For part (c), we need to find parametric equations that describe the path halfway around the circle counterclockwise, starting at $(0, 5)$ and covering an angle of $\pi$ radians (half a circle). Let's analyze the information provided.

The given circle equation is: $x^2 + (y - 1)^2 = 16$ This describes a circle with:

• Center at $(0, 1)$,
• Radius $4$ (since $\sqrt{16} = 4$).

Solution for Part (c)

To travel halfway counterclockwise, starting at $(0, 5)$, we set up parametric equations based on a counterclockwise rotation around the circle's center.

Parametric Formulation

Since the starting point is $(0, 5)$, which is located directly above the center $(0, 1)$ by a radius of $4$, we can write: $x = 4 \cos(t), \quad y = 1 + 4 \sin(t)$ For a half-circle path counterclockwise:

• Starting angle $t = \frac{\pi}{2}$ (pointing upwards),
• Ending angle $t = \frac{3\pi}{2}$ (pointing downwards).

Thus, the parametric equations for half a circle counterclockwise are: $x = 4 \cos(t), \quad y = 1 + 4 \sin(t), \quad \frac{\pi}{2} \leq t \leq \frac{3\pi}{2}$

This should solve part (c). Please let me know if you'd like further details or if there are any additional questions.

Here are 5 related questions for further exploration:

1. How would the parametric equations change if the path were clockwise instead?
2. What would the parametric equations look like if the center of the circle was shifted?
3. How do you determine the limits for $t$ when describing partial circular paths?
4. What if the circle had a different radius? How would the equations adjust?
5. How could these parametric equations be used to describe an ellipse instead of a circle?

Tip: When setting parametric equations for a circular path, start by identifying the radius and the center to simplify setting up the sine and cosine components.