Mathematical Models

Date: 07/17/2002 at 23:33:01
From: Navraj Singh
Subject: Purposes of Mathematical models

Dear Sir,

I'm taking part in the 2002 National Mu Alpha Theta Convention which 
is going to be held at Starkville, Mississippi, USA.  One of the 
rounds of the convention is called Chalk Talk where students have to 
talk about a topic related to Math.  For my division, the topic 
is 'Purposes of Mathematical Models'.  I'm a little confused about 
what exactly they mean by mathematical models.  Are they talking 
about literal mathematical models made out of materials, or 
something like the practical uses of math?  Can you help me with 
this?  If you understand my problem, or have any suggestions as to 
what I can do to prepare a good presentation on the topic, I'll be 
really thankful.

Sincerely,
Navraj Singh.

Date: 07/18/2002 at 09:58:14
From: Doctor Paul
Subject: Re: Purposes of Mathematical models

A mathematical model is not something made out of materials (like, 
for instance, a model airplane).  Rather, a mathematical model is 
some sort of formula or equation that is meant to model or simulate 
something in the environment.  Generally, the user inputs a number of 
variables and the mathematical model gives the user some sort of 
prediction about what will happen.

I can think of several examples:

1.  Most automobile insurance companies use mathematical models to 
determine your monthly payment.  They will ask you questions about 
your age, how far you drive to work or school, how many days a week 
you drive to work or school, what kind of car you have, what color it 
is, etc.  Then, based on your answers, the computer makes a judgment 
about how likely you are to file a claim on your insurance.  The more 
likely the computer thinks you are to file a claim, the higher your 
insurance rate.  Similar models are used for life insurance and home-
owners insurance policies.

2.  Mathematical models are used to predict the weather.  A 
weatherman might enter information about the jetstream, the current 
barometric pressure, etc.  Then the computer comes back with a model 
that shows how a particular weather system will react to the 
conditions entered.  The weather is notoriously hard to predict, as 
evidenced by the frequent inaccuracies in our daily weather 
forcasts.  But the models are getting better and research in this 
area is ongoing.

3.  One of the simplest examples of a mathematical model would be 
using a 2nd degree quadratic equation to model the path that a 
baseball takes when it is thrown into the air.  Of course, such a 
simulation would require the assumption that air resistance is zero.  
More sophisticated models (like those that simulate airplane flights) 
account for air resistance using techniques developed in college 
level physics classes.

Mathematical models are used in just about every industry you'll 
find.  Hopefully the above examples will help to clear up your 
confusion.  You might be interested in talking to a local weatherman 
or calling an insurance company to find out more about their models.  
My guess, though, is that the person you speak to at an insurance 
company will know little or nothing about how the computer turns your 
answers to their questions into an insurance rate.

I hope this helps.  Please write back if you'd like to talk about 
this some more.

- Doctor Paul, The Math Forum
  http://mathforum.org/dr.math/