To determine the length of an arc intercepted by a central angle in a circle, we use the formula for arc length:
[tex]\[ \text{Arc Length} = \text{Radius} \times \text{Central Angle in Radians} \][/tex]
Given:
- Radius ([tex]\(r\)[/tex]) = 6 feet
- Central Angle ([tex]\(\theta\)[/tex]) = [tex]\(\frac{\pi}{3}\)[/tex] radians
- Use [tex]\(\pi \approx 3.14\)[/tex]
First, we substitute the given values into the formula:
[tex]\[ \text{Arc Length} = 6 \times \left(\frac{3.14}{3}\right) \][/tex]
Next, simplify the expression inside the parentheses:
[tex]\[ \frac{3.14}{3} \approx 1.0466667 \][/tex]
Now, multiply the radius by the result:
[tex]\[ \text{Arc Length} = 6 \times 1.0466667 \approx 6.279999999999999 \][/tex]
After finding the arc length, we round it to the nearest tenth:
[tex]\[ 6.279999999999999 \approx 6.3 \][/tex]
Therefore, the arc length intersected by a central angle of [tex]\(\frac{\pi}{3}\)[/tex] radians in a circle with a radius of 6 feet, rounded to the nearest tenth, is:
[tex]\[ \boxed{6.3 \text{ ft}} \][/tex]
