00020001010008main.ACT0002020017Riverlog Experiment.EAC0100000030ab Riverlog Experiment.EAC main.ACT'*4 ;  ^E 2[)hProblem xloring With An$by Ravinder KumarAlcorn State Unsity8, USAQrkF@a?.eduj Object[i of this e-A#ity to ee a pwan] Well see how ClassPad c)be qgused to create an experiment, lore the problem, conjecture,vanEhen generalize. Here is=D:[[l"Th.L-shapwr channel which has a 90 n. A straight log of woody floating inŠO. tjOur aimo fiĆEngesM that couldM through. Assum% at widths horizontal$&vertical>s a`same".d*For sake of simplicity we will assume that e log is soin)it can be represented by a line. [ Strategy[mWo first create TexperimRth{ ituation whichvertica nd horizontpaths=e water u jchannel havewidthBuseto estimlength(longe flo$ rough. ThenFeverify our answer using C optimization technique of calculus. Finally, we raise some general=j?sNsldpthem. The follow~model consistssan L-shaped water chann&with a 90 turn\rCtant width throughoutrJisvtructjthat iH(both horizont هTti$ly)T1 uniq[\Ih to Explore'+7!1. Expand the strip Model). 3 >2>Resize it by clicking on E p inicToolbare. m"3Ru4animati[ as follows:]  a. CView# En 6 AWUI. Ano scrolMppearsb.tedlaZleft2sright arr to movelog.   #W4. If the log happens to step overur corn#even slightly change its length as _ follows: x X a. Dismisgwscr$bar bL licking onic (that s an X inl)sr. шbb. NextNKsolid/ arrow U*remeJ o geometry too.Y_c. Selec?7it, Pars-N area (indicator g22left.)0D d. Change the length to desired number by dctly keying it in.   5. RunOanimation again. %7 =6. Repeatprocess until you hfmaximum possibl.M{TY2 can get backɈ text windowclickÆ Swap or Rze while in Geometry >. t[[aNVopeg8 ical modelqgfollog strip(d experiment us`above in.ucPs.\ModelΈ  ' `ehIG  V DcvKr   A 'L'C uxy qq @   H@6 `!&1WYHCICrCUY]%@910Vu0h1Ft7Length: (,  @;6BG6!M xD 5 sh.i! v@J`0 "H" Ȩˎ   #C+`7`(!CD"zCމAo ahw @ "@"+B@269"`2T u5@!e Ȑ{h\ ywR# X ň`!0   &1WG74   |c!lGA&i`  ', )F@<6+`7`Zjs@  YF".H. ȐنB E͜O͈Y".!2( W2ghn -   ; [ Geometrical Solution[%gWhat is the posi of upper cornchannel relative to log aCmaximum lengthjust braces it?\Hint!0Why do you thinkat e ! upper cornofchannel E!should be right inBmiddle5mWmaximum-length log ( -wise) as" !braces io slipj othpart(from verticalJhorizontand vice-'sa)?\0 DiagramΉ>"BHMQY `F!dpY  U DvKr   A L C uxy   #J@! N$\ ]M@l6#w `%C UB6&!I'ވYE".(.H.) Lʶ*  +, O@<6+`7`O),s@  Y*(".-..H @K| / #U0 H1@!ވY-d"263 " @6  4B DC+`7R+5H Y3603ihw.7"@"8#J`9 N׈:ۈ`".; @!.&f@ 2?x!F@6&c`3 UB'BN+AH"".653(4BCY CD"  E( 1B@86?+`7`ZCFo@  YDBihw".G.H "  {IA͙͈J͈YG͈".u%Ȉ4+3?0&<89./)*$% [Expand the strip Diagram'M. It displays.maximum length log as it brace$upper corner. Observe { jrectangl-OJHK OLIlThey are both 11 squsmliealongdonal oftwo4Y*each[ is 2F in. Hence)= that can pas#rougeS channelj2k.[\Think About It!K 1.What if the width o "channel is 2 units? ? 2. IIH50 feetHI3H 100 meter4t yards?.[[$"hNow we give an algebraic solution2 problem i% general setting containedm last quesCinq"Think About It!"[[ fSuppose thickness of e water channel is t. We know-#at any maximum length log must bracTe _GvIcorner,@is " lienline representingd. me origincoordin systedatNw}soڈleft vertical baFalong <ithe y-axis and lower horizontal bank !long#x2. Therefore, coordinates ofIuppIcornP are (t,t).[[dM|g restmleft. Let|two endYbe(0,b)(a,0Rdelvar a,b,x,y,tRdone[ReseJ1iables] -Equationlinewhichlies,x4a<+yNb=1Rxa+yb*;[4 +Upper corn(t,t) lies on this ne. Thus,Rvntnn;[ Length ofe log  aP22+b$5, which must be maximized. ItP enough to aGY. But58a 09function of two variables a and b. So, we solve ta +b+ =1 for a.R1P(P =1,a)R}a=b0b-t[n'Next substitute thislue into a~2+bȍgetright(ans[1])an/rꆯ 2 b-t +b['?Set derivative of the abo expression with pect to b0. GR~z diff(ans,b)R24-63 t-4/t+3 3solve=0b=Nt-MXks,b=G+G -t 2 ,b=2tR&#getright(ans[2])bRDt- -3=\Na<tHix"3C[Clearly, the wer is: ja+bB length<O?q t+ -3t 2RlengthR02/3E!t-o-l2Ǝdelvar a,b,x,y,tdone[\ RemarkL XUOIn the starting problem, t was 1 and our exploration showed maximum length /2X2`2. O=maem=cal soluGabove confirmsis.  Thus, if*nidth of channel i00 yardLeَ, longest log at can floatroughS200 A 282.84 Et. ch[W[`g1We now ask you to consider the case when width of vertical wat*channel is s and;1ehorizont:4t. dra diagramjafor. Exphfollowing strip. [m\vDRΈM) p `A1@ DvKr  A 'L'C uxy qNL@  O F@6 &P1`  B' NQByR@Y4N".S.T˒E 5 UC # NYNV!hD. TW@  VUS".X.Yc Ȑ@r6t y~ Z#`CC+`57͈[#YX fo".\#2Q]4 ѕO^B A+`xш_ @  ^]\ _Q%ZVYTUOPR1^[Z \ What Do You Think?#`/;Once again revisit how yA] analyzed the model earlier and ImakIn inform/guess8isMgeneral case. se [Let (0,b)U(a,0) be two endpoints oflog. Coordinateuppcord are (s,t).&R22 var x,y,a,b,(RCPdone["1Equation of the line along whichoges is 9xAaJ+y\bd8=1. CoordinatFk upper corn are (s,t). Tpass|through&R sxRstyt=1H;s;t; solve(ans,a)Sa=`b_b-t[Length of the log is a 2+b$.R1getright(ans[1])aRQbs b-t:mF-Ze[++Now we apply an optimization technique to <2. ForXRWmust set@zeroqderivavemn,  and solve.[R (diff(ans,b)=0 R2b;3-t-3b2=t-s t+1J2=0We must the equation:[ ޤ=0 Por+G =s 2tR% factor(b*3-t>3-3bUD V+^ )Rb-: solve(ans,bQb=t+'1[ getright^[1])b\_aBsZ s 2t1&31R^d expand(ans)RKs+X3`JK =s+(s$)ɍD]a 2+b$lengthR>3sAtUV++ts{L1&3+~P߈POOВ[[] 2 +3s t%s"!tC1K3V+SPdPOtOД1$=3(st)2/8+-[t.3e\Remark 1: Verificationa N (y R5s+ s2t3.]EDs3F<1<]x[\2bN Verify R3s 2t*++tOs9Lm1u3+~PdPOOВ]RO+tMgsr/23 +3s!)t3]FR[S[S>lAs an example, suppose the width of vertical channel is 15 ft;d+horizont-0-. 6Thenalengbo longest log{ll be as follows:R 15sR150t0>approx(~)(", 35.1174119QA5z@7BBelow we create geometrical model to verify our answer.[\ G7V.AtionΈ(c,27 CfffffBN `W)x1  ` 33333Y Rr DvKr  A 'L'C uxy qd9 1@6f  33 Length: %"Se[  j Rb!p| vYT i8$y `f  G@%6, +FV#w%gXHCHC(4I78`Ch@ ei@ oe jH"gQnkbHl  XmU!A@7nCCB ` o'@  mlpAH#nAl* q5H fru N@68y  ywR#  s@#EC+ `rB`t ߞӈtהYu r2%7h Q&1WG7] ihegvb!bOrW2gw'x>"r@6 AUWt !Yy8!`ACC7BK z{@! Dz1HzC{C@ Cx|ox } 8]ڒՈh~ EFpۈoW2g@}~utrs|{xzypolnmH )19IQaifgedAq [eActaPP   tbsbbPP t tslength   t tss t ts ts  tss tss t  xs0yt0020012eActivity Save.EAC01000000309e Riverlog Experiment.EAC main.ACT'*4 ;  ^E 2[)hProblem xloring With An$by Ravinder KumarAlcorn State Unsity8, USAQrkF@a?.eduj Object[f of this e-A#ity to ee a pwan] Well see how ClassPad c)nc be used to create an experiment, lore the problem, conjecture,vanEhen generalize. Here isD:[[ e"Th.'L-shapwr channel which has a 90 n. A straight log of woodK floating inŽmcW . Our aimufiĆEngesM that couldM through. Assum% at widths2 horizontal&vertical>s a`same".d*For sake of simplicity we will assume that e log is soin)it can be represented by a line. [ Strategy[gWo first create TexperimRth{ ituation whichvertica nd horizontpaths=e ocwater channel havewidthBuseto estimlength(longe flo$rough. c$Then we will verify our answer using - optimization technique of calculus. Finally,Praise some dgeneralE?sNsldpthem.  follow~model consistssan L-shaped water chann&l]with a 90 turndCtant width throughoutrJisvtructjthat i=O(both horizont هTti$ly)T1 uniq[\Ih to Explore'+F!1. Expand the strip Model). 3 >2>Resize it by clicking on E p inicToolbare. m"3Ru4animati[ as follows:]  a. CView# En 6 AWUI. Ano scrolMppearsb.tedlaZleft2sright arr to movelog.   #W4. If the log happens to step overur corn#even slightly change its length as _ follows: x X a. Dismisgwscr$bar bL licking onic (that s an X inl)sr. шbb. NextNKsolid/ arrow U*remeJ o geometry too.Y_c. Selec?7it, Pars-N area (indicator g22left.)0D d. Change the length to desired number by dctly keying it in.   5. RunOanimation again. %7 =6. Repeatprocess until you hfmaximum possibl.M{TY2 can get backɈ text windowclickÆ Swap or Rze while in Geometry >. t[[aNVopeg8 ical modelqgfollog strip(d experiment us`above in.ucPs.\ModelΈ  ' `ehIG  V DcvKr   A 'L'C uxy qq @   H@6 `!&1WYHCICrCUY]%@910Vu0h1Ft7Length: (,  @;6BG6!M xD 5 sh.i! v@J`0 "H" Ȩˎ   #C+`7`(!CD"zCމAo ahw @ "@"+B@269"`2T u5@!e Ȑ{h\ ywR# X ň`!0   &1WG74   |c!lGA&i`  ', )F@<6+`7`Zjs@  YF".H. ȐنB E͜O͈Y".!2( W2ghn -   ; [ Geometrical Solution[%bWhat is the posi of upper cornchannel relative to log aCmaximum lengthjust braces it?\Hint!0Why do you thinkat e ! upper cornofchannel E!should be right inBmiddle5mWmaximum-length log ( -wise) as" !braces io slipj othpart(from verticalJhorizontand vice-'sa)?\0 DiagramΉ>"BHMQY `F!dpY  U DvKr   A L C uxy   #J@! N$\ ]M@l6#w `%C UB6&!I'ވYE".(.H.) Lʶ*  +, O@<6+`7`O),s@  Y*(".-..H @K| / #U0 H1@!ވY-d"263 " @6  4B DC+`7R+5H Y3603ihw.7"@"8#J`9 N׈:ۈ`".; @!.&f@ 2?x!F@6&c`3 UB'BN+AH"".653(4BCY CD"  E( 1B@86?+`7`ZCFo@  YDBihw".G.H "  {IA͙͈J͈YG͈".u%Ȉ4+3?0&<89./)*$% [Expand the strip Diagram'E. It displays.maximum length log as it brace$upper corner. saObserve rectangl-OJHK OLIM. They are both 11 squsmliealongdonal of 0two=*each[ is 2, in. Hence)=u &that can pas+roughachannelj2k.[\Think About It!K 1.What if the width o "channel is 2 units? ? 2. IIH50 feetHI3H 100 meter4t yards?.[[$"eNow we give an algebraic solution2 problem i% general setting containedm last quesCin "Think About It!"[[ `Suppose thickness of e water channel is t. We know-#at any maximum length log must bracTe pOfNorner,@is e  lienline representingd. ge origcoordin systedatNw soڈleft vertical baF6baloy-s dPhorizont:2x2 . Therefore,upper corn are (t,t).[[ aThe log rests along t eft bank andowR. Let\ ordinates of5two endYqb0,b)O(a,0Rdelvar a,b,x,y,tRdone[Rese1iables] -Equationlinewhichliesx"a*+ybD=1;+ Upper corn(t,t) lies on this ne. Thus,R ta+b*=1R8;[r Length ofe log  aP22+b5, which must be maximized. ItP enough to aGY. But58 2a functiQ wo variable a and b. Soe solve aέb =1 for a.Rsolve(ta+b9=1,a)RILa=Hb0b-t[n'Next substitute this value into a2+bȌgetright(ans[1])an/AꔯĒ[}?Set derivative of the abo expression with pect to b 0. [R  diff(ans,b)R'2b34-63R t-4/t+32_tb-3solve=0b=t-mb2,b=G+GF,b=;tf$getright(ans[2])bRt- -3t22R'-a<tHi2-x3C-[Clearly, the wer is: ja+b length<ʖ2+qi2t 3!t-?-=22+-3_2Rrxdelvar a,b,x,y,tRdone[\ RemarkLXUOIn the starting problem, t was 1 and our exploration showed maximum length /2X2`2. O=maem=cal soluGabove confirmsis.  Thus, if*nidth of channel i00 yardLeَ, longest log at can floatroughS200 A 282.84 Et. ch[W[`c1We now ask you to consider the case when width of vertical wat*channel is s and;1bPhorizont;5t. dra diagramkbfor. Expi following i strip. [x\DYΈM) p `A1@ DvKr  A 'L'C uxy qNW @  O  F@6 &P ` UB'sNQByRވY4N".S.T˒E 5= UC # YNV!hD`  TW*@  VUS".X.Yn Ȑ@}6t y~ Z#`CC57͈[#YX fo".\#2Q]? ѕO^B A+`xQ_@  ^]*\".%ZVYTUOPo1{[ \ What Do You Think?#`/;Once again revisit how yA] analyzed the model earlier and ImakIn inform/guess8isMgeneral case. se [Let (0,b)U(a,0) be two endpoints oflog. Coordinateuppcord are (s,t).&R2&delvar x,y,a,b,s,tRdone[!1Equation of the line along whichoges is HxPaJ+ykbs/=1. CoordinatFk upper cornare (). Tpass|through&RΎsxsty=1H;s;t; solve(ans,a)Sa=` bs b-t[ Length of the log is  a522+bP.Rb]getright(ans[1])aR}:mF-┮[++Now we apply an optimization technique to 1. ForVPVmust set?zerooderiv`vet +b 2 , and solve.[R,((diff(ans,b)=0 RNbW3-tk3-3b=t-s t+1J=0We must the equation:[ ޤ=0 Por+ t+3b2 =s/R%% factor(b>3>-t>3-jUiޅ )Rb-: solve(ans,bQb=t+ 183[ getright^[1])be_a s t+ s"2*t61>313Rv| expand(ans)RKs+Љ3`JK"=s+(&t<)ɍDt 2 13Ra"2+b=lengthR>3sAtUV++ts{LĒ+~P߈POOВ[[] 3s 2 t++t:s9L1&3V+~PdPOOД1B=3(st)2/8+-rt.3e2+t 2\Remark 1: Verification1a= N (y Rfs+ sst3.]EDs3<1<]ފ[2bN Verify R3s 2t*++tOs9Lm1u3+~PdPOOВ]RO+tMgsr/23 +3s!)t3]FR[S[S>eAs an example, suppose the width of vertical channel is 15 ft;d+horizont-=104. Thenalengbo longest log{ll be as follows:RI 15sR150t0>approx(~)(", 35.1174119QA5z@7BBelow we create geometrical model to verify our answer.[\ G7V.AtionΈ(c,27 CfffffBN `W)x1  ` 33333Y Rr DvKr  A 'L'C uxy qd9 1@6f  33 Length: %"Se[  j Rb!p| vYT i8$y `f  G@%6, +FV#w%gXHCHC(4I78`Ch@ ei@ oe jH"gQnkbHl  XmU!A@7nCCB ` o'@  mlpAH#nAl* q5H fru N@68y  ywR#  s@#EC+ `rB`t ߞӈtהYu r2%7h Q&1WG7] ihegvb!bOrW2gw'x>"r@6 AUWt !Yy8!`ACC7BK z{@! 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