{VERSION 6 0 "IBM INTEL NT" "6.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 261 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 266 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Tim es" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Warning" -1 7 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 2 2 2 2 2 1 1 1 3 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output " -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 } 3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Plot" -1 13 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT 256 62 "F. Advanced: Building a R oller Coaster using Cubic Polynomials" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 57 "In this Maple worksheet, we build a r oller coaster using " }{TEXT 265 2 "n " }{TEXT -1 54 "peak and valley \+ points. Our approach is to determine " }{TEXT 266 5 "(n-1)" }{TEXT -1 162 " cubic polynomials that connect these points and then to calcu late the angle of steepest descent/ascent of each hill and resulting t hrill factor for the coaster.." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 257 19 "I. Gett ing Started" }}{PARA 0 "" 0 "" {TEXT -1 113 "We clear all variables. \+ To avoid multiple solutions, we set the maximum number of solutions va riable equal to 1." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restar t:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "_MaxSols:=1:" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT 260 16 "II. Data Points" }}{PARA 0 "" 0 "" {TEXT -1 152 "We enter the number of peak and valley points (n) and th e x coordinates (xdata), y coordinates (ydata) and slope conditions (s lopes) for these points. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 136 "In this example, the first peak is at (0,75) the n a vally at (50,0), then another peak at (100,50) and the last valle y is at (200,0). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 267 114 "In your work, you should use the collected peak and val ley data points for Greyhound (The Devil, Steel Dragon II)." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "n:=4:" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 22 "xdata:=[0,50,100,200]:" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 19 "ydata:=[75,0,50,0]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "slopes:=[0,0,0,0]:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 0 "" }{TEXT 258 0 "" }{TEXT -1 0 "" }{TEXT 259 0 "" }{TEXT -1 34 "III. Connecting Cubic Polynomials" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 186 "In this example, we have 4 peak a nd valley data points. So we must determine 3 connecting cubic polyno mials. We do this by simply repeating the approach taken in part C) t hree times. ." }}{PARA 0 "" 0 "" {TEXT -1 50 "We determine cubic polyn omials of the form f(x) = " }{XPPEDIT 18 0 "a*x^3+b*x^2+c*x+d;" "6#,** &%\"aG\"\"\"*$%\"xG\"\"$F&F&*&%\"bGF&*$F(\"\"#F&F&*&%\"cGF&F(F&F&%\"dG F&" }{TEXT -1 236 " that connect consecutive peak and valley points. \+ Note that n peak and valley points will determine (n-1) hill pieces. \+ The \"do\" loop below is a generalization of the technique for determi ning one hill (from a previous Maple worksheet)" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 24 "for i from 1 to (n-1) do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "f:=x->a[i]*x^3+b[i]*x^2+c[i]*x+d[i]:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "fp:=D(f):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 159 " s[i]:=fsolve(\{f(xdata[i])=ydata[i],fp(xdata[i])=slopes[i],f(xdata[i+1 ])=ydata[i+1],fp(xdata[i+1])=slopes[i+1]\},\{a[i],b[i]=Pi/(xdata[i+1]- xdata[i]),c[i],d[i]\}):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "assign(s [i]):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "f:=x->a[i]*x^3+b[i]*x^2+c[ i]*x+d[i]:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "tplot[i]:=plot(f(x),x =xdata[i]..xdata[i+1]):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od:" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "And, we display the (n-1) hill pie ces." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }} {PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name changecoords has been r edefined\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "display(tplot [k] $k=1..(n-1));" }}{PARA 13 "" 1 "" {GLPLOT2D 195 150 150 {PLOTDATA 2 "6'-%'CURVESG6$7S7$$\"\"!F)$\"#vF)7$$\"3SLLL3x&)*3\"!#<$\"3S+y?M_Y*[ (!#;7$$\"3zmm\"H2P\"Q?F/$\"3+NLoY*HOY(F27$$\"3XLL$eRwX5$F/$\"3bq_]C_%o T(F27$$\"3=ML$3x%3yTF/$\"3UfWF$eW;N(F27$$\"3gmm\"z%4\\Y_F/$\"3m)y)Hu&) fpsF27$$\"34LLeR-/PiF/$\"3+MfXn*3!zrF27$$\"3;++DcmpisF/$\"3w;C)R=\\72( F27$$\"3vLLe*)>VB$)F/$\"3Oiq2C:oXpF27$$\"3o++DJbw!Q*F/$\"3:r\"zFjqq!oF 27$$\"3%ommTIOo/\"F2$\"3Ywew&[$Q^mF27$$\"3^LL3_>jU6F2$\"3xRMa*\\sR]'F2 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$\"3;LL$3s?6z\"Ffjl$\"3'**e5#Q6BLcF/7$$\"3#***\\7`Wl7=Ffjl$\"3H8#o)e5? 2YF/7$$\"3zmmm'*RRL=Ffjl$\"3S29%GVw6q$F/7$$\"3ammTvJga=Ffjl$\"3YI_fiJm jGF/7$$\"3[L$e9tOc(=Ffjl$\"3ES-P&Q!fF@F/7$$\"35+++&Qk\\*=Ffjl$\"3'*Q<> E<**Q:F/7$$\"3PLL3dg6<>Ffjl$\"35NIe*=I_t*Fiy7$$\"3vmmmw(Gp$>Ffjl$\"3\" oi.Q!G2;dFiy7$$\"3G+]7oK0e>Ffjl$\"316g.5g[lDFiy7$$\"39+](=5s#y>Ffjl$\" 3Ixoj3c%*ypFf[l7$$\"$+#F)F(Fgz-%+AXESLABELSG6%Q\"x6\"Q!Ffim-%%FONTG6#% (DEFAULTG-%%VIEWG6$;F(F`imF[jm" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 68 "The coaster plot looks good. It matches \+ the peak and valley points." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 261 52 "IV. Calculation of Angle of Steepest Descent/ Ascent" }}{PARA 0 "" 0 "" {TEXT -1 119 "We need to determine the angle s of steepest ascent/descent. We use what we learned from part VI of \+ module C. That is," }{TEXT 268 20 " for these functions" }{TEXT -1 129 " the x-coordinate of the point of steepest ascent/descent is the \+ x-coordinate of the midpoint between the peak and valley points." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "for i from 1 to (n-1) do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "f [i]:=x->a[i]*x^3+b[i]*x^2+c[i]*x+d[i]:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "fp[i]:=D(f[i]):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "mdpt[i]: =.5*xdata[i]+.5*xdata[i+1];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "slop e[i]:=abs(fp[i](mdpt[i]));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "rangl e[i]:=arctan(slope[i]):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "dangle[i ]:=evalf(rangle[i]*180/Pi):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od:" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 64 "The angles of st eepest descent/ascent or shown (in order) below." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "print(dangle[k] $k=1..(n-1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%$\"+,6v.m!\")$\"+YK*4j&F%$\"+k(*)po$F%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 263 41 "V. Safe ty Restrictions and Thrill Factor" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 262 15 "Safety Criteria" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 75 "For this coaster, the ma ximum angle of steepest descent/ascent is given by" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "maxhill:=max(dangle[k] $k=1..(n-1));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%(maxhillG$\"+,6v.m!\")" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 57 "which is less than 80 degrees and so the \+ coaster is SAFE." }}{PARA 0 "" 0 "" {TEXT -1 132 "We display a more co mplete picture of the coaster below. Numbers indicate degree measures of the angles of steepest descent/ascent." }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "warni ng:=\"SAFE\":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "for i from 1 to (n-1) do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "if dangle[i] > 80 then warning:=\"NOT SAFE\" end if;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "infoplot[i]:=textplot([mdpt[i],f[i](mdpt[i]),round(dangle[i])]);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "wplot:=textplot([40,70,warning]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "display(wplot,infoplot[k] $k=1..(n-1), tplo t[j] $j=1..(n-1),scaling=constrained);" }}{PARA 13 "" 1 "" {GLPLOT2D 247 225 225 {PLOTDATA 2 "6,-%%TEXTG6$7$$\"#S\"\"!$\"#qF)Q%SAFE6\"-F$6$ 7$$\"$]#!\"\"$\"++++]P!\")Q#66F--F$6$7$$\"$](F3$\"*+++]#!\"(Q#56F--F$6 $7$$\"%+:F3F=Q#37F--%'CURVESG6$7S7$$F)F)$\"#vF)7$$\"3SLLL3x&)*3\"!#<$ \"3S+y?M_Y*[(!#;7$$\"3zmm\"H2P\"Q?FR$\"3+NLoY*HOY(FU7$$\"3XLL$eRwX5$FR $\"3bq_]C_%oT(FU7$$\"3=ML$3x%3yTFR$\"3UfWF$eW;N(FU7$$\"3gmm\"z%4\\Y_FR $\"3m)y)Hu&)fpsFU7$$\"34LLeR-/PiFR$\"3+MfXn*3!zrFU7$$\"3;++DcmpisFR$\" 3w;C)R=\\72(FU7$$\"3vLLe*)>VB$)FR$\"3Oiq2C:oXpFU7$$\"3o++DJbw!Q*FR$\"3 :r\"zFjqq!oFU7$$\"3%ommTIOo/\"FU$\"3Ywew&[$Q^mFU7$$\"3^LL3_>jU6FU$\"3x RMa*\\sR]'FU7$$\"3E++]i^Z]7FU$\"3'HR$phIKFjFU7$$\"3/++](=h(e8FU$\"3\"3 (G@K=URhFU7$$\"3A++]P[6j9FU$\"3)3[4(\\r@\\fFU7$$\"3[L$e*[z(yb\"FU$\"3g ^'**Q*zUpdFU7$$\"3+nm;a/cq;FU$\"3aVb;RMwZbFU7$$\"3mmmm;t,mFU$\"3H1+L[U98 \\FU7$$\"3M+]i!f#=$3#FU$\"353+r!>\\\"zYFU7$$\"37+](=xpe=#FU$\"3e,Rp\"H tIX%FU7$$\"3-nm\"H28IH#FU$\"3%3(GU=kl9UFU7$$\"3%om\"zpSS\"R#FU$\"3c25O hr=%*RFU7$$\"3cLL3_?`(\\#FU$\"3[&p35)GbbPFU7$$\"3fL$e*)>pxg#FU$\"3Cr0r +&pw]$FU7$$\"3D+]Pf4t.FFU$\"3oNwpv+i#H$FU7$$\"3ZLLe*Gst!GFU$\"3#*\\KdG r*=1$FU7$$\"39+++DRW9HFU$\"39q1sUN/EGFU7$$\"3:++DJE>>IFU$\"3;**f9p5h)f #FU7$$\"35+]i!RU07$FU$\"3;=)f\\5aCQ#FU7$$\"3$)***\\(=S2LKFU$\"3m\\Rq;x &y9#FU7$$\"3nmmm\"p)=MLFU$\"3LPUMZRtU>FU7$$\"3U++](=]@W$FU$\"3tkx))*o< 0t\"FU7$$\"36L$e*[$z*RNFU$\"3*oT$)R5A]a\"FU7$$\"3e++]iC$pk$FU$\"3K)**e pR]/N\"FU7$$\"3Sm;H2qcZPFU$\"3qc6APL)f<\"FU7$$\"3Y+]7.\"fF&QFU$\"3sxuz n4N.5FU7$$\"3amm;/OgbRFU$\"3cdcJ2n%)\\%)FR7$$\"3I+]ilAFjSFU$\"35jAI)e, 3\"pFR7$$\"3)RLLL)*pp;%FU$\"3+*ph#\\Yw^bFR7$$\"3WLL3xe,tUFU$\"3%))y#*Q u%\\&H%FR7$$\"3Wn;HdO=yVFU$\"31/_0ybQ\">$FR7$$\"3a+++D>#[Z%FU$\"3B%f(G *e([3BFR7$$\"3)om;aG!e&e%FU$\"3u%GP%GXGg9FR7$$\"3wLLL$)Qk%o%FU$\"3]()= q0#4Td)!#=7$$\"3m+]iSjE!z%FU$\"3Uu60:!H#[QF\\\\l7$$\"3u+]P40O\"*[FU$\" 3U*o#HT=%o/\"F\\\\l7$$\"#]F)FL-%'COLOURG6&%$RGBG$\"#5F3FLFL-FH6$7SFg\\ l7$$\"3OLL$3x&)*3^FU$\"3a1Jz%>xJ-(!#>7$$\"3!pm\"H2P\"Q?&FU$\"3]+56@-nC CF\\\\l7$$\"3RLLeRwX5`FU$\"3@N&['*p^Oa&F\\\\l7$$\"3pLL3x%3yT&FU$\"3'p! 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Coaster Design" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 "Now, the FUN part..." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 161 "A. Desi gn your own cubic polynomial coaster by modifying the Maple code given above. Your design should be creative, interesting, and as thrilling as possible." }}{PARA 0 "" 0 "" {TEXT -1 73 "B. Can you design a cub ic polynomial coaster with maximum thrill factor?" }}}}{MARK "17 1 0" 23 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }