what is the length of the arc shown in red?\nthe length of the arc is $\\square$ cm.\n(simplify your answer…

Explanation:

Step1: Identify given values

The radius $r$ is $7\text{ cm}$ and the central angle $\theta$ is $30^{\circ}$.

Step2: State arc length formula

The formula for arc length $s$ using degrees is: $$s = \frac{\theta}{360^{\circ}} \cdot 2\pi r$$

Step3: Substitute values into formula

Substitute $r = 7$ and $\theta = 30^{\circ}$: $$s = \frac{30}{360} \cdot 2\pi(7)$$

Step4: Simplify the expression

Reduce the fraction and multiply the constants: $$s = \frac{1}{12} \cdot 14\pi = \frac{14\pi}{12}$$

Step5: Reduce to lowest terms

Divide the numerator and denominator by $2$: $$s = \frac{7\pi}{6}$$

Answer:

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