Question

An arctic village maintains a circular cross-country ski trail that has a radius of 3.7 kilometers. A skier started skiing from the position (-3.345, 1.581). measured in kilometers, and skied counter-clockwise for 4.07 kilometers, where he paused for a brief rest. (Consider the circle to be centered at the origin). Determine the ordered pair (in both. kilometers and radii) on the coordinate axes that identifies the location where the skier rested. (Start by drawing a diagram to represent this situation.)

Answer 1

To determine the location where the skier rested, we can use the concept of displacement. The skier started skiing from the position (-3.345, 1.581) and skied counter-clockwise for 4.07 kilometers. By calculating the final position using the displacement vector equations, we find that the skier rested at approximately (0.129, -3.537) kilometers or (0.035, -0.955) radii.

Step-by-step explanation:

To determine the location where the skier rested, we can use the concept of displacement. The skier started skiing from the position (-3.345, 1.581) and skied counter-clockwise for 4.07 kilometers. We can calculate the final position by adding the displacement to the initial position.

The displacement vector is given by: Δx = 4.07 km * cos(θ), Δy = 4.07 km * sin(θ), where θ is the angle of the displacement vector.

Using the formula for a circle, we can find the angle θ given the radius of the circular trail. The angle θ = arccos(-3.345/3.7) = 127.04°. Substituting this angle into the displacement vector equations, we find that the final position of the skier is approximately (0.129, -3.537) kilometers.

The ordered pair on the coordinate axes that identifies the location where the skier rested is (0.129, -3.537) kilometers or approximately (0.035, -0.955) radii.