To find the length of an arc intercepted by a central angle in a circle, you can use the formula:
[tex]\[ \text{Arc Length} = r \theta \][/tex]
where [tex]\( r \)[/tex] is the radius of the circle and [tex]\( \theta \)[/tex] is the central angle measured in radians.
Given:
- Radius, [tex]\( r = 26.9 \)[/tex] meters
- Central angle, [tex]\( \theta = \frac{9\pi}{5} \)[/tex] radians
Now, let's break down the steps:
1. Identify the Radius and Central Angle:
- Radius [tex]\( r = 26.9 \)[/tex] meters.
- Central angle [tex]\( \theta = \frac{9\pi}{5} \)[/tex] radians.
2. Substitute the Known Values into the Formula:
[tex]\[
\text{Arc Length} = 26.9 \times \left(\frac{9\pi}{5}\right)
\][/tex]
3. Perform the Multiplication:
- First, calculate the fraction of the angle:
[tex]\[
\frac{9\pi}{5} \approx 5.654866776
\][/tex]
- Then, multiply the radius by this value:
[tex]\[
26.9 \times 5.654866776 \approx 152.11591628681776
\][/tex]
Thus, the arc length is approximately [tex]\( 152.12 \)[/tex] meters.
So, the arc length intercepted by a central angle of [tex]\( \frac{9\pi}{5} \)[/tex] radians in a circle with a radius of 26.9 meters is about [tex]\( 152.12 \)[/tex] meters.
