To find [tex]\( m \angle BOC \)[/tex], which is the measure of the angle subtended by the arc BC at the center O in radians, follow these steps:
1. Identify the given values:
- The radius of the circle, [tex]\( r \)[/tex], is 15 units.
- The length of the arc BC, [tex]\( L \)[/tex], is [tex]\( 21 \pi \)[/tex] units.
2. Understand the relationship:
The length of an arc in a circle is calculated using the formula
[tex]\[
L = r \times \theta
\][/tex]
where [tex]\( L \)[/tex] is the arc length, [tex]\( r \)[/tex] is the radius, and [tex]\( \theta \)[/tex] is the central angle in radians.
3. Find the central angle [tex]\( \theta \)[/tex]:
Rearrange the arc length formula to solve for [tex]\( \theta \)[/tex]:
[tex]\[
\theta = \frac{L}{r}
\][/tex]
Substitute the given values [tex]\( L = 21 \pi \)[/tex] and [tex]\( r = 15 \)[/tex]:
[tex]\[
\theta = \frac{21 \pi}{15}
\][/tex]
Simplify the fraction:
[tex]\[
\theta = \frac{21 \pi}{15} = \frac{7}{5} \pi
\][/tex]
Therefore, the measure of [tex]\( m \angle BOC \)[/tex] in radians is [tex]\( \frac{7}{5} \pi \)[/tex].
The correct answer is:
D. [tex]\( \frac{7}{5} \pi \)[/tex]
