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Find the length of an arc of a sector if you are given a central angle.

If the central angle is [tex]67^\circ[/tex] and the radius is 11 inches, what is the length of the arc? Round the answer to the nearest tenth.

Answer:


Sagot :

To find the length of an arc of a sector when given the central angle and the radius, we can follow these steps:

1. Understand the given data:
- Central angle in degrees: 67°
- Radius of the circle: 11 inches

2. Convert the central angle from degrees to radians:
The formula for conversion from degrees to radians is:
[tex]\[ \text{radians} = \text{degrees} \times \left(\frac{\pi}{180}\right) \][/tex]
So, the central angle in radians is:
[tex]\[ 67^\circ \times \left(\frac{\pi}{180}\right) \approx 1.1694 \text{ radians} \][/tex]

3. Calculate the arc length:
The formula for the arc length (s) is given by:
[tex]\[ s = r \theta \][/tex]
where [tex]\( r \)[/tex] is the radius and [tex]\( \theta \)[/tex] is the central angle in radians. Substituting the known values:
[tex]\[ s = 11 \times 1.1694 \approx 12.8631 \text{ inches} \][/tex]

4. Round the arc length to the nearest tenth:
Therefore, 12.8631 rounded to the nearest tenth is:
[tex]\[ 12.9 \text{ inches} \][/tex]

So, the length of the arc is approximately 12.9 inches when rounded to the nearest tenth.