To find the arc length of an arc subtended in a circle with a given radius and a given angle, we use the formula for arc length. The formula for the arc length [tex]\( L \)[/tex] is:
[tex]\[
L = r \theta
\][/tex]
where:
- [tex]\( r \)[/tex] is the radius of the circle.
- [tex]\( \theta \)[/tex] is the angle in radians.
In this problem, we are given:
- The radius [tex]\( r \)[/tex] is 6.
- The angle [tex]\( \theta \)[/tex] is [tex]\(\frac{7\pi}{8}\)[/tex].
Let's plug these values into the formula:
[tex]\[
L = 6 \times \frac{7\pi}{8}
\][/tex]
First, we perform the multiplication of the constants:
[tex]\[
L = \frac{6 \times 7\pi}{8} = \frac{42\pi}{8}
\][/tex]
Next, we simplify the fraction [tex]\(\frac{42\pi}{8}\)[/tex]:
[tex]\[
L = \frac{42\pi}{8} = \frac{21\pi}{4}
\][/tex]
Therefore, the arc length is:
[tex]\[
L = \frac{21\pi}{4}
\][/tex]
Thus, the correct answer is:
E. [tex]\(\frac{21\pi}{4}\)[/tex]
