# Parametric Equations in Applied Contexts

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• 1:02 Circular Path
• 1:34 Finding the Parametric…
• 2:21 Solving the Equations
• 4:16 Lesson Summary
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Lesson Transcript
Instructor: Yuanxin (Amy) Yang Alcocer

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

Watch this video lesson to learn how you can use parametric equations to help you solve real world problems such as the path of a moving object. Learn how to form parametric equations to model the real world.

## Parametric Equations

In this video lesson, we will talk about parametric equations and how they apply in the real world. These equations take an equation in two or more variables and define each variable in terms of one variable called the parameter.

For example, we can take the equation x^2 + y^2 = 1, which has two variables, and turn it into a parametric equation by defining x = sin (t) and y = cos (t), where t is our one parameter. Why do we do this?

We do this because it turns an equation in two or more variables into a simple one variable problem. Of course, you will need to solve for each variable to get an exact point, but you only need to worry about your one parameter. We can also think of our parameter as time.

## Circular Path

Since we are talking about using parametric equations in the real world, let's talk about one real world situation where parametric equations can be applied. Say, for instance, we are watching horses being exercised in a circular corral. Each horse is making its exercise path along the same circle inside the corral. We can model the path our horses take with parametric equations. With our parametric equations, we can determine the location of our horses at any given time.

## Finding the Parametric Equations

So, how do we find our parametric equations? Well, we know that for a circle with a radius of r, the parametric equations are x = r sin (t) and y = r sin (t) where r is the radius and t is our parameter. In our case, our parameter represents time. The circle the horses are taking inside the corral has a radius of 10 feet. So, our parametric equations are x = 10 sin (t) and y = 10 cos (t). We will limit our t so that it begins at 0 and ends at 2pi. By limiting our t like we did, it tells us that our horse only makes one complete circle for its exercise routine.

## Solving the Equations

Now, let's solve some problems with our parametric equations.

Find the starting position of our horses.

To do this, we solve our parametric equations for t = 0. Doing that, we get x = 10 sin (0) = 10*0 = 0 and y = 10 cos (0) = 10*1 = 10. So, our starting point is at (0, 10). If we graphed out our corral with the center point at (0, 0), then our starting point, (0, 10), is the top side of our circle if we are standing at the bottom of the corral looking into the corral with the circle in the middle of it.

Now, find the position of the horses at t = pi/2.

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