### Carousel Motion ProblemA child enters a carousel that takes one minute to revolve once around. The child enters at the point \( (0, 1) \), that is, on the due north position. Assume the carousel revolves counterclockwise.**Question:** What are the coordinates of the child after 15 seconds? **Instruction:** Enter the exact answer as a point \( (a, b) \).**Answer Box:** \[ ( \_ , \_ ) \]---To find the coordinates of the child after 15 seconds:1. **Understand the Carousel's Motion:** - The carousel takes 60 seconds (1 minute) for a full rotation. - This rotation is counterclockwise.2. **Calculate Angle Rotated in 15 Seconds:** - A full rotation (360 degrees or \( 2\pi \) radians) occurs in 60 seconds. - In 15 seconds, the angle rotated is: \[ \theta = \left(\frac{15}{60}\right) \times 360^\circ = 90^\circ \quad \text{or} \quad \theta = \left(\frac{15}{60}\right) \times 2\pi = \frac{\pi}{2} \, \text{radians} \]3. **Determine the New Coordinates:** - Initial point: \( (0, 1) \) - After rotating 90 degrees counterclockwise, the coordinates move to a new point. - Using the unit circle, for a 90-degree (or \( \frac{\pi}{2} \) radians) counterclockwise rotation from \( (0,1) \), the new coordinates are \( (-1, 0) \).Thus, the coordinates of the child after 15 seconds are: \[ (a, b) = (-1, 0) \]

A child enters a carousel that takes one minute to revolve once around. The child enters at the point (0,1), that is, on the due north position. Assume the carousel revolves counterclockwise. What are the coordinates of the child after 15 seconds? Enter the exact answer as a point, (a, b).

A child enters a carousel that takes one minute to revolve once around. The child enters at the point (0,1), that is, on the due north position. Assume the carousel revolves counterclockwise. What are the coordinates of the child after 15 seconds? Enter the exact answer as a point, (a, b).

Trigonometry (MindTap Course List)

8th Edition

ISBN:9781305652224

Author:Charles P. McKeague, Mark D. Turner

Publisher:Charles P. McKeague, Mark D. Turner

Chapter7: Triangles

Section7.3: The Ambiguous Case

Problem 30PS: True Course A plane headed due east is traveling with an airspeed of 190 miles per hour. The wind...

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Question

### Carousel Motion Problem

A child enters a carousel that takes one minute to revolve once around. The child enters at the point \( (0, 1) \), that is, on the due north position. Assume the carousel revolves counterclockwise.

**Question:**
What are the coordinates of the child after 15 seconds?

**Instruction:**
Enter the exact answer as a point \( (a, b) \).

**Answer Box:**
\[ ( \_ , \_ ) \]

---

To find the coordinates of the child after 15 seconds:

1. **Understand the Carousel's Motion:**
- The carousel takes 60 seconds (1 minute) for a full rotation.
- This rotation is counterclockwise.

2. **Calculate Angle Rotated in 15 Seconds:**
- A full rotation (360 degrees or \( 2\pi \) radians) occurs in 60 seconds.
- In 15 seconds, the angle rotated is:
\[
\theta = \left(\frac{15}{60}\right) \times 360^\circ = 90^\circ \quad \text{or} \quad \theta = \left(\frac{15}{60}\right) \times 2\pi = \frac{\pi}{2} \, \text{radians}
\]

3. **Determine the New Coordinates:**
- Initial point: \( (0, 1) \)
- After rotating 90 degrees counterclockwise, the coordinates move to a new point.
- Using the unit circle, for a 90-degree (or \( \frac{\pi}{2} \) radians) counterclockwise rotation from \( (0,1) \), the new coordinates are \( (-1, 0) \).

Thus, the coordinates of the child after 15 seconds are:
\[ (a, b) = (-1, 0) \]

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