Question

how long is the arc intersected by a \\( \frac { 2 \pi } { 3 } \\) radian central angle in a circle with radius 7 feet?
\\( \frac { 2 \pi } { 21 } \\) feet
\\( \frac { 7 \pi } { 3 } \\) feet
\\( \frac { 21 } { 2 \pi } \\) feet
\\( \frac { 14 \pi } { 3 } \\) feet

Explanation:

Step1: Recall the arc length formula

The formula for the length of an arc \( s \) in a circle with radius \( r \) and central angle \( \theta \) (in radians) is \( s = r\theta \).

Step2: Substitute the given values

We are given that \( r = 7 \) feet and \( \theta=\frac{2\pi}{3} \) radians. Substituting these values into the formula, we get \( s = 7\times\frac{2\pi}{3} \).

Step3: Simplify the expression

Multiplying 7 and \( \frac{2\pi}{3} \), we have \( s=\frac{14\pi}{3} \) feet.

Answer:

\(\frac{14\pi}{3}\) feet (corresponding to the option \(\frac{14\pi}{3}\) feet)