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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 100 "This Maple Worksheet produces the solution to the rocket launc
h linear program, with air resistance." }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with(simplex):" }}{PARA 7 "" 1 "" 
{TEXT -1 87 "Warning, the protected names maximize and minimize have b
een redefined and unprotected\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 
"" }}{PARA 0 "" 0 "" {TEXT -1 117 "Define the parameters N (number of \+
intervals), T (total time), g (gravity), a0=maximum acceleration, h=de
sired height" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 5 "N:=5;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"NG\"
\"&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "T:=10;" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%\"TG\"#5" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 6 "g:=32;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gG\"#K" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "a0:=48;" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%#a0G\"#[" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
7 "h:=300;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"hG\"$+$" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "tau:=T/N;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%$tauG\"\"#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }
}{PARA 0 "" 0 "" {TEXT -1 34 "Extra constant for air resistance:" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 7 "k:=0.1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"kG$\"\"\"!\"\"" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "q:=exp(-k*tau);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%\"qG$\"+JvI(=)!#5" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 11 "p:=(1-q)/k;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>
%\"pG$\"+pCp7=!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "r:=(t
au-p)/k;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG$\"+5`2t=!\"*" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 116 "T
he constraints, with the added air resistance (See Equations 3.35 and \+
3.36 in the text for constraints CS2 and CS3)" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "CS1:=seq(x[
N+i]<=a0,i=1..N):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 103 "CS2:=
seq(sum( (r+(p^2*(1-q^(i-j))/(1-q)))*x[j],j=1..i) >=(g*p^2*(i-1-i*q+q^
i)/(1-q)^2)+i*g*r,i=1..N-1):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 91 "CS3:=sum( (r+(p^2*(1-q^(N-j))/(1-q)))*x[j],j=1..N )=(g*p^2*(N-1-
N*q+q^N)/(1-q)^2)+ N*g*r+h:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
30 "CS4:=seq(x[i]<=x[i+N],i=1..N):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 31 "CS5:=seq(-x[i]<=x[i+N],i=1..N):" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 26 "CS:=[CS1,CS2,CS3,CS4,CS5]:" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 26 "Obj:=sum(x[j],j=N+1..2*N);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%$ObjG,,&%\"xG6#\"\"'\"\"\"&F'6#\"\"(F*&F'6#\"
\")F*&F'6#\"\"*F*&F'6#\"#5F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 20 "Z:=minimize(Obj,CS);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"ZG<,
/&%\"xG6#\"\"'$\"+++++[!\")/&F(6#\"\")F+/&F(6#\"\"%$\"+(GaV[*!\"*/&F(6
#\"\"&$\"+]Cc2@!#=/&F(6#\"\"$F+/&F(6#\"\"(F+/&F(6#\"\"\"F+/&F(6#\"\"#F
+/&F(6#\"\"*F6/&F(6#\"#5$\"+n\\PQ?F?" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 94 "Now that we have the optimal va
lues, construct the position, velocity and acceleration values:" }}
{PARA 0 "" 0 "" {TEXT -1 81 "  The sequence of discrete values for acc
eleration, velocity and position follow:" }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "assign(Z); Obj;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+VN%[`\"!\"(" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 20 "A:=seq(x[i],i=1..N);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"AG6'$\"+++++[!\")F&F&$\"+(GaV[*!\"*$\"+]Cc2@!#=" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "V:=seq(p*(sum(q^(i-j)*A[j],
j=1..i)-g*((1-q^i)/(1-q)) ),i=1..N);" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#>%\"VG6'$\"+]zI+H!\")$\"+o#z[F&F($\"+LQ,>sF($\"+\"[M!H=F($!+E\"HJI%F
(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 96 "Y:=seq(sum((r+p^2*((1-
q^(i-j))/(1-q)))*A[j],j=1..i)-i*g*r-g*p^2*((i-1-i*q+q^i)/(1-q)^2),i=1.
.N);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"YG6'$\"+'\\?p*H!\")$\"+J27
D6!\"($\"+ph)4Q#F+$\"+ij$yE$F+$\"+\")******HF+" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 96 "For the acceleration
, velocity and position functions, we'll use Heaviside functions rathe
r than" }}{PARA 0 "" 0 "" {TEXT -1 70 "piecewise defined functions... \+
 Just a matter of taste, not substance." }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}{PARA 0 "" 0 "" {TEXT -1 39 "See Equations 3.30 and 3.31 in the t
ext" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 26 "C:=seq((A[j]-g)/k,j=1..N);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"CG6'$\"+++++;!\"(F&F&$!+rXc^AF($!+++++KF(" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 165 "Vall:=(C[1]*(1-exp(-k*t)) )*(Heaviside(t)-Heavisid
e(t-tau))+sum( ( (C[j]+(-C[j]+V[j-1])*exp(-k*(t-(j-1)*tau)))*(Heavisid
e(t-(j-1)*tau)-Heaviside(t-j*tau)) ),j=2..N):" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 209 "Yall:=(C[1]*t-((A[1]-g)/k^2)*(1-exp(-k*t)))*(He
aviside(t)-Heaviside(t-tau))+sum(((Y[j-1]+C[j]*(t-(j-1)*tau)+(1/k)*(V[
j-1]-C[j])*(1-exp(-k*(t-(j-1)*tau))))* (Heaviside(t-(j-1)*tau)-Heavisi
de(t-j*tau))),j=2..N):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "A
all:=sum(A[j]*(Heaviside(t-(j-1)*tau)-Heaviside(t-j*tau)),j=1..N):" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "plot(Yall,t=0..10,color=bla
ck);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "plot(Vall,t=0..10,c
olor=black);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "plot(Aall,t
=0..10,color=black);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}
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