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Find the exact length of the arc made by the central angle [tex]\(\theta=\frac{\pi}{12}\)[/tex] in a circle of radius [tex]\(r=8 \, \text{yd}\)[/tex].

Note: Enter the exact, fully simplified answer.

[tex]\[ s = \square \, \text{yd} \][/tex]

Sagot :

GinnyAnswer
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]
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