Mathematical Modeling: Course Introduction
Mathematical models are used extensively in the sciences, engineering, economics and finance, management, and even in music and the visual arts (though perhaps not consciously!). In this course you'll learn how models are built, and you'll learn how models are linked to real-world phenomena. This course makes extensive use of Maple 10 (Math software). You'll learn how to set up models using Maple and you'll learn how to use Maple to share your results and collaborate with your classmates.
This is a hands-on course, so without further ado let's look at an example of a simple model.
How many restaurants are there in the United States? Can you think of a method you might use to find an answer to that question?
One approach is to count the restaurants in your town and then figure out how many there are on a per capita basis. Then you can apply that per capita number to the entire U.S. population.
So start by counting the restaurants in your town. Depending on the size of the town this could be easy or it could be difficult. One way to do it would be to count the restaurant listings in the phone book, or if there are too many, you could estimate the number.
You'll also need to know the population of your town (try a Goggle search like this: population [your town name] statename or visit the U.S. Census Bureau web site). If you can't find a population number then you can estimate this as well.
Now take the population of your town and divide it into the population of the United States.
Next multiply the number of restaurants in your town by that number and you'll have an estimate of how many restaurants there are in the U.S.
If you set this up in a spreadsheet, it would look something like this:
This type of problem is known as a Fermi problem. And our approach to finding the answer represents a very simple mathematical model.
But how good is this estimate?
Even this very simple model is based on several assumptions. Maybe our estimate of the number of restaurants is off, or perhaps our town population figure is wrong. And what about the assumption that our town is representative of the entire United States in terms of the number of restaurants per capita. What if our town isn't typical?
Mathematical models represent an idealized view of the world, so we must always check our results and validate our assumptions. In this case, we've used the U.S. Census Bureau web site to check our results and found that according to the Census Bureau are 565,000 restaurants in the U.S. (2002).
Our estimate is almost twice that. Where do you think the discrepancy might lie?
One way to find out is to apply the model using other towns as the basis for the estimate. And you can give it a try. We've created an online worksheet you can use to apply the model to the town or city you live in. Try it for yourself here in a live worksheet (opens in a new window)
Did you come up with a different estimate? That would not be surprising. The town we used is a popular tourist destination and it is quite likely that it has more restaurants per capita than the U.S. average. Also, we included hotel restaurants, coffee shops, and fast food joints of all kinds in our count of restaurants. It's not clear whether the Census Bureau numbers also include these types establishments. We'd have to do some more digging to find out.
This exercise illustrates the process that underlies creating a mathematical model. We can define that process as:
- Understand the problem in the real-world
- Translate the real-world to the model-world
- Conduct a mathematical Analysis
- Interpret the model results in real-world terms
- Check and Refine
- Communicate the results and the model to others
Understanding and applying this process is what this course is all about.
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