The problem involves finding parametric equations for the path of a particle moving along a circle defined by:

$x^2 + (y - 2)^2 = 16$

This equation represents a circle centered at $(0, 2)$ with a radius of $4$.

For part (c), the goal is to describe the particle's motion halfway around the circle, counterclockwise, starting at the point $(0, 6)$.

Solution for Part (c)

1. Starting Point Analysis: The particle starts at $(0, 6)$, which is directly above the center $(0, 2)$ by a distance of $4$. This suggests the initial angle $t = 0$ aligns with the topmost point of the circle.

2. Parametric Equations for Counterclockwise Motion:

   • Since we want the particle to travel counterclockwise, we need to set up the parametric equations accordingly. For a circle centered at $(0, 2)$ with radius $4$, we can use: $x = 4 \sin(t), \quad y = 2 + 4 \cos(t)$ Here, $\sin(t)$ and $\cos(t)$ reflect the $x$ and $y$ offsets from the center of the circle.

3. Halfway Around the Circle:

   • To go halfway around the circle, the angle $t$ should range from $0$ to $\pi$.
   • This range will take the particle from $(0, 6)$ to $(0, -2)$, covering half the circle.

4. Answer: $x = 4 \sin(t), \quad y = 2 + 4 \cos(t), \quad 0 \leq t \leq \pi$

This setup should satisfy the requirements for part (c) by moving the particle halfway around the circle in a counterclockwise direction, starting at $(0, 6)$ and ending at $(0, -2)$.

Would you like further details or have any questions?

Related Questions:

1. How would the parametric equations change if the particle moved clockwise instead?
2. What would the parametric equations look like if the circle were centered at a different point, say $(1, 1)$?
3. Can you explain why the parameter $t$ range from $0$ to $2\pi$ represents a full revolution?
4. How would you set up the equations if the circle's radius was different, say $3$ instead of $4$?
5. What parametric equations would describe a particle moving along an ellipse instead of a circle?

Tip:

When dealing with circular motion, always identify the circle's radius and center first; this information will guide how you construct the parametric equations.