To find the radius of the circle, we start with the formula for the arc length of a circle, which is given by:
[tex]\[ \text{Arc Length} = r \times \theta \][/tex]
where [tex]\( r \)[/tex] is the radius of the circle, and [tex]\( \theta \)[/tex] is the central angle in radians corresponding to the arc.
Given the arc length [tex]\( \frac{115}{6} \pi \)[/tex], we need to solve for the radius [tex]\( r \)[/tex].
The problem implies that the arc length corresponds to the entire circumference of the circle, which means the central angle [tex]\( \theta \)[/tex] is [tex]\( 2\pi \)[/tex] (a complete circle).
Substitute the arc length and [tex]\( \theta \)[/tex] into the formula:
[tex]\[ \frac{115}{6} \pi = r \times 2\pi \][/tex]
To isolate [tex]\( r \)[/tex], divide both sides of the equation by [tex]\( 2\pi \)[/tex]:
[tex]\[ r = \frac{\frac{115}{6} \pi}{2\pi} \][/tex]
Simplify the fraction:
[tex]\[ r = \frac{\frac{115}{6}}{2} \][/tex]
[tex]\[ r = \frac{115}{12} \][/tex]
Convert this fraction to a decimal for better understanding:
[tex]\[ r \approx 9.5833 \][/tex]
Therefore, the radius of the circle is [tex]\(\boxed{9.5833}\)[/tex] inches.
