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{SECT 0 {PARA 260 "" 0 "" {TEXT -1 19 "Presidential Coffee" }}{PARA
256 "" 0 "" {TEXT -1 9 "Jim Herod" }}{PARA 257 "" 0 "" {TEXT -1 21 "Sc
hool of Mathematics" }}{PARA 258 "" 0 "" {TEXT -1 12 "Georgia Tech" }}
{PARA 259 "" 0 "" {TEXT -1 21 "herod@math.gatech.edu" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 ""
{TEXT -1 650 " The President and the Prime Minister sit down for a
conference in the Oval Office which is cooled to 76 degrees Fahrenhei
t. Each is served 6 ounces of coffee at 210 degrees. If they did not d
rink the coffee, one aide for the President has noticed, that in 30 mi
nutes the coffee will have cooled to nearly room temperature -- the co
ffee will be 80 degrees. The cream is served in ice, which keeps the o
ne ounce containers at 40 degrees. The Prime Minister pours in one oun
ce of cream immediately. They talk for 10 minutes, at which time the P
resident pours in his one ounce of cream, and they both drink. We ask:
who has the cooler cup of coffee?" }}{SECT 1 {PARA 3 "" 0 "" {TEXT
-1 26 "An analysis of the problem" }}{PARA 0 "" 0 "" {TEXT -1 380 " \+
This analysis of the coffee problem assumes that the cups of coffee \+
cool by following Newton's Law of cooling. (The extent to which such a
body behaves in this way can be examined from data in Newton's Method
Used for Newton's Law of Cooling -- a work sheet found in multidimens
ional calculus at www.math.gatech.edu
/~herod/twoyrs/temp2.mws.) " }}{PARA 0 "" 0 "" {TEXT -1 212 "There are
two situations. We examine each in turn. Each assumes that the temper
ature decreases in proportion to the difference in the temperature of \+
the coffee and the temperature of the surrounding air. That is," }}
{PARA 0 "" 0 "" {TEXT -1 35 " " }
{XPPEDIT 18 0 "diff(T(t),t) = k*(76-T(t)" "6#/-%%diffG6$-%\"TG6#%\"tGF
**&%\"kG\"\"\",&\"#wF--F(6#F*!\"\"F-" }{TEXT -1 37 ", with T(0) the i
nitial temperature." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA
3 "" 0 "" {TEXT -1 44 "The numerical value for the cooling constant" }
}{PARA 0 "" 0 "" {TEXT -1 256 "Fortunately, the astute President's Aid
e had observed that when the coffee was poured and not drunk in a thir
ty minute period, then the coffee cooled to 80 degrees. We can use the
se two data point -- 210 degrees at t = 0, 80 degrees at t = 30 -- to \+
find k." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "dsolve(\{diff(T(t
),t)=k*(76-T(t)),T(0)=210\},T(t));" }}}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 22 "Tk:=unapply(rhs(%),t);" }}}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 22 "k:=solve(Tk(30)=80,k);" }}}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 0 "" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 3
"" 0 "" {TEXT -1 27 "The Prime Minister's Coffee" }}{PARA 0 "" 0 ""
{TEXT -1 363 " The Prime Minister, still feeling the effects of th
e jet lag, is eager for another cup of coffee and pours in his cream. \+
But then, he gets engrossed in the President's reflections on the anti
cs of Congress and forgets the coffee until the President pauses. We c
ompute the temperature of his coffee with cream, taking into account t
hat he added cream at once." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0
19 "To:=(6*210+1*40)/7;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "
dsolve(\{diff(T(t),t)=k*(76-T(t)),T(0)=To\},T(t));" }}}{EXCHG {PARA 0
"> " 0 "" {MPLTEXT 1 0 23 "TPm:=unapply(rhs(%),t);" }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "plot(
TPm(t),t=0..10.25);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 22 "
The President's Coffee" }}{PARA 0 "" 0 "" {TEXT -1 322 " The Presi
dent, wanting to start the meeting with a little light humor, is so in
volved with his story, that he ignores the coffee for ten minutes. Whe
re upon, he pours in his cream, smiles at his pleasure that the Prime \+
Minister is amused, and picks up his coffee. We compute the temperatue
of his coffee at this time." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0
49 "dsolve(\{diff(T(t),t)=k*(76-T(t)),T(0)=210\},T(t));" }}}{EXCHG
{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "TPr:=unapply(rhs(%),t);" }}}{EXCHG
{PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "TPrC:=t->piecewise(t < 10,TPr(t),(6
*TPr(t)+1*40)/7);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "plot(T
PrC(t),t=0..10.25);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 52 "
Who has the perfect temperature for a cup of coffee?" }}{EXCHG {PARA
0 "> " 0 "" {MPLTEXT 1 0 49 "`Prime Minister's coffee temperature`=TPm
(10.25);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "`President's co
ffee temperature`=TPr(10.25);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1
0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 452 "Unnoticed by either of these t
wo, the President's Aide -- a Georgia Tech graduate -- has poured hims
elf a cup of coffee, too. He mentally computes exactly the correct tim
e to pour in the cream so that the temperature of the coffee at the en
d of ten minutes is 110 degrees. (He has heard the President's stories
so many times, he knows how long it will take!) At what time does he \+
pour in the cream so that the coffee will be 110 degrees at ten minute
s?" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 23 "The Aide's calculations" }}
{PARA 0 "" 0 "" {TEXT -1 134 "Here are the calculation the Aide made i
n his head. He thought, \"I want to find the correct time to pour in t
he cream. Call this time " }{XPPEDIT 18 0 "alpha" "6#%&alphaG" }{TEXT
-1 86 ". After I pour in the cream, the coffee, with cream, continues \+
to cool. I will choose " }{XPPEDIT 18 0 "alpha" "6#%&alphaG" }{TEXT
-1 84 " so that when t = 10, the coffee is 110 degrees. Just what I ne
ed after last night!\"" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "ds
olve(\{diff(T(t),t)=k*(76-T(t)),T(0)=210\},T(t));" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 22 "TA:=unapply(rhs(%),t);" }}}{EXCHG {PARA 0 ">
" 0 "" {MPLTEXT 1 0 24 "C:=(6*TA(alpha)+1*40)/7;" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 51 "dsolve(\{diff(T(t),t)=k*(76-T(t)),T(alpha)=C
\},T(t));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "Tnew:=unapply(
rhs(%),t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "fsolve(Tnew(1
0)=110,alpha,0..10);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}
}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1
23 "Exercise for the reader" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA
0 "" 0 "" {TEXT -1 1007 "\011 On January 2, 1980, the outside temp
erature was a bitter 12 degrees at about 2:00 AM. The inside thermosta
t was set on 60 degrees to save on fuel. The furnace and thermostat ar
e set to work this way: when the thermostat temperature is 58 degrees,
the thermostat signals the furnace to turn on. The furnace stays on u
ntil the thermostat reaches 62 degrees. It was observed that it took o
nly 20 minutes for the furnace to come on again after it had turned of
f. How long would the furnace have stayed off if the thermostat had be
en set for 50 degrees, instead of 60?\n Make two simplifying assum
ptions: that the thermostat has a four degree spread at any temperatur
e for cutting on and off the furnace, and that the house cools accordi
ng to Newton's Law of Cooling: that the rate of change of the temperat
ure is proportioned to the difference in the temperature of the house \+
and the temperature of the surrounding environment:\n\011\011\011 \+
T '(t) = k ( T(t) - 12 )." }}
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