To solve for the radius [tex]\( r \)[/tex] of the circle, we can use the relationship between the arc length [tex]\( s \)[/tex], the radius [tex]\( r \)[/tex], and the central angle [tex]\( \theta \)[/tex] (in radians). The formula that relates these quantities is:
[tex]\[
s = r \theta
\][/tex]
We are given:
[tex]\[
\theta = \frac{\pi}{18} \quad \text{and} \quad s = \frac{5\pi}{6}
\][/tex]
Our task is to find the radius [tex]\( r \)[/tex]. We can rearrange the formula to solve for [tex]\( r \)[/tex]:
[tex]\[
r = \frac{s}{\theta}
\][/tex]
Substitute the given values into the equation:
[tex]\[
r = \frac{\frac{5\pi}{6}}{\frac{\pi}{18}}
\][/tex]
To simplify the division of fractions, we multiply by the reciprocal:
[tex]\[
r = \frac{5\pi}{6} \cdot \frac{18}{\pi}
\][/tex]
Notice that the [tex]\( \pi \)[/tex] terms in the numerator and denominator cancel each other out:
[tex]\[
r = \frac{5 \cdot 18}{6}
\][/tex]
Further simplify the expression by performing the multiplication and division:
[tex]\[
r = \frac{90}{6}
\][/tex]
[tex]\[
r = 15
\][/tex]
Therefore, the exact length of the radius is:
[tex]\[
r = 15 \, \text{m}
\][/tex]
