The problem provides the radius of a circle as $21.5$ cm and a central angle as $316$ degrees. We need to find the length of the arc corresponding to this central angle.

1. Problem Description

The problem provides the radius of a circle as $21.5$ cm and a central angle as $316$ degrees. We need to find the length of the arc corresponding to this central angle.

2. Solution Steps

First, we need to find the circumference of the entire circle. The formula for the circumference $C$ of a circle is given by:

$C = 2 \pi r$

where $r$ is the radius of the circle.

Given the radius $r = 21.5$ cm, the circumference is:

$C = 2 \pi (21.5) = 43\pi$ cm

Next, we need to determine the fraction of the circle represented by the given central angle of $316$ degrees. Since a full circle is $360$ degrees, the fraction is:

$\frac{316}{360} = \frac{79}{90}$

Finally, to find the arc length $s$, we multiply the circumference of the circle by this fraction:

$s = C \times \frac{316}{360} = 43\pi \times \frac{79}{90} = \frac{3397\pi}{90}$

Approximate $\pi$ as $3.14159$:

$s = \frac{3397 \times 3.14159}{90} \approx \frac{10670.96}{90} \approx 118.566$

3. Final Answer

The arc length is approximately $118.57$ cm.