https://file.aiquickdraw.com/mathbot/file/1728461304220todoh9to.jpg

A game shows a chronometer that measures the time an attack takes. The arm starts upwardly as indicated by [ST1]. And it makes a whole tour in 1 minute. It has a diameter of 16 units. The arm starts upwardly as indicated by [ST1]. T2 indicates the position of the arm after 10 seconds. How many degrees is alpha?

To solve the problem, we can break it down into a few steps.

### Given Data:
- The circle has a **diameter** of 16 units, meaning the **radius** is 8 units.
- The arm of the chronometer makes a complete tour in 1 minute (60 seconds).
- After 10 seconds, the arm has moved from the starting point \(T_1\) to \(T_2\).

The key is to determine how many degrees (\(\alpha\)) the arm rotates after 10 seconds.

### Step 1: Full circle rotation
A full rotation around a circle corresponds to \(360^\circ\) in one complete revolution (which is 60 seconds in this case).

### Step 2: Proportionality calculation
Since the arm moves for 10 seconds, and we know the entire revolution takes 60 seconds, the proportion of the circle covered in that time is:

\[
\text{Proportion covered} = \frac{10 \text{ seconds}}{60 \text{ seconds}} = \frac{1}{6}
\]

### Step 3: Calculate \(\alpha\)
The angle \(\alpha\) covered by the arm in 10 seconds will be \(\frac{1}{6}\) of the total \(360^\circ\):

\[
\alpha = \frac{360^\circ}{6} = 60^\circ
\]

Thus, the angle \(\alpha\) between \(ST_1\) and \(ST_2\) is \(60^\circ\).

Would you like more details on this solution or further explanations?

Here are 5 follow-up questions to extend your understanding:
1. What is the relationship between the radius and the arc length subtended by an angle in a circle?
2. How would the angle change if the arm moved for 20 seconds instead of 10?
3. Can you calculate the length of the arc covered by the arm in 10 seconds?
4. How would this problem change if the diameter of the circle were different?
5. Can we use radians instead of degrees to measure the angle \(\alpha\)? How would that change the solution?

**Tip:** When dealing with circular motion, always relate the time spent rotating to the fraction of the full circle to find the corresponding angle.

Explore how to calculate the angle covered by a chronometer arm after 10 seconds, given it makes a complete rotation in 1 minute. This problem involves circle geometry, proportionality, and angle measurement and is suitable for students in Grades 8-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation