MATH 312 Section 3.1: Linear Models Prof. Jonathan Duncan Walla Walla College Spring Quarter, 2007
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
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: Growth of Decay? dp dt = λp Recall that the model will be a growth model if λ > 0 and a decay model if λ < 0.
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
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 T m is the constant ambient temperature, then: dt dt = k(t T m) The Constant k Recall that k will be negative for both warming and cooling objects.
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
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 dθ dt A = L(b a) 2m
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 dt = ( ) ( ) Amount of Amount of salt added salt removed
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 1 2 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 dt = 3 4A 100 + 2t A(t) = (50 + t) + A(30) = (50 + 30) C (50 + t) 2 100, 000 64.38 lbs (50 + 30) 2
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: ir L di dt Capacitor: 1 C q 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.
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 dq dt + 1 q = 200 5 10 6 q(t) = 1 1000 + Ce 200t i(t) = 200Ce 200t q(0.005) 0.00026 i(0.005) 0.1472
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