To find the radius of the circle, we start with the formula for the length of an arc, which is given by:
[tex]\[ \text{Arc Length} = r \cdot \theta \][/tex]
where [tex]\( r \)[/tex] is the radius of the circle, and [tex]\( \theta \)[/tex] is the central angle in radians.
For a full circle, the central angle [tex]\( \theta \)[/tex] is [tex]\( 2\pi \)[/tex] radians. Therefore, the formula becomes:
[tex]\[ \text{Arc Length} = r \cdot 2\pi \][/tex]
We're given that the arc length [tex]\( \frac{115}{6}\pi \)[/tex] inches, so we set the arc length equal to [tex]\( r \cdot 2\pi \)[/tex] and solve for [tex]\( r \)[/tex]:
[tex]\[ \frac{115}{6}\pi = r \cdot 2\pi \][/tex]
Divide both sides of the equation by [tex]\( 2\pi \)[/tex]:
[tex]\[ r = \frac{\frac{115}{6}\pi}{2\pi} \][/tex]
Simplify by canceling out [tex]\( \pi \)[/tex]:
[tex]\[ r = \frac{115}{6} \cdot \frac{1}{2} = \frac{115}{12} \][/tex]
When you calculate [tex]\( \frac{115}{12} \)[/tex], you get approximately:
[tex]\[ r \approx 9.583333333333334 \][/tex]
Thus, the radius of the circle is approximately [tex]\( 9.583333333333334 \)[/tex] inches.
So, the radius of the circle is [tex]\(\boxed{9.583333333333334}\)[/tex] inches.
