10.5. Applications of Parametric Equations http://www.ck12.org
After about 4.2588 seconds the ball hits the ground at (-20.29, 0). This means the person threw the ball from (30, 5)
to (-20.29, 0), a horizontal distance of just over 50 feet.
Example C
Kieran is on a Ferris wheel and his position is modeled by the parametric equations:
xK= 10 ·cos(π
5 t)
yK= 10 ·sin(π
5 t)
+ 65
Jason throws the ball modeled by the equation in Example A towards Kieran who can catch the ball if it gets within
three feet. Does Kieran catch the ball?
Solution: This question is designed to demonstrate the power of your calculator. If you simply model the two
equations simultaneously and ignore time you will see several points of intersection. This graph is shown below on
the left. These intersection points are not interesting because they represent where Kieran and the ball are at the
same place but at different moments in time.
When theTmaxis adjusted to 2.3 so that each graph represents the time from 0 to 2.3, you get a better sense that at
about 2.3 seconds the two points are close. This graph is shown above on the right.
You can now use your calculator to help you determine if the distance between Kieran and the ball actually does go
below 3 feet. Start by plotting the ball’s position in your calculator asx 1 andy 1 and Kieran’s position asx 2 andy 2.
Then, plot a new parametric equation that compares the distance between these two points over time. You can put
this underx 3 andy 3 .Note that you can find thex 1 ,x 2 ,y 1 ,y 2 entries in the vars and parametric menu.