00020002010018Parametric Equations.ACT00080200161 Parametric Inter.EAC01000000149bÉ 1 Parametric Inter.EAC†– Equations.ACT†6†9 ˆF†JŠ Ž ŽkE †(Œ'[ˆw¨]†Š\Ž † x=f(t) y=g”t:pŒÇ er with iˆËval–Uˆð <Žïer>[ŒSˆctŒ2Œ‡-3Žsˆm2t’ŸŽ7 Graph the e  for Œ’ˆgiven’‹.\ŒZ -2ˆ+‡‘eŠ(”aŒ¨ and identifyŠJgraph.?ˆºtŒÖ2ŒÞ-3ˆØ î’î’î’eqn(1ÚˆÔ2tŒö” 2  -3[ŒnEliminate the ”FŒ and identifyŠ"graph.ŽŽˆž2cos†¡†¯ î’î’î’eqn(1°ˆª2sinŒ"–#2)[‡A‰{ -Œþî2Ž¢ R†GŽ.2cos(t)îxRŒ2î’†ŒStš2sinŠ2y 2†œ2ŽOxŒ&2Ža+y’ŒyŽz4î’އ˜„ C+š+˜}–[’ê simplify(ans)ŠtŒr4—&¬N(°FinaForm¸$N†Graph2D†܆& 3Šð ˆLISTSYSˆü†@4ˆ< Modify 0ˆPˆ<STATCALC TˆdˆŒ< \\ˆx SequenceˆŒ,ˆxSheetˆOä|’ŠŒ`’o lveEqˆˆ´†~`wràˆ´(Upˆì’tupFLG1ø(Š<†Lis†{ D‰ˆPic†dÜViewWin†ˆˆŒ_osve†v‡4¤x䉉°Œä’¼ ä’È´ä’ÔÈäŽP à‰ †ä† † ‡^†— ’ø’†¬!’<’P’d(’†u4’ˆ2Š@’†rL’†rX’†rd’ Üp’!ð|† "‘ˆ† #‘”† $‘, † %‘@¬† &‘T¸† '‘hÄ’‰‹І)‘܆ 0*‘¤è¸†ä+†ô†  †,ˆŒ-( †„.<† 0P$† 1d†%†%2x<†3ŒH† 4 T† 5´`† EÈl† F†Ö x’ˆZŠó„†ŽI†Ö† J†Öœ† K†Ö¨† L†Ö´† M†ÖÀ† N†Ö̆ O‘|؆ P†Öä† Q†Öð‡0R‹¸†Öü††Ñ†å S††‡Oˆ˜†]‡[ †A ˆ^†ˆÖ_†(†ˆ´,†a†0†b†4†‰u8’Œ•†D† ͆P†Ά\†‰hh†ׇ¸t’؆€†Ù†Œ†Ú†˜†Û†K¤‘h Financial‡‚Forma†n †ˆ † †ˆŒE‡¥” systemä]listˆŠ^]=’_Œ†`Šaˆ " b~yÁŽŽž¾ þ@þ€þÀÿÏ@ÁŒy ÿ@ÿx‡¸°:ŒŽXŠ”XÖ¸Œ»† a†ö seq_h†­b† NewFolde‹ ˆ3 ’ ˆÔ’Ó†0’ŠÎ ” —H††‡î<ˆxÒ<ŒÄ ˆˆ †š<‡’“  ö<ö<þxþðÿ,ÿh†<N† ‰ †Œ †ˆo† †0†!@†0PSheet1úê¬^Š2ž3ž 4†5ž ŒH3DÊ|èš|š†††Š˜| “i™” ™†2††. Š Ž ˆ†’kE ‰9ŽÀ‰`†$«d«yŽ †j€ « ‰µ†J†8 ††† oŠŽ † `– P2Æ$´$’y– p”x” –<”  ™€–`’ œ0˜ä(1…0qy`ˆä#Y‡uYƒˆH˜–˜œšHª„Æ$ÎH™Œ™˜žˆÌ†v Œ˜ ’ˆQ– Â9ÂZÂ{œ½¶Þ˜Ø¬ð†ˆü †c’e5‰yˆ” – œ™8Vˆ»[‡ ˆÇThe equation is xŒ2Ž"+y’Œ5=4.[Œ@Ž!When - ‘ î ’“ŒœFind†>e Žo‰Že&’^ŒÐ A=(-2,2) a†< B=(3,4).\Œ’†‰ŒΈt‡ZÎi‘fG†`wA“Tˆˆ Y™%€dQa)†2ˆ Š †Ž1vŠ<ˆr† Ž A† ŽXLŽ'ˆeC˜ u˜x˜yŽ †qˆqˆuÊÈ `ˆ±B@†¸Ð6–°† ˆ ŠÇ† ˆ1 ’ ”L–CA˜CŠÃŠ&`ŒÏ¬CŠüˆcs”|[‰F[Œ Study the vector onŠŒ(line.\Œ5VŒ" analysisΈL‰œΊ•t†žŠ2%€e`ƒ‡ gw† Y™Œf† ' gtŠ †† †1v† †9†>r† ˆ#ˆ A† 'LŽ'Š C˜ u˜x˜yŽ †qˆqˆ~Ä@  Žu†´ÈÈ `Š]B@†ÄÐ6–°†2ˆ2Œ&ŠC ’ Œ†C†CŽU–Bq9v5ˆT Y™ (GfˆP†  ™Y—4sDˆ¬Šw ¢— ‡KÊ’—A˜—‰)Œ&`‹5¶—ˆ› ªô †µ ÈÀ ‹  t@Ð6–°†q9v5ˆ Y™ (GfˆP† ™Y—4sDx†*`†. † † †;Ä@  † †Œ † Žˆ"ˆ¢*ˆ@†{ ÈÀ Š@s˜‹S†‹_†‹@ކ¾ÊÈ Š™–͈až-¢‰¢ÉˆXˆŸˆ¹‡!Š"ˆªóœÉr™Tgf‹T– ¨ÉˆK£Tˆˆaˆ{‡¬Š"ˆŠû †Ž †††† †† †††0ˆ-Œ [†CVector AB on the line has Ž"ˆrelati†" OA+AB=OB. Œ@Then†>†-OA.[ŒV’_AddŠVpoint PeŒ}ˆl.\ŒŠ P=(x,y)ΈC†GÎ∊2%€e† `t5H€Š Œf‹ ˆG–”Dv‡@rˆ A† 'LŽ'‰$C˜ u˜x˜yŽ †qˆq‡¤Ä@  €‡¶ÅH! A@†Ð6–°† †††`Œ †’Ž †9 ÈÀ  Š#u–Cq9v5B†[ Y™ (GfˆT† ™Y—4sD€Š[††|ÄH ƒŒCP˜†‰eQrˆC `$y1¨†††GÄ@ ŠŠYˆ Žˆ"ˆª*‡ÊÀ  Š¤ÉA–ÉR–ÉyÉ¢‚‡WÅH!ŒÛB™a’¾ˆÕ žÛaŒ×t¥a8‹a (GfˆP ™Y—4sDy† `†† † †Ä@ ˆ †Œ†3ÊÿÿÿŽ"ˆ!¢*ˆ@"†] ÈÀŽ@s@†mÐ6–°†tSY™_†‹@Ž#† ËÈ Š™–Bž*ž9²Š‰$¢ÉˆXˆŸˆ¹‡Š"ˆ%ªó&œÉr˜ÉghŠÉŒ dŠÕ¦ÉˆK'£Tˆˆaˆ{‡ŽŠ"ˆˆ(£‚ˆÏ†a) Ä@  † † †††Ž†&Ž ˆˆ)†(†#†&†"†Ž-ŽNˆ5ŒZ[Š`When we have the point P=(x,y) and vector PA, Œ PA=tîAB.O T˜OŒcOA-OPˆ)(OB-OA)[Œ{Ž It follows†atR†Ÿ ާ-2Œ°Ž-š xŒÏyŒ×ˆ˜(œD3Œó4ªC¤c)[š!-x†-y+2B=’© 5î’t† 2† \Œ Calculatorˈ*†,߆†50RŠ ŒA-2ŒJ2Ž-˜ xŒiyRŒ’,ž&2¤#yŒ²+pŒŽÈtî(š™3Œâ4’˜°¹ )š¢ 5(î’Œit’Žù–‹Y5ÀS Graph2D,† †Š3Š@† ˆLISTSYSˆL4N†Modify†; €†($ˆ< STATCALC ¤†Uˆ(ˆŒ< ¬\ˆ< Sequence†<,ˆxSheet†Š4†|’ŠŒ°’olveEq†´ˆ´Lwr0†Í (Upˆ<’ tupFLG1H†<(Š<†Lis†{p†dD‰ˆPic†´Ü ViewWind؆‰,äб‡@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~[yÁŽŽž¾ ¤@yÁŽŽž¾ þ@þ€þÀÿÿ@ÿxyÁyŽŽž¾ þ@þ€þÀÿÿ@ÿx‡¸yÁŽŽž¾ þ@þ€þÀÿÿ@ÿx‡¸yÁŠŽž° †† a†F seq_histb† NewFoldeŠ]ˆ3 ’ systemä]l†0’^]=” –˜††î<ˆxö<þxìðÒ´ emyÁŠŒsystˆä]list’^]= ö<þxþ´<† ††† †’ ’,† †0†@†@PSheet1úê¬^Š2ž3ž 4ž 5ž ‡›ŠX3D´|PSheet1èê¬^Š2ž3ž 4ž 5ž ŒH3D Šˆ i™” ™†† Š Ž ˆ8¸Dkäûx† ††$’S’\¢œ†jD²<‡Žh†(žŽ `– PÝÆ$´$“$– p”x” ,ˆˆ  ™€ `” ˜$’.ˆ>(1…0qy`ˆH#Y‡uYƒ†U ˜–0– šH®Æ$ÊHš©š¶´¨5°´þaþ ‰¼†Œ˜ °˜’ ®`–$ ™†OŠq[Œ;ŽAs the result,Ž-x-2=5t’y+2=2 -î(1…0qy`ˆH#Y‡uYƒ†U ˜–0– šH®Æ$ÊHš©š¶´¨5°´þaþ ‰¼†Œ˜ °˜’ ®`–$ ™†OŠq[Ž;eAct0200115 OP=OA+tîAB.EAC01000000123bÉ 5 OP=OA+tîAB.EAC†Parametric Equations.ACT†1†4 ˆA†EŠ Ž ŽkÅ †(Œ'[ˆr¨]†Š\Ž † x=f(t) y=g”t:pŒ¯ er with in† val–UˆëŽ•ˆq— ’Œ™Find†;e Žl‰Že#’[ŒÍ A=(-2,2) a†< B=(3,4).\ŒêGeo‰byΈj‡MÎ\‘Y:2%€e†`h@Š 2%€e™™™™™YŠ †† †v† †!†&r† Ž A† 'LŽ'Š C˜ u˜x˜yŽ †qˆqˆuÆ@  ŽuŠÄH ƒŠP@†¬Ð6–°†‰eQrB†&`$y1ˆIоŽ>Š†ß ÈÀ ŠUv–Cv‡•46‡ Y™ `iTBtŠ žCˆóËÈ `Š˜… n ~–ãˆ,ĘމKˆŸˆü‡…ŠNˆ¢* ‡¤ÅH!‹A‰ Ð6–°†††`Œ †’Ž  †, ÈÀ  Š#u@†<ŽCq9v5B†N Y™ (GfˆT† ™Y—4sD€Š[††  † †wÄ@ ŠKˆaŽˆ"ˆ ª* †·ÊÀ  Š¤ŠA–ŠR–ŠyŠ¢‚‰ ÅH!ŒßB–U“Š€ Ÿ"‰3—"t¥"8–˜P¨˜‰ˆŠ£"ˆ‰”Êÿÿÿˆ"††Ä@  † ††† ÈÀŽs@†.Ð6–°†5)Y™5 ™@ŽˆˆK¢aˆˆaˆ{††Š"ˆ¢‹ †Ž‹Š¡r˜‹ghŠ‹Œ dŠ—¦‹ˆK¢ìˆˆaˆ{‡Š"ˆˆ« †£4ˆ^ˆˆˆ&ˆ‡bŽ ˜;ˆYyˆø‰‡ˆ- ††ˆ9’[‹›[Œ $The equation is  OP=OA+tîAB.[Œ It follows[Œ.xŒ6yŒ>=š-2ŒV2Ž1ˆZšB5"[ŽT =˜#5î’tJ2† +2\Œ® CalculatorˈÀ†Ä߈ʆ.´RˆÖô–RŒžµî’6tŒ[-’Z”-¤-+”e†€ŒºÀS Graph2D,† †Š3Š@† ˆLISTSYSˆL4N†Modify†; €†($ˆ< STATCALC ¤†Uˆ(ˆŒ< ¬\ˆ< Sequence†<,ˆxSheet†Š4†|’ŠŒ°’olveEq†´ˆ´Lwr0†Í (Upˆ<’ tupFLG1H†<(Š<†Lis†{p†dD‰ˆPic†´Ü ViewWind؆‰,äб‡@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~[yÁŽŽž¾ ¤@yÁŽŽž¾ þ@þ€þÀÿÿ@ÿxyÁyŽŽž¾ þ@þ€þÀÿÿ@ÿx‡¸yÁŽŽž¾ þ@þ€þÀÿÿ@ÿx‡¸yÁŠŽž° †† a†F seq_histb† NewFoldeŠ]ˆ3 ’ systemä]l†0’^]=” –˜††î<ˆxö<þxìðÒ´ emyÁŠŒsystˆä]list’^]= ö<þxþ´<† ††† †’ ’,† †0†@†@PSheet1úê¬^Š2ž3ž 4ž 5ž ‡›ŠX3D´|PSheet1èê¬^Š2ž3ž 4ž 5ž ŒH3D Šˆ i™” ™†† Š Ž ˆ8¸Dkäûx† ††$’S’\¢œ†jD²<‡Žh†(žŽ `– PÝÆ$´$“$– p”x” ,ˆˆ  ™€ `” ˜$’.ˆ>(1…0qy`ˆH#Y‡uYƒ†U ˜–0– šH®Æ$ÊHš©š¶´¨5°´þaþ ‰¼†Œ˜ °˜’ ®`–$ ™†OŠq[Œ;ŽAs the result,Žx=5t-2Ž,y=2t+ -îÀS Graph2D,† †Š3Š@† ˆLISTSYSˆL4N†Modify†; €†($ˆ< STATCALC ¤†Uˆ(ˆŒ< ¬\ˆ< Sequence†<,ˆxSheet†Š4†|’ŠŒ°’olveEq†´ˆ´Lwr0†Í (Upˆ<’ tupFLG1H†<(Š<†Lis†{p†dD‰ˆPic†´Ü ViewWind؆‰,äб‡@ôxäŠ ŠŒä(  ä<´äP$Èäd’ÜäŽx < †ä† † H’’T’’`’’l’’x’’„’’’’œ’’¨’ ’´’!’À’"’Ì’#’Ø’$’ä’%’ð’&’ü’'‘@’(‘T’)‘hˆ¾Œ*‘|,’+‘8’ˆ‹§D¸†ä-†P†  †.’\’0’h’1’t’2’€’3’Œ’4’˜’5’¤’E’°’F’¼’H’È’I’Ô’J’à’K’ì’L’ø’M‘,’N‘@’O‘T’‰^‹k(’Q‘|‰ŒR‘@’S‘¤ˆ–ŒT‹¸X† †ä]††d† ˆ^’h’_’l’`’p’a’t’b’x’”’|’Œ•’ˆ’Í’ˆ2ŒÎ’ ’Ð’¬Ü††ˆŒ!†” system‰list’‡]=’‡‰’†ü‰’†ø‰’†ô~[yÁŽŽž¾ ¤@yÁŽŽž¾ þ@þ€þÀÿÿ@ÿxyÁyŽŽž¾ þ@þ€þÀÿÿ@ÿx‡¸yÁŽŽž¾ þ@þ€þÀÿÿ@ÿx‡¸yÁŠŽž° †† a†F seq_histb† NewFoldeŠ]ˆ3 ’ systemä]l†0’^]=” –˜††î<ˆxö<þxìðÒ´ emyÁŠŒsystˆä]list’^]= ö<þxþ´<† ††† †’ ’,† †0†@†@PSheet1úê¬^Š2ž3ž 4ž 5ž ‡›ŠX3D´|PSheet1èê¬^Š2ž3ž 4ž 5ž ŒH3D Šˆ i™” ™†† Š Ž ˆ8¸Dkäûx† ††$’S’\¢œ†jD²<‡Žh†(žŽ `– PÝÆ$´$“$– p”x” ,ˆˆ  ™€ `” ˜$’.ˆ>(1…0qy`ˆH#Y‡uYƒ†U ˜–0– šH®Æ$ÊHš©š¶´¨5°´þaþ ‰¼†Œ˜ °˜’ ®`–$ ™†OŠq[Ž;eAct0200166 Throw a baseball.EAC010000002602É 6 Throw a baseball.EAC†Parametric Equations.ACT†6†9 ˆF†JŠ Ž ŽkE †(Œ'[ˆw¨]†Š\Ž$<Ššing–>\Œ% Tap ----†ˆ>Έ=†ÎŒŸŒœˆˆê("y† `vcy@Œ #! iv†³ y”VC‡`†˜DˆŒKr† ŽmA† 'LŽ'‰Z‰E’ u˜x˜yŽ †qˆqˆ¹0ÊH  ‘jˆÈH!Œˆ·Ð6–°†††– ’ Œˆ,ÊH! Š@†;Ð6–° Bˆ'®B’ œ$†Z0Ê@!ŒZˆRˆlŪl”¢ ™ X” ˜†Ö”Ö’Y ˜††^Å@ ”pˆµ ‡  ÈÀ Šàt–áP’s6†] Y™‚”‚f!“Šw!5rPŠŽ·ˆK ‡WĘaˆˆaˆ{‡tŠ"ˆ ¢*ˆ@ Ž‹s—•¢`†Š’ ˆ† † †$Ä@  ˆ †: [laŽ"ˆˆ.ª.†h ÈÀŽDr@†xÐ6–°†D#™x†Š Y˜ ™‰QA4g†– ™ VE!A#† Y™†CŽKˆœœaˆˆaˆ{¾ˆ†*1¢*†‹’‹‰-0.2î’(íëx-4)^2+4†9Ž+ˆ˜ɘ/-íëy†&3Ž‘ˆ™‹<ˆÃˆ3ªˆ„‰‰ e•e[‰The path of t† object is Œ! modeled by [Ž:í¸ í¹[Š =Žvìðîcos(ín)í´%-Ž.gŽ62Ž?†%Ž2ŒR+ŠEsinŒE+yìð[Œm †`: The initial velocityŽ" g: Grav” t†3 ime(second)ŽMŽU ÔWhenˆk=10 (ft/†9), ޳ín=45î, g=32’  and  †²=3ˆ:),[™ í¸ø¹ŸAP5î’î’t‘K-16î’t“4‡+¦8î’t+3\© Calculatorˆ߆† ,R†ˆ"10îcos(45î)í´Ž-Œ!32Œ*2Ž3†&ŒF+ˆGsinG+3RŽjŽm5Œuî’œB†Œ‘t–0-:16”;)ž…+àj8Œê†´†‡>ÀS Graph2D,† †Š3Š@† ˆLISTSYSˆL4N†Modify†; €†($ˆ< STATCALC ¤†Uˆ(ˆŒ< ¬\ˆ< Sequence†<,ˆxSheet†Š4†|’ŠŒ°’olveEq†´ˆ´Lwr0†Í (Upˆ<’ tupFLG1H†<(Š<†Lis†{p†dD‰ˆPic†´Ü ViewWind؆‰,äб‡@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~[ÃÁŽŽž¾ ¤@ÃÁŽŽž¾ þ@þ€þÀÿÿ@ÿxÃÁÃŽŽž¾ þ@þ€þÀÿÿ@ÿx‡¸ÃÁŽŽž¾ þ@þ€þÀÿÿ@ÿx‡¸ÃÁŠŽž° †† a†F seq_histb† NewFoldeŠ]ˆ3 ’ systemä]l†0’^]=” –˜††î<ˆxö<þxìðÒ´ emÃÁŠŒsystˆä]list’^]= ö<þxþ´<† ††† †’ ’,† †0†@†@PSheet1úê¬^Š2ž3ž 4ž 5ž ‡›ŠX3D´|PSheet1èê¬^Š2ž3ž 4ž 5ž ŒH3D Šˆ i™” ™†† Š Ž ˆ8¸Dkäûx† ††$’S’\¢œ†jD²<‡Žh†(žŽ `– PÝÆ$´$“$– p”x” ,ˆˆ  ™€ `” ˜$’.ˆ>(1…0qy`ˆH#Y‡uYƒ†U ˜–0– šH®Æ$ÊHš©š¶´¨5°´þaþ ‰¼†Œ˜ °˜’ ®`–$ ™†OŠq[Œ;\ŒGrapḧ†ÊŒ˜üN&$%FinaFormà$N†Graph2D†% 3Š ˆLISTSYSˆ$†@4ˆ< Modify XˆPˆ<STATCALC |ˆdˆŒ< „\ˆx SequenceˆŒ,ˆxSheetˆO |’ŠŒˆ’ olveEqˆ´†€`wrˆ´(Upˆ’tupFLG1 (Š<†Lis†{HD‰ˆPic†ŒÜViewWind°ˆŒ_osve†v‡?Ìdxt†_‰Ø †U†y”øˆð†i†äŽ'‡TŒ´ä’0ÈäŽO <‰* †ä† † ‡kH’ЦJT’’`’’l’’x’’„’’’’œ’’¨’’´’’À’†̆†؆ †ä†!†ð†"†ü†(#‘@† $‘T‰JŒ%‘hˆn† &‘|,†'‘8† !(‘¤D¸†ä)†P†  †*†W\†[+†h†ˆ=t†-†€†.†Œ†0†˜†1†¤†2†°†3†¼†4†Ȇ5†Ô†E†à†F†ì†H†ø†I‘,† J†È† K‘T† L‘h(† 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