Question

the circle below has center o, and its radius is 7 ft. given that m∠aob = 30°, find the length of the minor arc ab. give an exact answer in terms of π, and be sure to include the correct unit in your answer. length of minor arc ab:

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, convert the angle from degrees to radians. We know that to convert degrees to radians, we use the conversion factor $\theta_{rad}=\theta_{deg}\times\frac{\pi}{180}$. Given $\theta_{deg}=30^{\circ}$, then $\theta_{rad}=30\times\frac{\pi}{180}=\frac{\pi}{6}$ radians.

Step2: Identify radius

The radius of the circle $r = 7$ ft.

Step3: Calculate arc - length

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

Answer:

$\frac{7\pi}{6}$ ft