Population Growth Newton’s Law of Cooling Kepler’s Law Second Law of Planetary Motion Mixture Problems Circuit Analysis C
MATH 312
Section 3.1: Linear Models
Prof. Jonathan Duncan
Walla Walla University
Spring Quarter, 2008
Population Growth Newton’s Law of Cooling Kepler’s Law Second Law of Planetary Motion Mixture Problems Circuit Analysis C
Outline
1
Population Growth
2
Newton’s Law of Cooling
3
Kepler’s Law Second Law of Planetary Motion
4
Mixture Problems
5
Circuit Analysis
6
Conclusion
Population Growth Newton’s Law of Cooling Kepler’s Law Second Law of Planetary Motion Mixture Problems Circuit Analysis C
The Population Growth Model
In this section, we will solve examples of many of the linear models
we studied at the end of Chapter 1.
Population Growth
A naive view of population growth is that a population P(t) will
grow proportionally to its size. That is, for every x people in the
population at a given time, y new individuals will be added. The
differential equation for this model is:
dP
= λP
dt
Growth of Decay?
Recall that the model will be a growth model if λ > 0 and a decay
model if λ < 0.
Population Growth Newton’s Law of Cooling Kepler’s Law Second Law of Planetary Motion Mixture Problems Circuit Analysis C
A Population Growth Example
Next we will take a look at an example using this model.
Example
The population of an endangered species is declining. In 1998, the
population in a certain region was measured as 1230 individuals. In
2000, the population in that same region was only 1080
individuals. Find a function P(t) for the population size t years
after 1998 assuming this trend continues. What would the
population have be in 2005?
P(t) = 1230e λt
P(t) = 1230e −0.65t
P(2005 − 1998) = P(7) = 1230e −0.65(7) ≈ 780
Population Growth Newton’s Law of Cooling Kepler’s Law Second Law of Planetary Motion Mixture Problems Circuit Analysis C
Newton’s Law of Cooling Model
We now look at Newton’s Law of Cooling and an example of this
model.
Newton’s Law of Cooling
The rate at which the temperature of an object cools or warms to
the ambient temperature is proportional to the difference between
the object’s temperature and the ambient temperature. If T (t) is
the temperature of the object at time t, and Tm is the constant
ambient temperature, then:
dT
= k(T − Tm )
dt
The Constant k
Recall that k will be negative for both warming and cooling
objects.
Population Growth Newton’s Law of Cooling Kepler’s Law Second Law of Planetary Motion Mixture Problems Circuit Analysis C
A Cooling Example
We now look at a particular instance of this model.
Example
A thermometer is taken from an inside room to the outside, where
the air temperature is 5◦ F. After one minute, the thermometer
reads 55◦ F. After five minutes, the thermometer reads 30◦ F . What
was the temperature of the inside room?
T (t) = 5 + Ce kt
T (t) = 5 + 59.46e −0.173t
T (0) = 5 + 59.46 = 64.46◦ F
Population Growth Newton’s Law of Cooling Kepler’s Law Second Law of Planetary Motion Mixture Problems Circuit Analysis C
Planetary Motion Model
We now look at our first new mathematical model.
Example
The angular momentum of a moving body
of mass m is given, in polar coordinates,
by the expression L = mr 2 dθ
dt . Assume
that L is constant and prove Kepler’s
Second Law. Namely, show that the
radius vector joining the orbital focus to
the body of mass sweeps out equal areas
in equal time intervals.
L = mr 2
A=
dθ
dt
L(b − a)
2m
Population Growth Newton’s Law of Cooling Kepler’s Law Second Law of Planetary Motion Mixture Problems Circuit Analysis C
Mixture Model
We now revisit our mixture model and solve a specific example.
Mixtures
Recall that if a tank is kept at a constant volume and a solution is
added while the mixed contents are removed, then we can write a
differential equation for the amount of a substance in the tank at
time t as:
dA
Amount of
Amount of
=
−
salt added
salt removed
dt
Population Growth Newton’s Law of Cooling Kepler’s Law Second Law of Planetary Motion Mixture Problems Circuit Analysis C
A Mixture Example
Let’s examine a specific example of this model.
Example
A large tank is partially filled with 100 gallons of fluid in which 10 pounds
of salt is dissolved Brine, containing 21 pound of salt per gallon, is
pumped into the tank at a rate of 6 gallons a minute. The well-mixed
solution is then pumped out at a rate of 4 gallons per minute. Find the
number of pounds of salt in the tank after 30 minutes.
dA
4A
=3−
dt
100 + 2t
A(t) = (50 + t) +
A(30) = (50 + 30) −
C
(50 + t)2
100, 000
≈ 64.38 lbs
(50 + 30)2
Population Growth Newton’s Law of Cooling Kepler’s Law Second Law of Planetary Motion Mixture Problems Circuit Analysis C
Circuit Analysis Model
We finish with another new model, at least for our in-class work.
Circuit Analysis
In circuit analysis, the sum of the voltage drops across inductors,
resistors, and capacitors must equal the voltage source in the
circuit. The voltage drops are given by:
Resistor:
Inductor:
Capacitor:
iR
di
L dt
1
Cq
Where R is the resistance in ohms, L is the inductance in henrys,
C is the capacitance in farads, q(t) is the charge of the capacitor
at time t, and i(t) = dq
dt is the current at time t.
Population Growth Newton’s Law of Cooling Kepler’s Law Second Law of Planetary Motion Mixture Problems Circuit Analysis C
A Circuit Analysis Example
And now our final example.
Example
A 200-volt electromotive force is applied
to the RC series circuit shown. Find the
charge q(t) on the capacitor if i(0) = 0.3.
Determine the charge and current after
.005 seconds, and determine the long
term charge of the capacitor.
1000
q(t) =
dq
1
+
q = 200
dt
5 × 10−6
1
+ Ce −200t
1000
q(0.005) ≈ 0.00026
i(t) = −200Ce −200t
i(0.005) ≈ 0.1472
Population Growth Newton’s Law of Cooling Kepler’s Law Second Law of Planetary Motion Mixture Problems Circuit Analysis C
Important Concepts
Things to Remember from Section 3.1
1
Constructing linear models
2
Solving linear models such as:
Population growth/decay
Newton’s law of cooling
Planetary motion
Mixture problems
Circuit analysis