Let's solve this step-by-step.
1. Identify the given values:
- Arc length, [tex]\(s = 18\)[/tex] cm
- Central angle, [tex]\(\theta = \frac{7 \pi}{6}\)[/tex] radians
- Use [tex]\(\pi \approx 3.14\)[/tex]
2. Substitute the approximate value for [tex]\(\pi\)[/tex]:
[tex]\[
\theta = \frac{7 \cdot 3.14}{6}
\][/tex]
3. Calculate the central angle in radians:
[tex]\[
\theta = \frac{21.98}{6} = 3.6633 \text{ radians}
\][/tex]
4. Recall the formula for arc length of a circle:
[tex]\[
s = r \theta
\][/tex]
where [tex]\(s\)[/tex] is the arc length, [tex]\(r\)[/tex] is the radius, and [tex]\(\theta\)[/tex] is the central angle in radians.
5. Rearrange the formula to solve for the radius [tex]\(r\)[/tex]:
[tex]\[
r = \frac{s}{\theta}
\][/tex]
6. Substitute the given values into the equation:
[tex]\[
r = \frac{18}{3.6633}
\][/tex]
7. Calculate the radius [tex]\(r\)[/tex]:
[tex]\[
r \approx 4.9136 \text{ cm}
\][/tex]
8. Round the value to the nearest tenth:
[tex]\[
r \approx 4.9 \text{ cm}
\][/tex]
Therefore, the length of the radius of the circle, rounded to the nearest tenth, is [tex]\( \boxed{4.9 \text{ cm}} \)[/tex].
