Question

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the circle below has center o, and its radius is 6 cm. given that m∠aob = 50°, find the length of the major arc acb. give an exact answer in terms of π, and be sure to include the correct unit in your answer. length of major arc acb:

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 central angle from degrees to radians. The measure of the central angle of the major arc $ACB$ is $\theta=360^{\circ}-50^{\circ}=310^{\circ}$. To convert degrees to radians, use the conversion factor $\theta_{rad}=\theta_{deg}\times\frac{\pi}{180}$. So, $\theta = 310\times\frac{\pi}{180}=\frac{31\pi}{18}$ radians.

Step2: Apply the arc - length formula

The radius of the circle $r = 6$ cm. Using the arc - length formula $s=r\theta$, substitute $r = 6$ and $\theta=\frac{31\pi}{18}$ into the formula. Then $s=6\times\frac{31\pi}{18}$.

Step3: Simplify the expression

$6\times\frac{31\pi}{18}=\frac{31\pi}{3}$ cm.

Answer:

$\frac{31\pi}{3}$ cm