To determine the measure of the central angle [tex]\( \angle BOC \)[/tex] in radians, we can use the relationship between the arc length, the radius of the circle, and the central angle.
1. Given Data:
- Radius ([tex]\( r \)[/tex]) of the circle: 15 units
- Length of arc [tex]\( BC \)[/tex]: [tex]\( 21 \pi \)[/tex] units
2. Formula: The arc length ([tex]\( L \)[/tex]) of a sector of a circle is given by:
[tex]\[
L = r \cdot \theta
\][/tex]
where [tex]\( r \)[/tex] is the radius of the circle and [tex]\( \theta \)[/tex] is the central angle in radians.
3. Rearrange the formula to solve for the central angle [tex]\( \theta \)[/tex]:
[tex]\[
\theta = \frac{L}{r}
\][/tex]
4. Substitute the given values into the formula:
[tex]\[
\theta = \frac{21 \pi}{15}
\][/tex]
5. Simplify the fraction:
[tex]\[
\theta = \frac{21 \pi}{15} = \frac{7 \pi}{5}
\][/tex]
So, the measure of [tex]\( \angle BOC \)[/tex] in radians is [tex]\( \frac{7}{5} \pi \)[/tex].
Thus, the correct answer is:
C. [tex]\( \frac{7}{5} \pi \)[/tex]
