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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 46 "Dr. Sarah's Demo 4 on Ne
wton's Law of Cooling." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 256 132 "Write down a reas
onable differential equation which models the process of water in a fr
eezer.  Explain why your model is reasonable." }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT 257 47 "How would you solve this by hand \+
if you had to?" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 259 72 "What is a reasonable temp
erature for hot tap water?  For cold tap water?" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 258 360 "We would l
ike to plot the solutions to the differential equation obtained with t
he different initial condition.  First, using only one Maple command, \+
dsolve your differential equation with the hot tap water initial condi
tion, unapply it and call this solution \"hotsolution\".  When you are
 done with this step, call me over and show this to me before moving o
n!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 0 "" }{TEXT 260 161 "Now, using only one Maple command
, dsolve your differential equation with the cold tap water initial co
ndition, unapply it and call this solution \"coldsolution\"." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 80 "We will display both solutions in different colors, so we
 will need with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "
with(plots):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 261 60 "Plot the hotsolu
tion with the color of the curve being red. " }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 
262 121 "Set hotplot equal to the plot of  a red hotsolution - be sure
 to use \":\" instead of \";\" at the end of this Maple command." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }{TEXT 263 53 "Set coldplot equal to the plot of a blue \+
hotsolution." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 31 "Now we will display both plots:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "display(\{hotplot,coldplot\}
);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 264 38 "Which takes
 less time to freeze?  Why?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 265 102 "What are \+
some errors one might encounter in a real life experiment?  Could erro
rs reverse the results?" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 266 72 "Which freezes fast
est?  (ie which has the fastest rate of change)?  Why?" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "H
it enter at the end of the next line:" }}}{SECT 1 {PARA 3 "" 0 "" 
{TEXT -1 47 "The cafe-ole vs black coffee cooling experiment" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 205 "The fol
lowing is a reasonable model for the cooling of hot water in our exper
iment, since room temperature was about 25 degrees celcius, and the te
mperature of the hot water was about 73 degrees celcius.  " }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "T1:=unapply(rhs(dsolve(\{diff(T(t),
t)=k*(25-T(t)),T(0)=73\},T(t))),t);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 136 "In order to plot this, we will have to set a value for k.  I o
btained k=1/40 by looking at the resulting temperature at t=5 minutes \+
- I " }}{PARA 0 "" 0 "" {TEXT -1 121 "knew that I wanted this to be in
 the mid to upper 60s (degrees celcius) by the experiment we conducted
 with the CBL unit." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "k:=1/
40:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "plot(\{T1(t)\},t=0..
5,y=60..73);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 267 40 "W
hy does this look like a straight line?" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 47 "Let's look at dat
a from a real life experiment:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 17 
"EXPERIMENT SETUP:" }}{PARA 0 "" 0 "" {TEXT -1 16 "2 identical mugs" }
}{PARA 0 "" 0 "" {TEXT -1 4 "milk" }}{PARA 0 "" 0 "" {TEXT -1 16 "1/8 \+
cup measurer" }}{PARA 0 "" 0 "" {TEXT -1 45 "2 CBL units with temperat
ure measuring probes" }}{PARA 0 "" 0 "" {TEXT -1 48 "2 TI-83 calculato
rs with heat programs installed" }}{PARA 0 "" 0 "" {TEXT -1 32 "coffee
 urn filled with hot water" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "EXP
ERIMENT DIRECTIONS:" }}{PARA 0 "" 0 "" {TEXT -1 56 "Dispense hot water
 from the urn into the creamfirst cup." }}{PARA 0 "" 0 "" {TEXT -1 55 
"Dispense hot water from the urn into the creamlast cup." }}{PARA 0 "
" 0 "" {TEXT -1 107 "Insert the temperature probe into the CBL and sta
rt the program so that the unit measures every 13 seconds." }}{PARA 0 
"" 0 "" {TEXT -1 74 "Add 1/8 cup milk to creamfirst cup at time approx
imately equal to 1 minute" }}{PARA 0 "" 0 "" {TEXT -1 75 "Add 1/8 cup \+
milk to creamlast cup at time approximately equal to 5 minutes." }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "EXPERIMENT RESULTS:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 30 "times := [seq(13*i, i=1..36)]:" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 0 "" }{TEXT 268 14 "What is times?" }{TEXT -1 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 67 "CreamFirst is the temperature measurements from the creamfirst \+
cup." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 265 "CreamFirst := [74.
51, 77.98, 78.26, 77.64, 77.1, 73.25, 71.58, 71.28, 70.82, 70.59, 70.3
7, 70.14, 69.69, 69.47, 69.18, 68.75, 68.75, 68.31, 68.31, 67.89, 67.6
, 67.39, 67.18, 66.98, 66.77, 66.56, 66.42, 66.15, 65.95, 65.68, 65.68
, 65.27, 65.07, 64.87, 64.67, 64.67]:" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 59 "This changes the points into a desired format for plottin
g." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "CreamFirstPts := zip(
(t,y)->[t,y], times,CreamFirst):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 264 "CreamLast := [78.38, 80.66, 80.37, 79.44, 78.87, 78.
31, 77.97, 77.43, 76.89, 76.63, 76.3, 76.04, 75.78, 75.52, 75.01, 74.6
9, 74.44, 74.19, 73.94, 73.45, 73.14, 72.9, 72.66, 72.42, 72.19, 71.95
, 71.72, 65.08, 64.82, 64.82, 64.62, 64.62, 64.43, 64.23, 64.04, 63.59
]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "CreamLastPts := zip((
t,y)->[t,y], times,CreamLast):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 12 "with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 213 "pf
l := plot(CreamFirstPts, color=blue):\npfp := plot(CreamFirstPts, colo
r=blue, style=point, symbol=diamond):\npll := plot(CreamLastPts, color
=red):\nplp := plot(CreamLastPts, color=red, style=point, symbol=diamo
nd):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 270 122 "Guess at and describe w
hat you expect to happen to the CreamFirst and CreamLast graphs withou
t looking at the graph below." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "
" }}}{EXCHG {PARA 257 "" 0 "" {TEXT -1 160 "Creamfirst will be blue an
d creamlast will be red.  Without looking at the graph below, explain \+
how you could tell the plots apart without having looked at the " }
{TEXT 269 14 "Maple commands" }{TEXT -1 1 "?" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 271 
107 "Call me over and show me your answers to the last two questions B
EFORE executing the display command below." }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "plt1 := displ
ay([pfl,pfp, pll,plp], view=[0..480, 60..100]):\n%;" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 0 "" }{TEXT 272 30 "Which takes less time to cool?
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" 
{TEXT -1 12 "Extra Credit" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }
{TEXT 273 116 "Extra Credit:  Describe what I am doing in the followin
g commands.  What does the graph tell us about our experiment" }
{XPPEDIT 18 0 "%?;" "6#%#%?G" }}{PARA 0 "" 0 "" {TEXT -1 62 " Hint:  Y
ou will need ?blah to understand some of the commands" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "fd := x -> x[2..nops(x)] - x[1..nop
s(x)-1];\ndd := (x,y) -> zip( (x,y)->y/x, fd(x), fd(y));" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "df := dd(times,CreamFirst):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "dl := dd(times,CreamLast):" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 124 "plot(zip((x,y)->[x,y], times[1..nops(times)-1], df))
:\nplot(zip((x,y)->[x,y], times[1..nops(times)-1], dl)):\ndisplay([%,%
%]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}}{MARK "0 0 0
" 0 }{VIEWOPTS 1 1 0 1 1 1803 }