[ŽÍStep 3.ŒÅˆō substitue bˆŋpî’qaŒ/ intoŽ‡ŒĮ†W‰;b:=Ļ8ŠÕĻQ—ŒŨŠĮķ€+qŦ2Žy+‡ū a+p† 2R†Œ [ˆStep 4.Œ$%E Find the critical points by settingŠderivative of f equals to 0.–hsolve(diff(f(a,b),a)=0†R†€†‡Š–a=Ž pî’q’īŽŋ1ŽĮ3ŒÏ,a=-pŠk!ŒãŒcŽ)aø|a=ŽeĪdŠžŸ6î’p“IQ†Ļ”ŽÉ+4ãî’Ō‡ +2î’† pî’qŒ 2Œ4Œ3Ð)3R†[ŒT[†6ˆ`Note.Žl&$ Since p,q>0, it is not hard to see !Œ˜f(a,b)ŒĨaĄˆX†Ä|a=ŽÃ’›1ÄŒę>0 so f indeed†us a Q minimum at â[.[’úŒø Numerically:Œþ Suppos‡ˆø4 and q‰6:—= p:=4RM† “Zq:=6R† †Œa:=Ž pî’qŽ'2Ž11Ž93ŠFŽ92î’18Ļ$†Ž bŠ1Ž#12Žt°8ŒĢf(a,b)Šĩ"Š( 14.046965@F–GXD†5–l[ŒÞ,Conclusion: The above is t† minimum length.†4‰ DiscusŠ4ŽGFWhenŠ3GH reaches itsB, willŠT triangle B†+b†qn†noceles?Ž•ˆą‡(taníü((y+q)/(x+p))/î)*180˜ų49.596534•–SCywR†† x+pR"Š8.545144TQDXˆ/˜/y+q–/ 10.206051ff’_Œ_[†^ Remark:ŒwH Since †~ is not same as †b which implies that† e triangle3an ŒÆ isoceles.[ŒnŽ'Exa†F e 2. Repla†tˆJwall by a circle\ļ$ΈÓ=†Ũ΋<ŠâŠí‰M 2U‰C† `4(iS‡p Y™bV )I‰|PhDvv‰ˆD‰‰ˆ‹8‡Ēr† ‡Ŧ A†† † L† †Œ C˜ u˜x˜yŽ †N† >†WÅH  ŽN?†iĀH ŠJ@†yÐ6–°†2…a€€"–†& 'R)5x†— —Ž;Ž‘ˆ3 ČĀ ŒU–Br%hFTB†ÍY…ˆÆŒ1ŽNw‡†NYˆNˆéˆ…Š—B˜…9y5w˜”…wˆ‘Ē…ˆdĒCC˜Č3 C0#–Č!ĻȉUÄ@ČŽ˜‰DĒˆÓ@mŽˆÕE‹U˜8?F†ČČ  † @†Ð6–°†XW`ˆ”†% ™)E2…Rˆ$ ™ƒE”q!†.`†A?† G†FČČ"ŒF†(†\(‚Y…ˆS phpŽ 2CŠ+†eg”I$’€Ž<!9 "`† Šy†T$(  &€0Žl'ˆ”! )ƒs'l1‰“„4!'65€€ŽĻ8yĻA75"īD˜‚lGa‚0PSQB–ŽðW5Ļ`•w—eif‘iaSPŽðtI`u yĖ…a€€‘,‘™‚Ž`™…`0€89 Ž GuwPŽ% e@Ž$6qH€Ž0HfE2Ž<b™„3<yb`ŽT™ƒDŒ`"4"†ŽHPCpŽT†(†‚(†—saŽt……‘Ž€„wWŽPƒupƒpŒ°‚ytŽŒ1ˆĮŠ ch$€2)Ž€ydV<y€v0xU’cTxw†T)whuŠĮŠcS lwrv’0•DDx5&Q4x“—Lypqf€p&d” LƒD—™d…&”‡AŽü‰”60’•ˆpŽ –6“ Œ9IdPŒ$4ˆ QŒ<ƒa™ˆG†H$!3`ŒT2ƒ”c<B—†”ŽlT–DA`i#‰ql†9€EŽ&84Žœ37dŒ†H†ēČČ"€Š7ˆ(†Ä(‚Y…@ŒÎphŽÎ2CÎegŽæ”æI$’€ !9 "`† Î$( Ú&€:'ˆ”!F)ƒs'‘:1‰“R4!'^65€€‘:8yvA75"‚D˜‚lGa‚‘jP^SQB–W5`Ž `•w—eifpŽ$iaSPŽ0tI`u y0Ž$…a€€ ŽT‘™‚`™…`Ž$€89Ž$GuwŽT% e@Œ6qH€ŒœHfE2ŽĻb™„3Ž`ybŽī™ƒDŒĖ"4"†ŽHPCpŽT†(†î(†—saŽČ……‘ŽÔ„wWƒupƒŽø‚ytŽŒ1‰3Š ch€2)Ž€ydV4y€v0xU’cLx(w†T)@whuŠĮŠcS lwrvXw•DDx5&Q@ “—PŽypqf`Ž €p&” <ƒD—™€ŽH…& ŽT‡ApŽ0‰”60Žl’•ˆ–6“09IdŽx4ˆŽ„QŒĻƒa™ˆģ†ī$!3Žœ2ƒ”cŽīB—†”ŽlT–DAŽi#‰qŽœ†9€EŽ&84Žœ37d†I‡9ĘČ ŠÓ3‰"Ð6–°‡5E‚Dsa?"XG“Su7ˆ;Ž î IJB: ĸĸü†ž  †F† ‰€–J‡œ9ČČ"‹Q2˜~•‚DsaPf4ty4Š ††Œ† Length IJ: ĸĸü†ž$†K†LÄČ ƒ †UI@†\Ð6–°†9y5w˜ˆ&f#5’ˆjĒnL†ÅH ŒCJ˜Cc5—p€fŠąU†TƒC†― ™ ē†M†Ö9ČČ"ŒG1˜ŠE’ø°ø Coord of īųŽķ‡HN†w†w€‹‡Z(†r(‚Y…‰ ‡jphpŽ 2CŠ+eg”I$’€Ž<!9Ž "`† `$(  &€0Ž'ˆ”! )ƒs' Ž01‰“pŽ<4!' 65€€ŽT8y$A75"0D˜‚lGa‚0PSQB–ŽœW5Ļ`•w—eif„iaSPŽĖtI`u yĖ…a€€Ā‘™‚`™…`Žð€89ŽäGuwŽT% e@Œä6qH€8HfE2ŽĻb™„3 yb\™ƒD"4"†ŽHPCpŽT†(‡Š(†—saŽČ……‘ŽÔ„wWŽƒupƒp‚yt€Ž 1ˆˆch$€2)Ž0ydV<y€v0xU’cTx Ž`w†T)0ŽlwhuŠGˆxwcS lwrv’0•DDŽœx5&Q@ŽĻx“—PŽīypqf`Ž €p&” <ƒD—™Ø…&‡Aü‰”6œ’•ˆ‘–6“09IdŽx4ˆŽ„Qƒa™8$!3Žœ2ƒ”cŽīB—†”T–DA i#‰q,†9€E,&84837d†O Å@  † † B†P†ČĀ@†%Ð6–°†,r‡ia€˜†7Y…4 'R)5w‡†OY—†QˆXĘXAŽXˆ\R†v5Å@!’K†S†v ČĀ˜v˜jQ1E˜w†đ ‡9y5w˜†`†ÉT†XÅH ŠÎA–CŒ+Š­)G$wCŠđY–§UĒ―ˆYˆŸV‰/ÄH!œŨSÜ:‰5‡P ™˜Q‘Y ™VAV‰  ‰BWˆhÆ@–NL†FŽXĢ›‰%ˆˆY†<ƆŠ †F†Z† ĀČ Œ@†Ð6–°†#—)5w˜† €'R)5x†: ™uA ™†[†OČČ `ˆUF–C°C‚Ž‰\†’ÄH Š›GēC… —žC]†Õ%ĘH!ŽČŽ˜ˆÞ^†]Ä@âŽoˆ_‡ Æ@ ’ˆÃĀ’‰H—)A X‹r†…3‹ž‰‰U‰YaŠ]ˆwˆbĒ{ˆ•c‡K ‡KĨ‘ĩ‡†Š—)5w˜† `ˆ[† \†d†$Ä@  +ˆc†b˜Z„Ž"ˆe†NĮ *f†d ČĀ Œ@†sÐ6–°† š~’ €'R)5x†Y™†g†ĶÃH#Š‚E–C9yŽĩ˜7 ˆ7˜^ˆ‰hŽÉŽ‰ŽYˆŸˆđˆˆ"ˆ]†^†Z†i‡#ÆH–6‰ˆÆj‡=ŘPˆk†ï†ïĪï”ã‹a§yFl†Ž!šŽ‰8 kW–2g”‰fŠƒm Å@  † † [†k††n†Ęg†0[u- ™†oˆDš&ˆDp†ZČČ Œ@†iÐ6–°†pr%hFTB†{Y…x €'R)5x†XYˆXˆFq† !Æ@ ’\ˆIˆ\°ŠrĒĮˆĐˆ'ˆ+sĒĄˆSŽ&tĒcˆ@W–2g•Š™€†D† CŽoV†UŽCT†A†ˆÏ–ˆˆäˆ1Ž‘]\†Z†f†c†`†F†K†L†P†Q†O†N†M†J†I†H†G†?‡ŧ(†R† ††[†Given conditions:Ž -AB is perpe† cular to BC.Ž# *-The circleˆ+adjacentˆ&both †Band–2[ˆyKey:Œƒ R Let GJ=x>0Š0 IE=y>0, th†šIH=ˆŠI†$&JH=x. So we are minimizing JI=x+y but ŒÜ@subjecŠt†Špoi†H lyˆ/onˆaŽŋ : (x-r)^2+(yŠ=r^2.‹.( Clear_a_z RŽdone–&‘V Note. Conside‡7‡(triang‡*IBJ:‹Š^IB‹2“+BJš =JI† 2Ž.[Œ So (IE+EB) & +(GJ+BG) @ĻT and ’X(y+rĒS+(xšŽ†=†y)’§RŠ5ŽĢsimplify(solve(šZ ëŽZĄ ,y))R y=r2†U -rŠ—K[ŽŽŒģŒLy:ČMžs x-rRˆ[Œ .We define th† istance function IJ as follows:–BŽ7f(x,r)=x+yRŽdone–o simplify(Œ/)Š,ŒŽxŒ—2Ž+r’Œ$œÁdiffŽN,xÄP-ržQ -2î’rî’x”b’ˆ–―solve(œq=0Œttx=r-žĮ†b,x=’Œ†y+ŒÎ!’I‘žý|x= 2Œ î’r+rRžr[Œ/ Note that !ŒAf(x,r)Ž3xVŽ0|x= k8Šk is positive so we n’\the distance function Ž„-achieves its minimum Ž6cr†Rcal†\int Ī~Šé.[ŒĸWe let x0=GJ and y0=IERŠÅŒÔx0:=r+Ą1‡1‹/°#”J y†J simplify(Ž:rî’Œlx0+r’ -Ž )Ķi2ˆî’rRŠŒ taníü(Œ y0+rŒ+x’ )RŽ$îŒM4[ŽH=This shows that† e ladder Œ rtest whenŠ angle IJBˆĪZŒŊ or”4tri”7ŒÐ†van†c oceles.[†Ļ†KŽA Challenging Problem.\ WˆŠabouˆ°is?Ή%u‡1Ή7‰<ˆP‰@ tPb)1† `e6gY€Š ™Dd6†ž wQuw`†D‰|v’K†r† Ž  A† 'LŽ'‰°C‰―†u† † Œ x˜ yŽ †+† v†4Å@  Ž4w†FÁH ŠI@†VÐ6–°†]i“‚“ˆ4 Y™’%!e†2Ž:ŽAˆsČĀ  Š–B ™˜™6c)†B ™HbqXe †ķ —Rycr!bŠ†3ˆĻÄHž—zXŒ–šFWV46ŠX7@—A•WŠdwt&“€†.`F{Š\˜ó|‡9Ā’óH˜ó08†%&Šį…y“#7Š―˜žó}Š›ČĻóHYT†!†įY—ˆĸ8BŒĸyX†gfŠĸ†3~ Å@  † † |†zŽˆĘ€†ÄH Š0G@†@Ð6–°†H7C† %hVq0†^ ˜<ŽC}†ĒY‚œYF˜Y’•f$H™†Ŧ Y™•1Yˆ†2žXxŽGˆaˆ ƒ†ÜČČ `ŠšŦxCQ†ü 00bPŒ žR„ŽBōB…Œ„!„Ķ„€3“y‰"ˆ„Q 2Š§ĒÖ†īB€3“y0Q 2`ˆ ˆˆŒ†‡†"ČČ Š@†1Ð6–°†8U08 ˆBY™p—–€ŒNžBˆīBY—ˆIpŽB—h–Š=ĒB‰ŒB"ô„Š†čÉøÆ‹†„ČČ"Šņˆ†ÂWIfQŽøVƒWƒ‹A‡RU)RPjHx6Šō‡vC)$46‘Ž)y—H!r`2<v"—’ ’Bb0Ž `Ži2Œ#ˆ$V’e—PŽ0D&‰ 0‡p€ŽH9—EpŽTt)ŽH ˆˆk ™†† †’6‡Ž€V(•Œ€9 !ŒPSg…”ŒŒ$ yh “hŒČ hE„Ž€ xg@ŒāQ7uŒ˜(T Œørƒr°YBy(e8E•2ue(cw%Žā ”‘!6G€Œ`yi—@˜ä†Œ‡nČČ ‹_ˆÐ6–°‡„c@‹vY™†uRžB†B9ČĀ"€ 1@†Ð6–°† 1&5e…p† ˆ‚H9C(Š ††Œ†5 î : ĸĸü†ž †z† Ž†a ČČ" Š–j a10h† Y™u‘˜ d† ™u„D†B&Š †3†BČČ  †ŽA˜­ !PŒšŠ+ Ĩ †æÄČ Š…E˜ðV’e—HŒð)$1@cĻð‘†CÄ@ ŒČŽ˜ˆÞ’ŠH"”ˆoŽ[ˆą–Dˆ*ˆHˆ6“†Ö•†‡‹WIfQ€EVƒWƒ`QU)RŠũHx6pC)$€Ž 6‘Ž)y—`Ž$!r`2PŽ0vŽ<—’ Ž0’Bb0ŒTŽ<i2Œ#†lV’e—ŽHD&‰ŽT0‡pŽ„9—EŽœt)Ž„ ˆˆģ ™†† †’6‡Ž€V(•Œž9 !ŒPSg…”ŒŒ$ yh “hŒø hE„ xg@(Q7uŽø(T @rƒr°YBy(e8EL•2ueXcw%Žā ”‘!6G€Xyi|˜ä†ˆŋ5Ä@  ††•† ĀĀ !PŒ ™'ˆ/WIfQƒ8† F5‰1y†F — L ™Ž:ˆ$Ž †–†bĀH ŠxB@†Ð6–°†5 "CĶ?`˜zˆ—œCC˜C°šžC˜†õ ČĀ Œþ–… ™˜™8ˆ— ™HYT†!ˆĀ†Ü X1@9†Y‰ Ė‰ƙHˆˆ`šŽ’ˆ0ˆˆ ›‡w ËŠ‚Œv†vY—šŽvi5–qÐ‰N—œ†Ä@  † ††›†Ē—Ž†7ž†<&Č 8•†Ÿ†V'Į˜–Ž †R'ĒRˆĄ†!ÄH!”˜†œ›W–2g”‰f@ 4!&18†ī ™&†ÁPš ™1W‰G6„†A ˆAĒ†Ĩ.šĨĢ†3ČĀ–ˆ†z†ĪĒ0ˆˆ4ĨĒJˆ4ˆˆdˆ<ˆ"ˆ@ˆˆĶ†Ó1šíˆ2ˆPHŽ §Œ"!’"Ļ†ÄH ‹D@‡ĄÐ6–°‡ĻRWB&ˆY™$"g ˜˜†u”“@ˆ ™†•†H†Ą††œ†›†š†ˆ ˜† †Ÿ†žŽ0†—†–†Ļ†€†YŽˆˆˆ)ˆ†z†“†Ž††Œ†‹†Š†‰†ˆ†‡†††…†„†ƒ†x†}†‚Že|†w††Éˆq”Ž [†ßHere a†given conditions:Ž"-We draw two lines DF and DG.Ž%/’%!an ellipse that is tangent to FD †D.[ŒZ -We construct the tangent line aŒpoi†E, which is lying onŠ3 ellipse. T–@1ŠGinterse†b FD and DGˆ[IŠ H resp† ively.[Œ;1. † ˆG %Equation of˜rˆŠ as followsŒrŽz>6.25î’x^2+2.768î’y† 0.03124† î’y-11.39†$-†2847Š2.11=0RˆūŒÄJdef‰ k(x,y)=øXRŽVdone[&[ô2’ôL‰eųDG:R3 y=4.486îėë3î’x-4.088Š. We define it as f(x,y)Rˆ(ŽŒ=y†>–L+”LRŽ4done[Œ@[ŒI†n† ˆULˆq equation FD:Œl,†Ŧ-222.9ˆ§167.8˜ĒthisˆĪg Ī!ŽĪŒ†Ī(žF†ŅŠ 4Objective:‰ want to †Md†ne shortest distance IH.˜Ų‰#Step 1.ŒÄH‰S”Alope ofŠN†Bge†bŽ ellipse a‡y g†„n poi†‚‹ƒ:‘@RˆBeq:=6.25î’x^2+2.768î’y† 0.03124† î’y-11.39†$-†2847Š2.11=0RŒSŠMŒ 2Œh4Ž+(346î’yž)125’+ŒĶ 781Œ‰’c000ŒÂ-'11ŠĨJ00Ž“-ŽIŽžŒ"†"†H’%H1211”E*ˆę“8 impdiff(eq,x)Ry'=‘ -‡@Œï‡N†ķ‡h+ŠÎy-‰H5ŽÆl+5536ˆ&Œ–r[†Uˆ -Step 2.1 We define the slope of IH to b˜above.Rˆ7Œ>Ž:m(x,y)=Ž-4î’Œa312500î’x+781î’y-284750Œ††,x+5536ˆ&ŒRŒcdoneŠtŒą[Ž[†nˆÄŠŅ3ŽŅQ If E=(a,b) is a point onŠĘ ellipse and†ã, whereˆíŠ+tange†-l‰(‡‰)@Ž:) atŽSE. Now,‹8Š2equation‰=can‰> written asŽļ y-b=‡0-a). ŽĖ:-We define the following function to representŠ$lˆ- IH:RˆŽCh(x,y)=y-b-m(a,b)*(x-a)RŽ*done–:[ŒBONotˆƒat kŒE0”wsŠœellipse. Since E is ˆ ”,†<ˆv=0.ŽW_g[† Step 4.ŒÁ J Now we †Į d H, whichˆb‰interse between DG a†,IH or fŠųŠ  : —''temp1:=solve({Œ5=0,70},{‡@}0ˆŠ‹cx= 6250000†00î’a 2ˆ +27680000†0î’bŒ2Œ!6+3124a†"-569493614544î’a-2919416Š?-5819268Œ^*625007007132î’a+2803724†qŽI 506385821Ž,y=Ž Š/†Bˆša’™Žŧ+†™1’N đ 4+1401426ˆĪŠ·10977††|a-1277†7†ó+23281ˆÁ0â·R‰NKH_x:=getright(temp1[1])R’ÔŠ\“pĒÖő+3124000000î’aî’b-569493614544†-2919416ˆ †819268*625007007132†+ +28037248Ž, 69506385821Rˆ0Œ6 H_y:=getright(temp1[2])RŒ\ ŠL†_Š˜Œ2Ž+†ķ1’k’ Œ 4+1401426ˆÁŠÔ10977†Šˆå-1277† 7î’b+23281ˆÞ0øÔ[ˆļ‰Step 5."7J Now we find I, which is the intersection between IH a†, DF or h(x,y)ŠsgŠ:‘P#temp2:=solve({h(x,y),gŠ},{† })R†1x=  12500000î’aŒ 2Œ( +5536Šb  -6147928†9+bî’Œ]6248ˆN+†912330Œ1 -4777266Œ… ˆ\604†x-ˆ&94316†† b-5044037Ž­,y†ŪŒC-Ä 2786Œŧ†š’đŽâ+ˆy97†IŠ―žŲ2+13926†ū†öˆs441331-1103†šˆ‹1911242‡!ÎĩR‹`‹fI_x:=getright(‹Ķ[1]Œ€ 12500000î’a†2Œ +5536Šb  -6147928†9+bî’Œ=6248ˆN+†912330Œ1 -4777266Œe ˆ\604†x-ˆ&94316†† b-5044037R†žŒ‘I_y:=getright(temp2[2])RŽķ-Ž  2786ŒØ†Ũ’ÖŽ)+ˆ–97†fŠÚžö2+13926†Û‡ˆ441331#-1103†ŨˆĻ1911242‡>äŌ[†‰mStep 6.z6= Set up the distance function and subject to k(a,b)=0:Rˆ 4define Lˆ,L1)=(I_x-H_x)^2+† y† yˆ L1*(ŒB)RŽ@done–P eq1:=diff(’S,a†uŠ4Œw L1î’Œ†25î’aŒ 2Œš+78†.b’+000ŽG-ŽA1139Œ 1’ +2w 28037ˆA0î’aqŽ/ +12417248ˆb’ ++14014264î’aî’b-5109777ŒM-1277† 7î’b+2328116†Đf*5625007007132î’a+2803724800î’b-569506385821+Ž  278Š@ˆ-aŽ 2Ž) +12339744ŠL’ Œ 2+1392679ˆwˆl441331ŒL-1103891ˆ„ 1911242000Œ 5536604Š'ˆb4316ŒŪ0†l37ŒĻ î’Œ6#557ŒŦ†Ųa–|–yÎ`‘ - ’’†pĸ‰c2+13926792î’aî’b-441331000†-1103891ˆ1911242† 5536604Š)23394316†7 b-5044037Œ(2Œ2+Œ;$56075†eŠg+1401426†J b-5109777†Œf*62500†713ˆĪ +28037248ˆĒb-569506385821f-Žpœ<ˆ ŠDŠ€†ö”§ŒĶ+†é1’c’ĮŒ4–Ŋ‹2”ē‹3277† 7î’b+23281‡•4ÖΠ—-ŒÃC+1250000††î’a+3124b-569493614544*6ˆ27007132ˆ128037248’1 506385821Œ2-Ž œ<ŽŠ”…Œk2ŽB+2768”Ķb’"Œ!6š·a š†§-2919416ŽŨ819268âԔ›”ÞŒ‘J‰F6‰‰A‡;7928< 553660Š1233943†’‹h044‡GT+‘2 ’2“- ‘ģ‡Ū‡2Š +5536000î’bŒ-6147928î’a+bî’Œ56248†2a+†912330Œ1 -4777266* ˆ^604î’a-ˆ(94316ˆl-5044037’rŒˆfŽ 125†ˆŸa žþŧ°ŧÎđŽ°-’Ž6ŽŽŽ° Û+2768”!bĄo6+3124000000î’aî’b-569493614544†-2919416ˆ †819268*625007007132†+ +28037248Ž, 69506385821Œ1=0Rˆ9Œ?eq2:=diff(L(a,b,L1),b)ˆ$Œf L1î’Œu78†aŒ‚ˆ0ŒŽ+’#69†bŒĨ125ŽJ-ŽD2847Œ1Šýl+†Č} ŠÓ†æ‹Œ2Ž/+‡=1’ō’ Œ 4+1401426‰H‹[10977‡1‰l-1277† 7î’b+23281‰eŽÍ*5625007007132î’a+2803724800î’b-569506385821+Ž  278Š@ˆ-aŽ 2Ž) +12339744ŠL’ Œ 2+1392679ˆwˆl441331ŒL-1103891ˆ„ 1911242000Œ 5536604Š'ˆb4316ŒŪ0†l37ŒĻ î’Žļ ˜,†þÃîÃÎÅĄN‡w€"14014264î’a+2483449600î’b-1277†7*625007007132Š-8037†2Œ-569506385821Œ3+Œ<!1392679Œ84† 48Ž81103891Œd 553660ˆŒ†~3394316ŠŒ5044†oY-Žc ˜†y Šœ†ŊˆÉaŒ2Ž—+1241’ŧ’ Œ4+—ˆĢ510977†ú†Šˆ˜ î’b+23281†Ē00ã& —+‡W†Ã46‡Ž‡ˆa+‡#72ŠËb+92‡-‡0• *5536604î’a-1233943160î’b-5044037+Ž  28†2480†'Ž 6250000†ˆaŽ72Ž@+2768”!b’"Œ!6+3124’@Š‚ 6949361454Š 29194†žŒŸ819268Œ€*Š€7007132†-+˜ĶŠK 506385821”›Ü-Œå'šƒ+‰”Ób”€âsĨE™l“E 1E‡ŽĄ  ‰<™=-6147928î’a+bî’ˆ62480ˆ†912330Œ -4777266* 5536604†A-ˆ(94316†: b-5044037ŒR2Œ[ˆfŽ 1250000ˆm’,Œ† +ˆ]Šb ôŧÎđŽ°-’Ž6ŽŽŽ° Û+2768”!b Ņ6+3124’@‹&69493‡†54‹D29194‡BC8192689*4625007007132î’a+2803724800î’b-569506385821=0RŠŠeq3:=diff(L(a,b,L1)†)ˆ%Œ7 Šh†TaŒH2Ž +276Œo’Œ&+3124†–ˆ‹1139ŒB-2847ˆž1211000Œ™ˆ  œĶ[ŽŽ–Âsolve({eq1,eq2†3},{ŒË})ŠÆŒųERROR:Overflow†K‰ Remark:[ŽčQ1) It is inevitable to switchˆ an external compu†  algebra system for further wŒ+ation†Cd†Galysis.=[C2) By switching to Maple 10, we get a = 1.784381519,b†2.123030625[ŒM 3) We dra†Hhe point Ž# –HŒ‚ –JŒ”3in††ˆAGeometry Strip above†–hich will see this ŒÎ4ma†Į es closely †Ũ†*hatˆËhad cojectured earlier. Žž‰,eActDGXXFy €ĸHč à Ąx125F_x  €EũL €xzŒIb \{Ta€ €2åH  Ę^7 ôab yTb €XÏ]a  žūaF_y € €=ëq €Øb ˆÔba €ôŒIb €8-Na ÍĶa €EũL €xzŒIb \{Ta€yG_x  €Ï9™„ r§b œR…‘a€ €„ËX €āŽfb u4šba Аr@b €Ðïv˜„a *į„‘aG_y  €Ï9™„ r§b œR…‘a€  ;Ċ €ÍßÂb 8ŨÕba €7Jb €h‘0a pÔ§aH_x%ûH_yPP  oPx€^3a44 (ans1 X((Fx  €Ï9™„ r§b œR…‘a€ €„ËX €āŽfb u4šba Аr@b €Ðïv˜„a *į„‘a$Fy  €Ï9™„ r§b œR…‘a€  ;Ċ €ÍßÂb 8ŨÕba €7Jb €h‘0a pÔ§ab44 qpa€e1$$F   €sd  Ļab aL1 €  €œR…‘ €Ï9™„ r§b œR…‘a€ €„ËX €āŽfb u4šba Аr@b €Ðïv˜„a *į„‘a  €Ï9™„ r§b œR…‘a€ €Ðïv˜„ u4šb TÎ #a € €XÏ] ôb @x}a €EũL €xzŒIb \{Ta€ \{T €EũL €xzŒIb \{Ta€ €2åH  Ę^7 ôab yTb €XÏ]a  žūa  € €Ï9™„ r§b œR…‘a€ €„ËX €āŽfb u4šba Аr@b €Ðïv˜„a *į„‘a  €EũL €xzŒIb \{Ta€ €2åH  Ę^7 ôab yTb €XÏ]a  žūa   €œR…‘ €Ï9™„ r§b œR…‘a€  ;Ċ €ÍßÂb 8ŨÕba €7Jb €h‘0a pÔ§a  €Ï9™„ r§b œR…‘a€ €h‘0 8ŨÕb āĻ;Na €\{T €=ëq €Øb ˆÔba €ôŒIb €8-Na ÍĶa €EũL €xzŒIb \{Ta€  €8-N ˆÔb  š%La €EũL €xzŒIb \{Ta€   €=ëq €Øb ˆÔba €ôŒIb €8-Na ÍĶa €EũL €xzŒIb \{Ta€  €Ï9™„ r§b œR…‘a€  ;Ċ €ÍßÂb 8ŨÕba €7Jb €h‘0a pÔ§a  €  €œR…‘ €Ï9™„ r§b œR…‘a€ €„ËX €āŽfb u4šba Аr@b €Ðïv˜„a *į„‘a  €Ï9™„ r§b œR…‘a€ €Ðïv˜„ u4šb TÎ #a € €XÏ] ôb @x}a €EũL €xzŒIb \{Ta€ \{T €EũL €xzŒIb \{Ta€ €2åH  Ę^7 ôab yTb €XÏ]a  žūa  € €Ï9™„ r§b œR…‘a€ €„ËX €āŽfb u4šba Аr@b €Ðïv˜„a *į„‘a  €EũL €xzŒIb \{Ta€ €2åH  Ę^7 ôab yTb €XÏ]a  žūa   €œR…‘ €Ï9™„ r§b œR…‘a€  ;Ċ €ÍßÂb 8ŨÕba €7Jb €h‘0a pÔ§a  €Ï9™„ r§b œR…‘a€ €h‘0 8ŨÕb āĻ;Na €\{T €=ëq €Øb ˆÔba €ôŒIb €8-Na ÍĶa €EũL €xzŒIb \{Ta€  €8-N ˆÔb  š%La €EũL €xzŒIb \{Ta€   €=ëq €Øb ˆÔba €ôŒIb €8-Na ÍĶa €EũL €xzŒIb \{Ta€  €Ï9™„ r§b œR…‘a€  ;Ċ €ÍßÂb 8ŨÕba €7Jb €h‘0a pÔ§aeqF €ŧd €  †y  Ļayx Z}y €sdx x1eq1 Ø ØF   €sd  Ļab aL1 €  €œR…‘ €Ï9™„ r§b œR…‘a€ €„ËX €āŽfb u4šba Аr@b €Ðïv˜„a *į„‘a  €Ï9™„ r§b œR…‘a€ €Ðïv˜„ u4šb TÎ #a € €XÏ] ôb @x}a €EũL €xzŒIb \{Ta€ \{T €EũL €xzŒIb \{Ta€ €2åH  Ę^7 ôab yTb €XÏ]a  žūa  € €Ï9™„ r§b œR…‘a€ €„ËX €āŽfb u4šba Аr@b €Ðïv˜„a *į„‘a  €EũL €xzŒIb \{Ta€ €2åH  Ę^7 ôab yTb €XÏ]a  žūa   €œR…‘ €Ï9™„ r§b œR…‘a€  ;Ċ €ÍßÂb 8ŨÕba €7Jb €h‘0a pÔ§a  €Ï9™„ r§b œR…‘a€ €h‘0 8ŨÕb āĻ;Na €\{T €=ëq €Øb ˆÔba €ôŒIb €8-Na ÍĶa €EũL €xzŒIb \{Ta€  €8-N ˆÔb  š%La €EũL €xzŒIb \{Ta€   €=ëq €Øb ˆÔba €ôŒIb €8-Na ÍĶa €EũL €xzŒIb \{Ta€  €Ï9™„ r§b œR…‘a€  ;Ċ €ÍßÂb 8ŨÕba €7Jb €h‘0a pÔ§aeq2 Ô ÔF   €  † ī}b  ĻaaL1   €r§ €Ï9™„ r§b œR…‘a€  ;Ċ €ÍßÂb 8ŨÕba €7Jb €h‘0a pÔ§a  €Ï9™„ r§b œR…‘a€ €Íß o”b 8ŨÕa xzŒI €=ëq €Øb ˆÔba €ôŒIb €8-Na ÍĶa €EũL €xzŒIb \{Ta€  €EũL €xzŒIb \{Ta€ €Ø é“b ˆÔa   €=ëq €Øb ˆÔba €ôŒIb €8-Na ÍĶa €EũL €xzŒIb \{Ta€  €Ï9™„ r§b œR…‘a€  ;Ċ €ÍßÂb 8ŨÕba €7Jb €h‘0a pÔ§a   € €Ï9™„ r§b œR…‘a€ €„ËX €āŽfb u4šba Аr@b €Ðïv˜„a *į„‘a  €EũL €xzŒIb \{Ta€ €2åH  Ę^7 ôab yTb €XÏ]a  žūa  r§ €Ï9™„ r§b œR…‘a€ €„ËX €āŽfb u4šba Аr@b €Ðïv˜„a *į„‘a € €Ï9™„ r§b œR…‘a€ €āŽf  !å€b u4ša xzŒI €EũL €xzŒIb \{Ta€ €2åH  Ę^7 ôab yTb €XÏ]a  žūa  €EũL €xzŒIb \{Ta€ Ę^7 ōĻb ôaeq3āāF  † €xz € b 4 ba @9b €8aa h‰ a^fH$a,b(p+a)^2+(q+b)^2f1H$x,y(p+x)^2+(q+y)^2lineFGF  €by €  €8a € €rb 4 a€ 4  € €rb 4 a€b Ð € €rb 4 a€a  €axp q temp1 X((Fx  €Ï9™„ r§b œR…‘a€ €„ËX €āŽfb u4šba Аr@b €Ðïv˜„a *į„‘a$Fy  €Ï9™„ r§b œR…‘a€  ;Ċ €ÍßÂb 8ŨÕba €7Jb €h‘0a pÔ§atemp2 HFx  €EũL €xzŒIb \{Ta€ €2åH  Ę^7 ôab yTb €XÏ]a  žūa(Fy € €=ëq €Øb ˆÔba €ôŒIb €8-Na ÍĶa €EũL €xzŒIb \{Ta€x44 qx0DD  rry0DD  rr03010b00200008000000001e4dxœíÝpŨāņЀ0#il ›82–‘5ÂóW]Ȗ8þĮˀž‹ą…ę0ąä%.ēK†AVؑB0Ī8ŧåÝóe}ĐÔĩëÚÍåĻŽ/åËÆYöĘwvŨ.w5{įŦd}Ū;ŨžkÏũ{ýįMÏHâOŒ6`?IKÝŋî~ý^wOĸ^wē9/óĄ9/Ôo6-jļü†írãjxtøĀĀӆ!ûā°ņ„aėÜķ?·TĶgÕüž 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