To determine the arc length of a circle given a radius and a central angle, we can use the formula for arc length:
[tex]\[ \text{Arc Length} = \text{radius} \times \text{angle in radians} \][/tex]
In this case, the radius of the circle [tex]\( r \)[/tex] is 26.9 meters and the central angle [tex]\( \theta \)[/tex] is [tex]\( \frac{9 \pi}{5} \)[/tex] radians.
Step-by-Step Solution:
1. Convert the central angle to a numerical value using [tex]\(\pi \)[/tex]:
- We use [tex]\( \pi \approx 3.14 \)[/tex].
- The central angle in radians is given by [tex]\(\theta = \frac{9 \pi}{5}\)[/tex].
- Substitute [tex]\(\pi\)[/tex] with 3.14:
2. Calculate the central angle in radians:
[tex]\[ \theta = \frac{9 \times 3.14}{5} \][/tex]
[tex]\[ \theta = \frac{28.26}{5} \][/tex]
[tex]\[ \theta = 5.652 \, \text{radians} \][/tex]
3. Use the arc length formula:
[tex]\[ \text{Arc Length} = r \times \theta \][/tex]
[tex]\[ \text{Arc Length} = 26.9 \, \text{m} \times 5.652 \][/tex]
4. Calculate the arc length:
[tex]\[ \text{Arc Length} = 152.0388 \, \text{m} \][/tex]
5. Round the answer to the nearest hundredth:
[tex]\[ \text{Arc Length} \approx 152.04 \, \text{m} \][/tex]
Therefore, the arc length is approximately [tex]\(\boxed{152.04}\)[/tex] meters.
