Linear & Exponential Modeling from Real World Situations

Instructor: Laura Pennington

Laura received her Master's degree in Pure Mathematics from Michigan State University. She has 15 years of experience teaching collegiate mathematics at various institutions.

In this lesson, we will start with an example of using linear and exponential models in a real-world situation. We will then explore these models and discuss how to know which model to use in a scenario, based on a phenomenon's rate of change.

Linear and Exponential Modeling

Suppose you and your friend, Mary, are at the carnival and are going to play a mechanical horse race game. You both are observing a few rounds before choosing which horse you want to predict will win the game. You notice that horse one always travels at a constant speed from start to finish. You also notice that horse two speeds up more slowly at first and then quite quickly after a bit. You might not realize it, but these horses can be modeled with mathematical models called linear and exponential models.

When a phenomenon increases or decreases at a constant rate, like horse number one, it can be modeled using a linear model. A linear model is a mathematical model in which the highest exponent of the variables in the model is one, and when this type of model is graphed, the graph is a line. For example, horse one can be represented with the following linear model:

y = 8.4x

where y is the distance the horse has traveled, in inches, and x is the amount of time that has passed, in seconds. Do you notice that the highest exponent on both y and x is one? This tells us this is a linear model, so its graph should be a line. Let's take a look!


linexp1


Sure enough! The graph of the model is a line.

Now, let's consider horse two. It turns out that the pattern that horse two takes on is exactly the pattern that can be modeled by an exponential model. An exponential model is a mathematical model in which the variable is in the exponent. The graph of an exponential model increases slowly at first, then more quickly, or decreases quickly at first and then more slowly.

We use the following exponential model for horse two:

y = 1.57x - 1

where y is the distance traveled, in inches, and x is the time, in seconds. We see that the variable x is in the exponent, so we know this is an exponential model. Let's take a look at the graph and see if it matches with the pattern we've observed in horse two.


graph2


Yes! It starts off slowly increasing and then goes more quickly, just as we expected!

Using Linear and Exponential Models to Solve Real-World Phenomena

Now that we know what linear and exponential models are, and how they can be used to model real-world phenomena (like a horse race), let's look at how we use these models to solve problems. To do this, let's go back to the races!

Suppose the horse race at the carnival is such that it goes for a varying amount of time and whichever horse is in first at the end of that time wins. Basically, this says that you are guessing at how long the race will be, since the horses' speeds take on the same pattern for each race.

You and Mary decide to give it a go. You bet on horse one to win, and Mary bets on horse two to win. The race ends up going for 10 seconds. Who won, you or Mary? Well, all we have to do is plug 10 in for x in each of the models, and whichever one has gone farther wins.


linexp2


Looks like your horse went 84 inches in 10 seconds, and Mary's horse went 89.99, or about 90, inches. Well, you lost this time, so you tell Mary you want to try again. Once again, you bet on horse one, and Mary bets on horse two. However, this time, the race only lasts for 5 seconds. Who wins this time? Let's plug x = 5 into both models to find out!


linexp4


This time, your horse went 42 inches, and Mary's horse went only 8.54 inches. You won! Now that you're even with the wins, you decide to go celebrate with an elephant ear at the food booth. Yum!

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