Answer :
To find the exact length of an arc in a circle, we can use the formula:
[tex]\[ s = r \theta \][/tex]
where [tex]\( s \)[/tex] is the length of the arc, [tex]\( r \)[/tex] is the radius, and [tex]\( \theta \)[/tex] is the central angle in radians.
Given the values:
- [tex]\(\theta = \frac{\pi}{12}\)[/tex] (central angle in radians)
- [tex]\(r = 8\)[/tex] yards (radius of the circle)
We substitute these values into the formula:
[tex]\[ s = 8 \cdot \frac{\pi}{12} \][/tex]
Now, simplify this fraction:
[tex]\[ s = 8 \cdot \frac{\pi}{12} = \frac{8\pi}{12} = \frac{2\pi}{3} \][/tex]
Therefore, the exact length of the arc is:
[tex]\[ s = \frac{2\pi}{3} \text{ yards} \][/tex]
So, the exact, fully simplified answer is:
[tex]\[ s = \frac{2\pi}{3} \text{ yards} \][/tex]
[tex]\[ s = r \theta \][/tex]
where [tex]\( s \)[/tex] is the length of the arc, [tex]\( r \)[/tex] is the radius, and [tex]\( \theta \)[/tex] is the central angle in radians.
Given the values:
- [tex]\(\theta = \frac{\pi}{12}\)[/tex] (central angle in radians)
- [tex]\(r = 8\)[/tex] yards (radius of the circle)
We substitute these values into the formula:
[tex]\[ s = 8 \cdot \frac{\pi}{12} \][/tex]
Now, simplify this fraction:
[tex]\[ s = 8 \cdot \frac{\pi}{12} = \frac{8\pi}{12} = \frac{2\pi}{3} \][/tex]
Therefore, the exact length of the arc is:
[tex]\[ s = \frac{2\pi}{3} \text{ yards} \][/tex]
So, the exact, fully simplified answer is:
[tex]\[ s = \frac{2\pi}{3} \text{ yards} \][/tex]
