Question

what is the length of $widehat{fg}$?
a. $\frac{2pi}{3}$ units
b. $\frac{4pi}{3}$ units
c. $\frac{2pi^{2}}{3}$ units
d. $\frac{8pi}{3}$ units
e. $\frac{4pi^{2}}{3}$ units

Explanation:

Step1: Identify the radius and central - angle

The circle has a center at the origin $(0,0)$ and a point $(4,0)$ on the circle, so the radius $r = 4$. The central - angle $\theta$ corresponding to arc $\widehat{FG}$ is $60^{\circ}$ or $\frac{\pi}{3}$ radians (since the angle is given as $60^{\circ}$ and we convert to radians using the formula $\theta_{rad}=\frac{\pi}{180}\times\theta_{deg}$).

Step2: Use the arc - length formula

The formula for the length of an arc of a circle is $s = r\theta$, where $s$ is the arc - length, $r$ is the radius of the circle, and $\theta$ is the central - angle in radians. Substitute $r = 4$ and $\theta=\frac{\pi}{3}$ into the formula: $s=4\times\frac{\pi}{3}=\frac{4\pi}{3}$.

Answer:

B. $\frac{4\pi}{3}$ units