For this Critical Thinking Assignment, you will explore a real-world scenario that can be modeled using parametric equations.

Begin by reading the following:

For the following exercises, use this scenario: A dart is thrown upward with an initial velocity of 65 mm per second at an angle of elevation of 52. Consider the horizontal and vertical positions of the dart at any time t. Neglect air resistance.

Part I: Complete the following steps:

1. Find parametric equations that model the problem situation.
2. Use the GeoGebra tool to graph the parametric equations. (Refer to this tutorial(Links to an external site.) as needed.)
3. Find all possible values for t that represent the situation.
4. State the parametric equations.
5. State the domain and range for each parametric equation.
6. Plot the minimum and maximum heights of the dart.
7. Plot the minimum and maximum horizontal positions of the dart.
8. Find the time at which the dart reaches the maximum height.
9. Save your GeoGebra work as a .pdf file for submission.

Part II: Based on your work in Part I, discuss the following:

1. Discuss why this situation can be modeled using parametric equations.
2. Discuss how you determined all possible values for t that represent the situation.
3. Discuss what the domain and range of the parametric equations mean in the context of this problem.
4. What do maximum and minimum values of the parametric equations represent in this context?
5. Discuss how your answers to Part I would be affected if:
6. The initial velocity is increased.
7. The angle of elevation is decreased.
8. Provide at least two other real-world situations that can be modeled using parametric functions and respond to the following:
9. What common characteristics do the real-world scenarios you chose share?
10. What did you look for in the way that the real-world scenario can be modeled?
11. How can you verify that the real-world scenarios you chose can be modeled by parametric functions?