{VERSION 5 0 "IBM INTEL NT" "5.0" }
{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 Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 }
{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 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 "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" 0 13 1 {CSTYLE "" -1 -1 "
" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }}
{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 55 "First, enter the distance \+
as a function of the angle..." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "D = 25 + 10*tan(theta);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#/%\"DG,&\"#D\"\"\"*&\"#5F'-%$tanG6#%&thetaGF
'F'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "Next, solve for theta..." 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "s
olve(%,theta);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'arctanG6#,&%\"DG#
\"\"\"\"#5#\"\"&\"\"#!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 86 "Now
, we know that the density function is proportional to the derivative \+
of the above." }}{PARA 0 "" 0 "" {TEXT -1 35 "So, first we find the de
rivative..." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 10 "diff(%,D);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"
\"\"F%,&*$),&%\"DG#F%\"#5#\"\"&\"\"#!\"\"F/F%F%F%F%F0F+" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 68 "Next we integrate the above for the entir
e range of values, D=0..50." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "> " 0 "" {MPLTEXT 1 0 15 "int(%,D=0..50);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%$intG6$,$*&\"\"\"F(,&*$),&%\"DG#F(\"#5#\"\"&\"\"#!\"
\"F2F(F(F(F(F3F./F-;\"\"!\"#]" }}}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" 
}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "Hmm, Maple can't integrate." }}
{PARA 0 "" 0 "" {TEXT -1 127 "To get the density function we will want
 to multiply the above by the proportionality constant k and set the r
esult equal to 1." }}{PARA 0 "" 0 "" {TEXT -1 55 "First, we tell Maple
 to approximate the above result..." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(%);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#$\"+**)z0Q#!\"*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 69 
"Now multiply this by k, set the result equal to 1, and solve for k...
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 
"solve(k*%=1,k);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+hrl+U!#5" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 53 "This gives us our much-sought afte
r density function!" }}{PARA 0 "" 0 "" {TEXT -1 158 "Unfortunately the
 d(theta)/dD expression is the fourth one up, and Maple only allows th
ree % references. So I copied and pasted the expression into the below
." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
28 "%*1/10*1/((1/10*D-5/2)^2+1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$
*&\"\"\"F%,&*$),&%\"DG#F%\"#5#\"\"&\"\"#!\"\"F/F%F%F%F%F0$\"+hrl+U!#6
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 111 "Let's plot this to make sure
 it agrees with our amusement park cannon intuitions about the probabi
lity density:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 16 "plot(%,D=0..50);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 
300 300 {PLOTDATA 2 "6%-%'CURVESG6$7en7$$\"\"!F)$\"3]z8Cx)4Sz&!#?7$$\"
3SLLL3x&)*3\"!#<$\"3as!H=A<QD'F,7$$\"3zmm\"H2P\"Q?F0$\"3c\\(zJ!H(pp'F,
7$$\"3XLL$eRwX5$F0$\"3Yp()4_D*)\\sF,7$$\"3=ML$3x%3yTF0$\"3?t:`v>+tyF,7
$$\"3gmm\"z%4\\Y_F0$\"3oy<lxCDp&)F,7$$\"34LLeR-/PiF0$\"3wN'RP$*)[#H*F,
7$$\"3;++DcmpisF0$\"3;(H]Q&R:85!#>7$$\"3vLLe*)>VB$)F0$\"3*G#o?xx'46\"F
Q7$$\"3o++DJbw!Q*F0$\"3\"48p;bh7A\"FQ7$$\"3%ommTIOo/\"!#;$\"3ImC**GB'*
\\8FQ7$$\"3^LL3_>jU6Fin$\"3#pdXlxIyZ\"FQ7$$\"3E++]i^Z]7Fin$\"3Jg3\"y(4
/S;FQ7$$\"3/++](=h(e8Fin$\"3^F\"e3E[W#=FQ7$$\"3A++]P[6j9Fin$\"3H4'esW&
GC?FQ7$$\"3[L$e*[z(yb\"Fin$\"3>E\"Qz%HSDAFQ7$$\"3+nm;a/cq;Fin$\"3]`_Df
_e)[#FQ7$$\"3mmmm;t,m<Fin$\"3f'HL-v\\*HFFQ7$$\"3=+]iSj0x=Fin$\"3,A8j\\
=GEIFQ7$$\"3#omm;pW`(>Fin$\"3C3US6J&RH$FQ7$$\"3M+]i!f#=$3#Fin$\"3O2h0M
]()yNFQ7$$\"37+](=xpe=#Fin$\"3(RCrJ$\\PBQFQ7$$\"3-nm\"H28IH#Fin$\"3'G+
I4xz!GSFQ7$$\"3%p;a8d3AM#Fin$\"3g^2GI%4')4%FQ7$$\"3%om\"zpSS\"R#Fin$\"
37X]`&*ep^TFQ7$$\"3q$ek`1OzT#Fin$\"3?'QU%>rbsTFQ7$$\"3?+v$41oWW#Fin$\"
3e*ypZ,Vx=%FQ7$$\"3q;/^c++rCFin$\"3;(*4i!RFr>%FQ7$$\"3cLL3_?`(\\#Fin$
\"3kc@$fdJ1?%FQ7$$\"3R$3-)Q84DDFin$\"3!*Gc-&>9!)>%FQ7$$\"3eL3_D1l_DFin
$\"3#4M_(eZ/*=%FQ7$$\"3w$eRA\"*4-e#Fin$\"39\"R.uW/Q<%FQ7$$\"3fL$e*)>px
g#Fin$\"3oCWR2+V_TFQ7$$\"3#pmm\"z+vbEFin$\"3=zm&pgq65%FQ7$$\"3D+]Pf4t.
FFin$\"3k$Q&*py^K.%FQ7$$\"3ZLLe*Gst!GFin$\"3t\\zy&>Z!QQFQ7$$\"39+++DRW
9HFin$\"3#o$=3_5!\\e$FQ7$$\"3:++DJE>>IFin$\"37TFmEyu3LFQ7$$\"35+]i!RU0
7$Fin$\"3>XV0<f!G.$FQ7$$\"3$)***\\(=S2LKFin$\"3g#GoJ'oJKFFQ7$$\"3nmmm
\"p)=MLFin$\"3iE5m08*pZ#FQ7$$\"3U++](=]@W$Fin$\"3W\"*Q#\\WS`A#FQ7$$\"3
6L$e*[$z*RNFin$\"3[,%G20O!=?FQ7$$\"3e++]iC$pk$Fin$\"3_nnA*)G=9=FQ7$$\"
3Sm;H2qcZPFin$\"3Ljx%yPxJk\"FQ7$$\"3Y+]7.\"fF&QFin$\"3%Q%G&eg`V[\"FQ7$
$\"3amm;/OgbRFin$\"3GN$ROM!*oM\"FQ7$$\"3I+]ilAFjSFin$\"3@E;z'pm(>7FQ7$
$\"3)RLLL)*pp;%Fin$\"3LcN1A6k66FQ7$$\"3WLL3xe,tUFin$\"3Z\"4'[XNx85FQ7$
$\"3Wn;HdO=yVFin$\"3?5k\")))G%zF*F,7$$\"3a+++D>#[Z%Fin$\"3guXz?o!Hd)F,
7$$\"3)om;aG!e&e%Fin$\"3sKMowk@_yF,7$$\"3wLLL$)Qk%o%Fin$\"3=nlwDD!oF(F
,7$$\"3m+]iSjE!z%Fin$\"35*45\"[!)3EnF,7$$\"3u+]P40O\"*[Fin$\"3#=&RyVdF
_iF,7$$\"#]F)F*-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%+AXESLABELSG6$Q\"D6\"
Q!F[^l-%%VIEWG6$;F(F^]l%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 
45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
20 "Yup! We're on track." }}{PARA 0 "" 0 "" {TEXT -1 64 "Can we genera
te a graph of the cumulative distribution function?" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "int(%%,D=0..x);" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%'arctanG6#,&%\"xG$\"+++++5!#5$\"
+++++D!\"*!\"\"$\"+hrl+UF+$\"+-+++]F+\"\"\"" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 16 "plot(%,x=0..50);" }}{PARA 13 "" 1 "" {GLPLOT2D 
400 300 300 {PLOTDATA 2 "6%-%'CURVESG6$7S7$$\"\"!F)$\"3yc(pxjV(*>\"!#F
7$$\"3SLLL3x&)*3\"!#<$\"3)fuB%RBpgl!#?7$$\"3zmm\"H2P\"Q?F0$\"39him<)z(
p7!#>7$$\"3XLL$eRwX5$F0$\"3cj'[)=#3H,#F97$$\"3=ML$3x%3yTF0$\"35\">\"y)
)p*R#GF97$$\"3gmm\"z%4\\Y_F0$\"3a**4t+Li,PF97$$\"34LLeR-/PiF0$\"3wOoo?
)4ce%F97$$\"3;++DcmpisF0$\"3Iyl:H@*3e&F97$$\"3vLLe*)>VB$)F0$\"3)*[\\'=
>(R1nF97$$\"3o++DJbw!Q*F0$\"3\\L&HGMm\"QzF97$$\"3%ommTIOo/\"!#;$\"3L*)
o(4O**[L*F97$$\"3^LL3_>jU6Fjn$\"31WN<P*>)o5!#=7$$\"3E++]i^Z]7Fjn$\"3>G
(3&RwvO7Fbo7$$\"3/++](=h(e8Fjn$\"3VS#Q=uJTU\"Fbo7$$\"3A++]P[6j9Fjn$\"3
y-?SE=vC;Fbo7$$\"3[L$e*[z(yb\"Fjn$\"3MTxF\"**ff#=Fbo7$$\"3+nm;a/cq;Fjn
$\"3]PLD(=984#Fbo7$$\"3mmmm;t,m<Fjn$\"3TG'=^$eESBFbo7$$\"3=+]iSj0x=Fjn
$\"3i\"z;>6L(fEFbo7$$\"3#omm;pW`(>Fjn$\"3_6d2I,MqHFbo7$$\"3M+]i!f#=$3#
Fjn$\"3?#=S(R$y5M$Fbo7$$\"37+](=xpe=#Fjn$\"3X<GLN'[9s$Fbo7$$\"3-nm\"H2
8IH#Fjn$\"3,4jOpjiUTFbo7$$\"3%om\"zpSS\"R#Fjn$\"3-X*=+Q1ca%Fbo7$$\"3cL
L3_?`(\\#Fjn$\"3Uq%e*>Ij*)\\Fbo7$$\"3fL$e*)>pxg#Fjn$\"3y&\\=M)4'4X&Fbo
7$$\"3D+]Pf4t.FFjn$\"3d=fO)*)\\U%eFbo7$$\"3ZLLe*Gst!GFjn$\"3_sWKUNm_iF
bo7$$\"39+++DRW9HFjn$\"3I31;1,U]mFbo7$$\"3:++DJE>>IFjn$\"3f^jC_ci6qFbo
7$$\"35+]i!RU07$Fjn$\"3+#35O9#*HL(Fbo7$$\"3$)***\\(=S2LKFjn$\"3Qy='[[_
sl(Fbo7$$\"3nmmm\"p)=MLFjn$\"3;))orBpZ?zFbo7$$\"3U++](=]@W$Fjn$\"3Kq-!
oi-T<)Fbo7$$\"36L$e*[$z*RNFjn$\"3+RUl??]\"Q)Fbo7$$\"3e++]iC$pk$Fjn$\"3
=&GgvxEie)Fbo7$$\"3Sm;H2qcZPFjn$\"3U[bgt$G+w)Fbo7$$\"3Y+]7.\"fF&QFjn$
\"3*4dJ#3QNC*)Fbo7$$\"3amm;/OgbRFjn$\"3O^!fF5+)p!*Fbo7$$\"3I+]ilAFjSFj
n$\"3Z.fy@'Hy?*Fbo7$$\"3)RLLL)*pp;%Fjn$\"3&R(*o7-(fG$*Fbo7$$\"3WLL3xe,
tUFjn$\"3*fz[Xr'=T%*Fbo7$$\"3Wn;HdO=yVFjn$\"3]uACu<>V&*Fbo7$$\"3a+++D>
#[Z%Fjn$\"3!\\Z'y!G%QH'*Fbo7$$\"3)om;aG!e&e%Fjn$\"3w=F]aZE?(*Fbo7$$\"3
wLLL$)Qk%o%Fjn$\"332wk`7:&z*Fbo7$$\"3m+]iSjE!z%Fjn$\"3?._5P)[!p)*Fbo7$
$\"3u+]P40O\"*[Fjn$\"3]$4G7h4Y$**Fbo7$$\"#]F)$\"3yD+G++++5F0-%'COLOURG
6&%$RGBG$\"#5!\"\"F(F(-%+AXESLABELSG6$Q\"x6\"Q!Fg[l-%%VIEWG6$;F(Fhz%(D
EFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curv
e 1" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 6 "Voila." }}}}{MARK "11 0 0
" 6 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }