Certainly! To determine the measure of angle [tex]\( \angle BOC \)[/tex] in radians, given that the radius of the circle is [tex]\(15\)[/tex] units and the length of the arc [tex]\( BC \)[/tex] is [tex]\(21\pi\)[/tex] units, we can follow these steps:
1. Recall the formula for the arc length:
The length of an arc [tex]\( s \)[/tex] of a circle can be calculated using the formula [tex]\( s = r\theta \)[/tex], where [tex]\( r \)[/tex] is the radius of the circle and [tex]\( \theta \)[/tex] is the central angle in radians.
2. Identify the given values:
- Radius [tex]\( r = 15 \)[/tex] units
- Arc length [tex]\( s = 21\pi \)[/tex] units
3. Set up the equation:
Given the formula [tex]\( s = r\theta \)[/tex], substitute the values:
[tex]\[
21\pi = 15\theta
\][/tex]
4. Solve for [tex]\( \theta \)[/tex]:
Divide both sides of the equation by [tex]\( 15 \)[/tex]:
[tex]\[
\theta = \frac{21\pi}{15}
\][/tex]
5. Simplify the fraction:
Simplify [tex]\( \frac{21\pi}{15} \)[/tex]:
[tex]\[
\theta = \frac{21}{15}\pi = \frac{7}{5}\pi
\][/tex]
Therefore, the measure of [tex]\( \angle BOC \)[/tex] in radians is [tex]\(\frac{7}{5}\pi\)[/tex].
So, the correct answer is:
A. [tex]\(\frac{7}{5}\pi\)[/tex]
