To solve the problem, follow these steps:
1. Understand the problem: We need to find the measure of the central angle, [tex]\( m \angle B \)[/tex], in radians that subtends an arc [tex]\( BC \)[/tex] whose length is given as [tex]\( 21 \pi \)[/tex] units. The circle has a radius of 15 units.
2. Use the formula for arc length: The formula for the length of an arc ([tex]\( L \)[/tex]) subtended by a central angle [tex]\(\theta\)[/tex] in a circle of radius [tex]\( r \)[/tex] is given by:
[tex]\[
L = r \theta
\][/tex]
Here [tex]\( L = 21 \pi \)[/tex] and [tex]\( r = 15 \)[/tex]. We can set up the equation:
[tex]\[
21 \pi = 15 \theta
\][/tex]
3. Solve for [tex]\(\theta\)[/tex]: To find the central angle [tex]\(\theta\)[/tex], divide both sides of the equation by the radius [tex]\( r \)[/tex]:
[tex]\[
\theta = \frac{21 \pi}{15}
\][/tex]
4. Simplify the expression: Simplify the fraction:
[tex]\[
\theta = \frac{21 \pi}{15} = \frac{7 \pi}{5}
\][/tex]
5. Compare with the options: The measure of the central angle [tex]\( m \angle B \)[/tex] is [tex]\(\frac{7 \pi}{5}\)[/tex] radians.
So, the correct answer is:
D. [tex]\( \frac{7}{5} \pi \)[/tex]
