We are asked to find the parametric equations for a particle moving along the circle given by the equation:

$x^2 + (y - 3)^2 = 16$

This is the equation of a circle with center at $(0, 3)$ and radius $4$, since $\sqrt{16} = 4$.

Step-by-step approach:

1. Standard form of the parametric equations of a circle:

   The general parametric equations for a circle with center $(h, k)$ and radius $r$ are:

   $x(t) = h + r \cos(t)$ $y(t) = k + r \sin(t)$

   For this circle:

   • Center: $(h, k) = (0, 3)$
   • Radius: $r = 4$

   Therefore, the parametric equations for this circle are:

   $x(t) = 0 + 4 \cos(t) = 4 \cos(t)$ $y(t) = 3 + 4 \sin(t)$

2. Halfway around counterclockwise:

   The particle moves halfway around the circle starting at $(0, 7)$. The starting point corresponds to $t = 0$, where:

   • $x(0) = 4 \cos(0) = 4$
   • $y(0) = 3 + 4 \sin(0) = 3$

   Since the particle moves halfway around the circle, this corresponds to a $t$-range from $t = 0$ to $t = \pi$, which takes us halfway around the circle in a counterclockwise direction.

3. Final parametric equations:

   Therefore, the parametric equations for the particle's motion are:

   $x(t) = 4 \cos(t)$ $y(t) = 3 + 4 \sin(t)$

   with the parameter $t$ ranging from $0$ to $\pi$, representing the counterclockwise movement from $(0, 7)$ halfway around the circle.

Final Answer:

$x(t) = 4 \cos(t), \quad y(t) = 3 + 4 \sin(t), \quad 0 \leq t \leq \pi$

Would you like further details or have any specific questions? Here are some related questions:

1. How can we describe parametric equations for a full revolution around the circle?
2. How does the center of the circle influence the parametric equations?
3. How do we convert parametric equations to the general form of a circle equation?
4. What would the parametric equations be for moving clockwise instead of counterclockwise?
5. What happens if we change the radius of the circle in the parametric equations?

Tip: When deriving parametric equations for circular motion, always use trigonometric functions ($\cos$ and $\sin$) to capture the cyclical nature of the path.