Let’s analyze the problem step by step. Here’s the situation:

1. Given Data:
   • The ski trail is a circular path with a radius of 3.4 km.
   • Yvette starts at the 3 o’clock position (eastmost point of the circle).
   • Yvette travels in the counterclockwise direction.
   • Yvette stops when her position is described as being -3.143 km east and 1.298 km north relative to the trail's center.

Part (a): Finding the Angle Swept in Radians

1. Coordinates for her position:

   • Her position is given as $(-3.143, 1.298)$ in terms of coordinates relative to the center of the circle.

2. Using the angle formula in polar coordinates: The angle $\theta$ (in radians) is determined using the arctangent function: $\theta = \tan^{-1}\left(\frac{y}{x}\right)$ Substituting $x = -3.143$ and $y = 1.298$: $\theta = \tan^{-1}\left(\frac{1.298}{-3.143}\right)$ This gives $\theta$, but since $x < 0$ and $y > 0$, the point lies in the second quadrant. So, we adjust $\theta$ accordingly to be in the range $(0, 2\pi)$.

3. Calculate $\theta$.

Part (b): Distance Traveled by Yvette

1. Arc Length Formula: The distance $s$ traveled along the circle is related to the angle $\theta$ (in radians) by the formula: $s = r \cdot \theta$ Substit