https://file.aiquickdraw.com/mathbot/file/172577793683291znvzk5.jpg

The image shows a problem related to determining the coordinates of the hands of a clock, as well as finding the angular movement of the minute hand after a certain time.

Here is the task outlined in the image:

**Module 13 Application Problem #1: Clock Face**
1. **Given:**
   - The length of the minute hand is 4 cm.
   - The length of the hour hand is 3 cm.
   - The center of the clock is located at a coordinate plane origin.

   You are asked to determine the **coordinates** of the end of the minute hand at a specific time (likely shown on the clock image), and then find how much it will move after 15 minutes.

2. **Questions:**
   - At the current time on the clock face, where are the coordinates of the minute hand’s tip?
   - How far will the minute hand rotate after 15 minutes?

Let’s solve these one by one.

### Part 1: Coordinates of the Minute Hand
The problem is asking for the coordinates of the tip of the minute hand, which is 4 cm long. The clock’s minute hand rotates 360° in 60 minutes, with 6° per minute.

We’ll assume the current position of the minute hand is shown in the clock face image.

- First, determine the angle of the minute hand with respect to the 12 o'clock position.
- Using trigonometry, you can calculate the coordinates of the tip as:
  \[
  x = r \cdot \cos(\theta)
  \]
  \[
  y = r \cdot \sin(\theta)
  \]
  where:
  - \( r \) is the length of the minute hand (4 cm),
  - \( \theta \) is the angle (in radians or degrees) the minute hand makes with the vertical (12 o’clock position).

### Part 2: Movement after 15 minutes
- In 15 minutes, the minute hand will rotate by:
  \[
  15 \times 6^\circ = 90^\circ
  \]
  This is a quarter of the full clock rotation, so we’ll calculate the new coordinates after this 90° movement.

Let me know if you want a full worked-out solution for the specific time on the clock or further clarifications!

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Here are five follow-up questions to expand on the concept:
1. What is the angular velocity of the minute hand in radians per minute?
2. How would the coordinates of the hour hand change as time progresses?
3. How do you calculate the angular displacement of the hour hand after a given time?
4. Can you generalize the formula for the position of any hand at any given time?
5. What would be the total distance traveled by the tip of the minute hand in one full revolution?

**Tip:** When working with angles on a clock, always remember the correspondence between time and degrees: 1 minute equals 6°, and 1 hour equals 30°.

Module 13 Application Problem #1: Clock Face - Determine the coordinates of the end of the minute hand at a specific time and the angular movement of the minute hand after 15 minutes.

Solve a trigonometry problem by calculating the coordinates of the tip of a clock’s minute hand and determining its angular movement after 15 minutes. The problem involves applying trigonometric functions, including cosine and sine, to determine the position of the minute hand. Perfect for students in Grades 9-12.

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