What is the arc length of the arc subtended in a circle with radius 6 and an angle of [tex]\frac{7 \pi}{8}[/tex]?

A. [tex]\frac{7 \pi}{4}[/tex]
B. [tex]36 \pi[/tex]
C. [tex]\frac{126 \pi}{4}[/tex]
D. [tex]\frac{14 \pi}{8}[/tex]
E. [tex]\frac{21 \pi}{4}[/tex]



Answer :

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]