To solve this problem, let's break it down step by step.

1. Understanding the problem:

   • The child starts at the point $(0,1)$, which represents the "north" position on a unit circle.
   • The carousel makes a complete revolution counterclockwise in 1 minute (60 seconds).
   • The question asks for the coordinates of the child after 15 seconds.

2. Key Concepts:

   • The unit circle is parameterized as $(x, y) = (\cos(\theta), \sin(\theta))$, where $\theta$ is the angle in radians measured counterclockwise from the positive x-axis.
   • One full revolution corresponds to $2\pi$ radians.

3. Calculate the angle after 15 seconds:

   • Time for one full revolution: 60 seconds.
   • Proportion of revolution completed in 15 seconds: $\frac{15}{60} = \frac{1}{4}$.
   • Corresponding angle: $\frac{1}{4} \times 2\pi = \frac{\pi}{2}$ radians.

4. Determine the child's new coordinates:

   • Initially, the child is at $(0, 1)$, which corresponds to $\theta = \frac{\pi}{2}$ on the unit circle.
   • After rotating counterclockwise by $\frac{\pi}{2}$ radians, the child moves to $(-1, 0)$.

Thus, the coordinates of the child after 15 seconds are $(-1, 0)$.

Explanation of Work:

• The rotation corresponds to $\frac{\pi}{2}$ radians since the carousel takes 60 seconds for one revolution.
• Using the unit circle, the angle $\frac{\pi}{2}$ points to the leftmost position $(-1, 0)$.

Would you like additional clarification or help with related questions?

Follow-Up Questions:

1. How do the coordinates change for different times, such as 30 or 45 seconds?
2. How would the solution differ if the carousel rotated clockwise?
3. What happens if the child starts at a different point on the circle?
4. How does the problem change if the carousel has a radius other than 1?
5. Can you explain how the sine and cosine functions are used for circular motion?

Tip: Visualizing problems on the unit circle often simplifies calculations involving rotations and trigonometric functions.