To solve part (c) of this problem, we need to derive the parametric equations for the particle moving halfway around the circle counterclockwise, starting from the point $(0, 5)$, over the interval $0 \leq t \leq \pi$.

Step-by-Step Solution

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

• Center: $(0, 1)$
• Radius: $4$

For a counterclockwise path, the parametric equations can generally be expressed as: $x = 4 \cos(t), \quad y = 1 + 4 \sin(t)$

Adjusting for Starting Point

To start at $(0, 5)$, we need to determine the value of $t$ that corresponds to this point:

• Plugging $y = 5$ into $y = 1 + 4 \sin(t)$: $5 = 1 + 4 \sin(t) \Rightarrow 4 = 4 \sin(t) \Rightarrow \sin(t) = 1$ This implies $t = \frac{\pi}{2}$.

Thus, to go halfway around counterclockwise, we start from $t = \frac{\pi}{2}$ and move to $t = \frac{3\pi}{2}$ (which corresponds to $180^\circ$ or half a circle).

Parametric Equations for Part (c)

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

This should be the correct answer for part (c).

Do you want further details, or do you have any questions about the solution?

Here are five related questions that might deepen your understanding:

1. How would the parametric equations change if the path went in a clockwise direction starting from $(0, 5)$?
2. What would the equations be for one complete revolution counterclockwise starting from $(0, 5)$?
3. How does changing the starting point affect the limits for $t$ in parametric equations?
4. What are the parametric equations for a circle with a different radius?
5. How can we derive parametric equations for ellipses with a given center?

Tip: When constructing parametric equations, always verify the starting point by solving for the appropriate initial value of $t$.