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Sec 1.3 Differential Equations as Mathematical
ModelsSec 3.1 Linear Model
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Where it appear
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)('2'' 2 tfxxx
Mechanical Eng.
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salt) of rateOutput ( - salt) of rateInput (dtdA
outin RRdtdA
Chemical Eng.
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)(12
2
tEqCdt
dqRdt
qdL
Electrical Eng.
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0)(1)/( 22
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dxdyTw
dxyd
Civil Eng.
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)( mTTkdtdT
Heat Transfer
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)()(2
2
tgxFdt
xdm
Civil Eng.
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Sec 1.3 Differential Equations as Mathematical ModelsSec 3.1 Linear Model
Behavior of some real-life system
or phenomenonMathematical Model
STEP 1: Identification of the variables
STEP 2: We make some reasonable assumptions (physical laws)
Construction of a Mathematical model:
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AssumptionsMathematicalFormulation
ObtainSolutions
Check model Predictions with
Known facts
Express DE
Solve D
E
Display graphically
If necessary alter assum
ptions
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1.Population Dynamics*2.Radioactive Decay*3.Spread of Disease*4.Chemical Reactions 5.Newton’s Law of Cooling*6.Mixtures*7.Draining a Tank8.Series Circuits9.Falling Bodies10.Falling Bodies and Air Resistance11.Suspended Cables
Applications
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When a pizza is removed from an oven. Its temperature is measured at 300 F. Three minutes later its temperature is 200 F. How long will it take for the pizza to cool off to a room temperature of 70 F?
EXAMPLE:
Newton’s Law of Cooling-Warming: the rate at which the tempof a body changes is proportional to the difference between the temp of the body and the temp of the surrounding.
changes pizza theof tempheat which t the
tat time pizza theof temp time
dtdT
T(t)t
)( mTTdtdT
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When a pizza is removed from an oven. Its temperature is measured at 300 F. Three minutes later its temperature is 200 F. How long will it take for the pizza to cool off to a room temperature of 70 F?
EXAMPLE:
Newton’s Law of Cooling-Warming: the rate at which the tempof a body changes is proportional to the difference between the temp of the body and the temp of the surrounding.
0 5 10 15 20 25 30 35 40
80
100
120
140
160
180
200
220
240
time in minutes
230 exp(-0.19018 t)+70
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A thermometer is taken from a inside room to the outside, where the air temperature is 5F. After I minute the thermometer reads 55F, and after 5 minutes it reads 30F. What is the initial temperature of the inside room?
#14/p99:
Newton’s Law of Cooling-Warming: the rate at which the temp of a body changes is proportional to the difference between the temp of the body and the temp of the surrounding.
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The population of a community is known to increase at a rate proportional to the number of people present at time t. If an initial population P_0 has doubled in 5 years, how long will it take to triple? Quadruple?
#1/p99:
Population: The population of a community is known to increase at a rate proportional to the number of people present at time t
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Population: The population of a community is known to increase at a rate proportional to the number of people present at time t
2002023,500,000
2003124,250,000 0314.023502425ln
000,500,23
k
c
tetP 0314.0000,500,23)(
000,495,27)5(52007 Pt
Interested:See #39/101Read sec 3.2
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Output rate
3 gal/min
input rate
3 gal/min
A large tank held 300 gallons of brine solution. Salt was entering and leaving the tank; a brine solution was being pumped into the tank at the rate of 3 gal/min. it mixed with the solution there, and then the mixture was pumped out at the rate of 3 gal/min. The concentration of the salt in the inflow was 2 lb/gal. If 50 pounds of salt were dissolved initially in the 300 gallons, how much salt is in the tank after 10 min?
300 gallons
changes A(t)at which rate thedtdA
enterssalt at which rateinput inR
leavessalt at which rateoutput outR
tat timesalt ofamount )( tA
salt ofrateoutput
salt of
rateinput
dtdA
outin R R
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Output rate
3 gal/min
input rate
3 gal/min
A large tank held 300 gallons of brine solution. Salt was entering and leaving the tank; a brine solution was being pumped into the tank at the rate of 3 gal/min. it mixed with the solution there, and then the mixture was pumped out at the rate of 3 gal/min. The concentration of the salt in the inflow was 2 lb/gal. If 50 pounds of salt were dissolved initially in the 300 gallons, how much salt is in the tank after 10 min?
300 gallons
outin R R dtdA
brine ofrateinput
inflow
salt ofionconcentrat
R in
lb/min 6 gal/min 3 lb/gal 2 R in
brine ofrateoutput
outflow
salt ofionconcentrat
R out
gal/min 3 lb/gal R in ?
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Output rate
3 gal/min
input rate
3 gal/min
A large tank held 300 gallons of brine solution. Salt was entering and leaving the tank; a brine solution was being pumped into the tank at the rate of 3 gal/min. it mixed with the solution there, and then the mixture was pumped out at the rate of 3 gal/min. The concentration of the salt in the inflow was 2 lb/gal. If 50 pounds of salt were dissolved initially in the 300 gallons, how much salt is in the tank after 10 min? how much salt in the tank after a long time?
300 gallons
100A 6
dtdA
100600t
CeA(t)
50)0( A
Sol:
100550600t
eA(t)
55050)0( CA
102.3394
10010
55060010
e)A(
0 100 200 300 400 500 600
100
150
200
250
300
350
400
450
500
550
600
Time (minutes)
Amountof
Salt A
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Output rate
4 L/min
input rate
4 L/min
A tank held 200 liters of fluid in which 30 grams of salt is dissolved. Brine containing 1 gram of slat per liter is then pumped into the tank at a rate of 4 L/min; the well-mixed solution is pumped out at the same rate. Find the number A(t) of grams of salt in the tank at time t.
200 liters
#19p99
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Chapter-Summary Chapter # 1 and Chapter # 2
Next week (mond –wed)
GROUPMembersChapter Date of submission
Expected Datefor presentation
GROUP 1
1)AL-GHAMDI, MOHAMMAD TURKI2)AL-ZAHRANI, AHMAD ALI AHMAD3)L-HADI, MOHAMMAD SALEH HADAJ4)AL-MUTAIRI, ABDUL-LATIF THUWAINI
1Mon 2/25/2008 11:58 AM
GROUP 2
1)Khalid Abdulghani2)Khaldoon Al-Azzah3)Yousef Al-Shaheen4)Abdulrahman Al-Saggaf5)Mohammad Bawazeer6)Ibraheem Alsufyani
2Mon 2/25/2008 12:24 PM