Examples of Cooling, Growth and Decay
Date: 06/03/99 at 20:45:31 From: Anonymous Subject: Newton's cooling method, logistic growth, exponential growth and decay Dear Dr. Math, My calculus teacher, Mr. William Lambert, recommended you. I am doing a project on Newton's cooling method [dy/dt=k(Ly)], logistic growth [dy/dt=ky(L-y)], and dy/dt=ky. I have to show how to integrate and give real life examples. I have no problem with the integration, it is the real life problems. I need real life applications of Newton's cooling method, logistic curves and the growth model. The usual dead body, and spreading rumor and interest are not considered acceptable because of their lack of originality. I was hoping that I might be able to inquire as to some real life problems from your vast experience. Thank you, Dr. Math. Sincerely, Anonymous
Date: 06/04/99 at 04:37:24 From: Doctor Mitteldorf Subject: Re: Newton's cooling method, logistic growth, exponential growth and decay Hi there, You have quite an ambitious teacher, to throw you into the thick of current research while you're still in high school. On the subject of cooling, it is a very good approximation, and very generally true, that the rate of heat transfer between two things is proportional to their temperature difference. (There are exceptions where the heat transfer is primarily through radiation, like the transfer of heat from the sun to the earth, or from the heating element in an oven.) Notice that the heat transfer is not proportional to the temperature but to the temperature difference. What difference does this make to your differential equation? Can you still solve it? One of the best applications is for home heating. How much heat is lost through the walls of a house during the winter? How much fuel is saved by adding insulation in the walls? Here's a problem you might set up using these ideas: one house stays at 70 degrees all the time while the temperature outside is 30. Another stays at 70 degrees during the day, but turns the heat off at 10 o'clock at night and back on (thermostat set to 70) at 6:00 in the morning. How much of the house fuel is saved by the "set-back thermostat"? You'll need to do some modeling and make some more assumptions to come up with an approximate answer to this. Logistic growth models are popular in evolutionary biology. It is assumed that there is a maximum number of individuals that can be supported in a given environment. The death rate is some increasing function of the total population. Whatever that function is, you can show that the logistic equation is a first approximation to the differential equation for population. Demonstrating this might be a good project for you. Here's another project: Think about two competing species sharing an environment such that the SUM of their two populations is the limiting factor in the logistic equation. Write down a pair of logistic equations describing their competition. What determines who wins? (There's a surprise in here, which I don't want to reveal until you've thought about this some yourself. Why don't you send me your initial thoughts on the subject?) - Doctor Mitteldorf, The Math Forum http://mathforum.org/dr.math/
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