The equation provided, $x^2 + (y - 1)^2 = 16$, describes a circle with center at $(0, 1)$ and radius $4$ (since $16 = 4^2$). Let's find parametric equations for the particle's path in each scenario:

1. Standard parametric form: For a circle with radius $r$ and center at $(h, k)$, parametric equations are: $x = h + r \cos(t)$ $y = k + r \sin(t)$ Here, $h = 0$, $k = 1$, and $r = 4$.

Solutions

(a) Once around clockwise starting at $(4, 1)$, $0 \leq t \leq 2\pi$

Since the particle moves clockwise, we’ll use: $x = 4 \cos(-t) = 4 \cos(t)$ $y = 1 + 4 \sin(-t) = 1 - 4 \sin(t)$ So the parametric equations are: $x = 4 \cos(t), \quad y = 1 - 4 \sin(t), \quad 0 \leq t \leq 2\pi$

(b) Three times around counterclockwise starting at $(4, 1)$, $0 \leq t \leq 6\pi$

For counterclockwise motion, we use the usual positive sine and cosine: $x = 4 \cos(t)$ $y = 1 + 4 \sin(t)$ The parametric equations are: $x = 4 \cos(t), \quad y = 1 + 4 \sin(t), \quad 0 \leq t \leq 6\pi$

(c) Halfway around counterclockwise starting at $(0, 5)$, $0 \leq t \leq \pi$

Starting at $(0, 5)$, which corresponds to $t = \frac{\pi}{2}$ in the circle's parametric form. Moving counterclockwise: $x = 4 \cos(t - \frac{\pi}{2}) = 4 \sin(t)$ [ y = 1 + 4 \sin(t - \frac{\pi}{2}) = 1