To determine the length of an arc intersected by a central angle in a circle, we can use the formula for arc length:
[tex]\[ \text{Arc Length} = r \times \theta \][/tex]
where [tex]\( r \)[/tex] is the radius of the circle and [tex]\( \theta \)[/tex] is the central angle in radians. Let's break down the steps to find the arc length given a radius [tex]\( r = 7 \)[/tex] feet and a central angle [tex]\( \theta = \frac{2\pi}{3} \)[/tex] radians:
1. Identify the radius ([tex]\( r \)[/tex]) and the central angle ([tex]\( \theta \)[/tex]):
- Radius [tex]\( r = 7 \)[/tex] feet.
- Angle [tex]\( \theta = \frac{2\pi}{3} \)[/tex] radians.
2. Substitute the values into the arc length formula:
[tex]\[ \text{Arc Length} = 7 \times \frac{2\pi}{3} \][/tex]
3. Perform the multiplication:
[tex]\[ \text{Arc Length} = \frac{14\pi}{3} \][/tex]
So, the length of the arc is [tex]\(\frac{14\pi}{3}\)[/tex] feet.
Thus, the correct answer is:
[tex]\[ \frac{14\pi}{3} \text{ feet} \][/tex]
