Question

the circle below has center v, and its radius is 6 ft. given that m∠wvx = 150°, find the length of the major arc wyx. give an exact answer in terms of π, and be sure to include the correct unit in your answer. length of major arc wyx :

Explanation:

Step1: Recall 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. First, we need to convert the given central angle from degrees to radians. The conversion formula is $\theta_{rad}=\theta_{deg}\times\frac{\pi}{180}$. Given $\theta_{deg}=150^{\circ}$, then $\theta_{rad}=150\times\frac{\pi}{180}=\frac{5\pi}{6}$ radians. But we want the length of the major arc. The total angle around a circle is $2\pi$ radians. The central angle of the major arc $WY\overset{\frown}{X}$ is $\theta = 2\pi-\frac{5\pi}{6}=\frac{12\pi - 5\pi}{6}=\frac{7\pi}{6}$ radians, and the radius $r = 6$ ft.

Step2: Calculate the length of the major arc

Using the arc - length formula $s=r\theta$, substitute $r = 6$ ft and $\theta=\frac{7\pi}{6}$ radians. So, $s=6\times\frac{7\pi}{6}=7\pi$ ft.

Answer:

$7\pi$ ft