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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "Byron C Phipps and Melinda
 Chauvin" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 14 "with(DEtools):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
27 "Problem set B, number 14:  " }}{PARA 0 "" 0 "" {TEXT -1 330 "A tan
k contains Q pounds of salt dissolved in 100 gallons of water.  Water \+
containing 1/4 lb. of salt per gallon enters at a rate of 3 gal./minut
e and the solution leaves the tank at the same rate.  The problem is m
odeled by the differential equation Q'(t) = 0.75 - 0.03Q(t).  Solving \+
the equation in terms of t (time) results in:" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 53 "A:=dsolve(\{diff(Q(t),t)=.75-.03*Q(t), Q(0)=c\},
 Q(t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG/-%\"QG6#%\"tG,&$\"#D
\"\"!\"\"\"*&-%$expG6#,$F)$!+++++I!#6F.,&$!#DF-F.%\"cGF.F.F." }}}
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t,c));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#A2GR6\"F&6$%)operatorG%&a
rrowGF&,($\"#D\"\"!\"\"\"-%$expG6#,$%\"tG$!+++++I!#6$!#DF-*&F/F.%\"cGF
.F.F&F&F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 44 "plot(\{A2(t,c) $c=0..40\}, t=0..190, Q=0..40
);" }}{PARA 13 "" 1 "" {GLPLOT2D 580 482 482 {PLOTDATA 2 "6M-%'CURVESG
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45.000000 0 }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 181 "The value for t r
epresents time; Q(t) represents the amount of salt in the tank at time
 t.  The constant in the initial equation represents the initial amoun
t of salt in the tank.  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 401 "Plotting the result with respect to Q results in \+
a critical point at Q(t)=25:  regardless of the initial proportion of \+
salt in the tank, the influx of solution will eventually \"flush\" the
 original contents of the tank and replace it with the solution of 1/4
lb. per gallon.  Because the capacity of the tank will be maintained a
t 100 gallons, the amount of salt in the tank will level out at 25 pou
nds." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 61 "dfieldplot(diff(Q(t),t)=.75-.03*Q(t),Q(t),t=0..190, Q
=0..50);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 13 "" 1 "" 
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}}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 276 "Plotting the direction field \+
of the differential equation illustrates the tendency of all solution \+
curves to approach Q=25.  Since the initial solution will never have a
 negative amount of salt, and the time will never drop below 0, the gr
aph will not involve negative values." }}{PARA 0 "" 0 "" {TEXT -1 3 " \+
  " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "B:=unapply(rhs(dsolve(\{diff
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{XPPMATH 20 "6#>%\"BGR6$%\"tG%\"cG6\"6$%)operatorG%&arrowGF),&$\"#D\"
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{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "sol1:=plot(B(t,0), t=0..190)
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..190, color=green):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "sol
3:=plot(B(t,10), t=0..190, color=orange):" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 42 "sol4:=plot(B(t,30), t=0..190, color=blue):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "sol5:=plot(B(t,40), t=0..190
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G-%%VIEWG6$;F(F\\[l%(DEFAULTG" 1 2 0 1 0 2 9 1 4 2 1.000000 45.000000 
45.000000 0 }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 105 "Plotting speci
fic solution curves illustrates the tendency for the solution in the t
ank to approach Q=25." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 
"" {TEXT -1 63 "Solutions to the original equation for different value
s of c:  " }}{PARA 0 "" 0 "" {TEXT -1 69 "(illustrated above as part o
f the graph of specific solution curves) " }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 53 "C:=dsolve(\{diff(Q(t),t)=.75-.03*Q(t), Q(0)=5\}, Q(
t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"CG/-%\"QG6#%\"tG,&$\"#D\"
\"!\"\"\"-%$expG6#,$F)$!+++++I!#6$!#?F-" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 54 "F:=dsolve(\{diff(Q(t),t)=.75-.03*Q(t), Q(0)=10\}, Q(t
));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FG/-%\"QG6#%\"tG,&$\"#D\"\"
!\"\"\"-%$expG6#,$F)$!+++++I!#6$!#:F-" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 54 "G:=dsolve(\{diff(Q(t),t)=.75-.03*Q(t), Q(0)=40\}, Q(t
));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"GG/-%\"QG6#%\"tG,&$\"#D\"\"
!\"\"\"-%$expG6#,$F)$!+++++I!#6$\"#:F-" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}}{MARK "7 3 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 }