Certainly! Let's work on this problem step by step:
1. Understand the Problem:
- We need to determine the distance [tex]\( s \)[/tex] that a point travels along the circumference of a circle over a given period of time.
- The given parameters are:
- Time [tex]\( t = 68 \)[/tex] seconds
- Angular speed [tex]\( \omega = \frac{\pi}{12} \)[/tex] radians per second
- Radius [tex]\( r = 17 \)[/tex] centimeters
2. Use the Formula:
- The distance traveled [tex]\( s \)[/tex] in a circular motion can be calculated using the formula:
[tex]\[
s = \omega \times r \times t
\][/tex]
- Here:
- [tex]\( \omega \)[/tex] is the angular speed
- [tex]\( r \)[/tex] is the radius
- [tex]\( t \)[/tex] is the time
3. Plug in the Values:
- Substitute the given numerical values into the formula.
[tex]\[
s = \left( \frac{\pi}{12} \right) \times 17 \times 68
\][/tex]
4. Perform the Calculation:
- First, compute the product of the numbers within the parentheses.
[tex]\[
\frac{\pi}{12} \times 17 \times 68
\][/tex]
- This computation gives:
[tex]\[
\frac{\pi}{12} \times 1156 = \frac{1156 \pi}{12}
\][/tex]
Simplifying further:
[tex]\[
\frac{1156 \pi}{12} \approx 302.6400922958167
\][/tex]
5. Round the Result:
- The problem asks us to round the answer to three significant digits.
[tex]\[
s \approx 302.64 \, \text{cm}
\][/tex]
So, the distance [tex]\( s \)[/tex] that the point travels along the circumference of the circle over 68 seconds is approximately:
[tex]\[
s \approx 302.64 \, \text{cm}
\][/tex]
This step-by-step solution details the problem-solving process and demonstrates how the final rounded distance [tex]\( s \approx 302.64 \)[/tex] cm was obtained.
