# Descriptive Modeling in Mathematics

Instructor: Yuanxin (Amy) Yang Alcocer

Amy has a master's degree in secondary education and has taught math at a public charter high school.

After reading this lesson, you'll learn how you can use your math skills to solve real world problems. Learn how you can mathematically model your problem so you can use appropriate equations to find the solution.

## A Real World Problem

Did you know that you can use math to help you model a real world problem making it easier to solve? That's right. Once you've created your mathematical model, you'll be able to use the associated mathematical equations for that model to help you solve your problem. This process of creating a mathematical model is referred to as descriptive modeling.

Let's take a look at how this process can make your life a whole lot easier.

Imagine that you are a forensic scientist who has just been called out to the scene of an accident. When you arrive at the scene of the accident, you see the two cars. A blue car has hit a red car in the side. The red car's side has an indent that's 1.2 inches deep.

Your job now is to determine just how fast the other car was going before it hit the other car. This information will help the car insurance companies figure out just who is responsible and how much to pay out.

## Descriptive Modeling

To help you, you create a descriptive model of the situation. This is what you do.

You first find a mathematical model for your problem. You ask yourself, just what kind of math is involved here. You decide on a physics model that shows the force of impact when objects collide with each other.

Next, you label your variables according to the model. In your case, you have the mass of the two cars, the force of impact, and the speed of travel for each car.

Then you use your variables and other information to solve the problem. Once you've labeled your variables and other information, you can then follow the model to solve the problem. In your case, you'll follow the physics model for determining the force of impact of a traveling object. You use the appropriate equation, plug everything in, and solve.

## Solving the Problem

For your problem, you'll use this equation to determine just how fast the car was going when it collided.

The F stands for the force of impact, the d stands for the distance the car crumpled or dented (this could be either the car itself or the object that got hit), the m stands for the mass of the car that caused the crash, and the v stands for the velocity or directional speed of the car.

You decided to use this equation because you have all the information except the velocity of the car which you want to find. So, you can plug in all your known values and then solve for the velocity. From studying the scene and from prior experience, you can tell that a force of impact of roughly 250,000 foot pounds per second squared was needed to indent the red car. The distance that the red car dented is 1.2 inches (0.1 feet). The mass of the blue car is 3000 pounds. Plugging this information into the formula and then solving for the velocity, you get this.

So, the blue car was traveling at 2.78 miles per hour when it hit the red car causing a dent of 1.2 inches.

## Example

Now, you try picking a mathematical model to use to help you solve a problem.

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