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/