Let's solve the problem step-by-step using the given data.
### Step 1: Identify the given values.
- Angular speed, [tex]\(\omega = \frac{\pi}{10}\)[/tex] radians per second.
- Radius of the circle, [tex]\(r = 4.4\)[/tex] inches.
- Time, [tex]\(t = 7\)[/tex] minutes.
### Step 2: Convert time from minutes to seconds.
Since there are 60 seconds in a minute, we need to convert [tex]\(t\)[/tex] from minutes to seconds.
[tex]\[
t = 7 \text{ minutes} \times 60 \text{ seconds/minute} = 420 \text{ seconds}
\][/tex]
### Step 3: Use the formula for the distance traveled along the circle.
The formula to find the distance [tex]\(s\)[/tex] that a point travels along the circle is:
[tex]\[
s = \omega \cdot r \cdot t
\][/tex]
### Step 4: Substitute the given values into the formula.
[tex]\[
s = \left(\frac{\pi}{10}\right) \cdot 4.4 \cdot 420
\][/tex]
### Step 5: Perform the multiplication step.
We'll break down the calculation:
[tex]\[
s = \left(\frac{\pi}{10}\right) \cdot 4.4 \cdot 420
\][/tex]
First, calculate the product of [tex]\(\frac{\pi}{10}\)[/tex] and 420:
[tex]\[
\frac{\pi}{10} \cdot 420 = 42\pi
\][/tex]
Next, multiply this result by 4.4:
[tex]\[
42\pi \cdot 4.4
\][/tex]
### Step 6: Evaluate the expression using the value of [tex]\(\pi \approx 3.14159\)[/tex].
[tex]\[
42\pi \cdot 4.4 = 42 \cdot 3.14159 \cdot 4.4 \approx 42 \cdot 13.823 \approx 579.666
\][/tex]
### Step 7: Round the result to three significant digits.
The distance [tex]\(s\)[/tex] is approximately:
[tex]\[
s \approx 580.566 \text{ inches}
\][/tex]
Therefore, the distance [tex]\(s\)[/tex] a point travels along the circle in 7 minutes, with an angular speed of [tex]\(\frac{\pi}{10}\)[/tex] radians per second and a radius of 4.4 inches, is 580.566 inches when rounded to three significant digits.
