a circle has a central angle measuring $\frac{7pi}{6}$ radians that intersects an arc of length 18 cm. what…

Explanation:

Step1: Recall arc - length formula

The formula for the length of an arc $s$ of a circle is $s = r\theta$, where $r$ is the radius of the circle and $\theta$ is the central - angle in radians. We know that $s = 18$ cm and $\theta=\frac{7\pi}{6}$ radians.

Step2: Solve for the radius $r$

From $s = r\theta$, we can express $r$ as $r=\frac{s}{\theta}$. Substitute $s = 18$ and $\theta=\frac{7\pi}{6}$ into the formula: $r=\frac{18}{\frac{7\pi}{6}}$. Using the rule of dividing by a fraction ($a\div\frac{b}{c}=a\times\frac{c}{b}$), we get $r = 18\times\frac{6}{7\pi}$. Since $\pi\approx3.14$, then $r = 18\times\frac{6}{7\times3.14}$. First, calculate $7\times3.14 = 21.98$. Then, $18\times6 = 108$. So, $r=\frac{108}{21.98}\approx4.9$ cm.

Answer:

4.9 cm