The problem provides the radius of a circle, $r = 21.5$ cm, and the angle of a sector, $\theta = 316^\circ$. We need to find the arc length of the sector.

1. Problem Description

The problem provides the radius of a circle, $r = 21.5$ cm, and the angle of a sector, $\theta = 316^\circ$. We need to find the arc length of the sector.

2. Solution Steps

First, convert the angle from degrees to radians. Since $180^\circ = \pi$ radians, we have

$$\theta_{radians} = \theta_{degrees} \times \frac{\pi}{180^\circ}$$

Substituting $\theta_{degrees} = 316^\circ$,

$$\theta_{radians} = 316^\circ \times \frac{\pi}{180^\circ} = \frac{316\pi}{180} = \frac{79\pi}{45} \text{ radians}$$

The arc length $s$ of a sector is given by the formula

$$s = r\theta$$

where $r$ is the radius and $\theta$ is the angle in radians.

Substituting $r = 21.5$ cm and $\theta = \frac{79\pi}{45}$,

$$s = 21.5 \times \frac{79\pi}{45}$$

$$s = \frac{1698.5\pi}{45} \approx \frac{1698.5 \times 3.14159}{45} \approx \frac{5335.42}{45} \approx 118.56 \text{ cm}$$

3. Final Answer

The arc length of the sector is approximately $118.56$ cm.