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Arc BC on circle A has a length of [tex]\frac{115}{6} \pi[/tex] inches. What is the radius of the circle?

The radius of the circle is [tex]\square[/tex] inches.


Sagot :

To find the radius of the circle, we start with the formula for the length of an arc, which is given by:

[tex]\[ \text{Arc Length} = r \cdot \theta \][/tex]

where [tex]\( r \)[/tex] is the radius of the circle, and [tex]\( \theta \)[/tex] is the central angle in radians.

For a full circle, the central angle [tex]\( \theta \)[/tex] is [tex]\( 2\pi \)[/tex] radians. Therefore, the formula becomes:

[tex]\[ \text{Arc Length} = r \cdot 2\pi \][/tex]

We're given that the arc length [tex]\( \frac{115}{6}\pi \)[/tex] inches, so we set the arc length equal to [tex]\( r \cdot 2\pi \)[/tex] and solve for [tex]\( r \)[/tex]:

[tex]\[ \frac{115}{6}\pi = r \cdot 2\pi \][/tex]

Divide both sides of the equation by [tex]\( 2\pi \)[/tex]:

[tex]\[ r = \frac{\frac{115}{6}\pi}{2\pi} \][/tex]

Simplify by canceling out [tex]\( \pi \)[/tex]:

[tex]\[ r = \frac{115}{6} \cdot \frac{1}{2} = \frac{115}{12} \][/tex]

When you calculate [tex]\( \frac{115}{12} \)[/tex], you get approximately:

[tex]\[ r \approx 9.583333333333334 \][/tex]

Thus, the radius of the circle is approximately [tex]\( 9.583333333333334 \)[/tex] inches.

So, the radius of the circle is [tex]\(\boxed{9.583333333333334}\)[/tex] inches.