# *Mathematical Modeling

Featured Courses: Mathematical Modeling

Mathematical models are used extensively in the sciences, engineering, economics, finance, and many other fields. In this course you’ll learn how models are built, and you’ll learn how models are used to expand our understanding of real-world phenomena.

This course makes extensive use of software (Maple) and students learn how to build models using Maple (provided to ESC students at low cost).

The course starts with the general principles on which mathematical modeling is based:

- Understand the problem in thereal-world
- Simplify and translate the real-world to the model-world
- Formulate the model and conduct a mathematical Analysis
- Interpret the model results in real-world terms
- Check and Refine
- Communicate the results and the model to others

How Many Restaurants are there in the United States?| top

The best way to learn about math modeling is to create models. One easy-to-understand type of model is called a Fermi estimate. The University of Maryland maintains a Fermi Problems website that explains the concept and provides numerous examples.

To understand how this works let’s use Fermi estimation to try and answer the question; How many restaurants are there in the United States?

Before reading on, can you think of a method you might use to answer to that question?

One approach is to count the restaurants in some “representative” town or city and then extrapolate from that count to calculate how many restaurants there are in the entire U.S. Start by counting the restaurants in some town you know about; depending on the size

of the town that might be easy, or it might be hard. If the task is daunting you could count the restaurant listings in the phone book. You can estimate this number if counting will take too long. You’ll also need to know the population of the town (try a Goggle search like this: population [your town name] statename or visit the U.S. Census Bureau web site.

Now divide the town population into the U.S. population; use can 300 million as U.S. population. The result of this calculation is the number you can multiply the town population by to get the U.S. population. Multiplying the number of restaurants in the town by this multiplier will yield an estimate of the number of restaurants in the U.S. This is a roundabout description of the concept of a ratio. In math notation it will look 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 now we’d like to know, how good is our 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 the number of restaurants in our town is representative of the entire United States. 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 went to the U.S. Census Bureau web site 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 would be to apply the model using other towns towns or cities 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. Student in the course try this out using a worksheet.

Did you come up with a different estimate? That wouldn’t be surprising. The town we used for the estimate above 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 number from Census Bureau also includes these types establishments. We’d have to do some more digging to find out.

Still, 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