To find the area of a sector given the central angle in radians and the radius, we use the formula for the area of a sector:
[tex]\[ \text{Area} = \frac{1}{2} \times r^2 \times \theta \][/tex]
where:
- [tex]\( r \)[/tex] is the radius of the circle,
- [tex]\( \theta \)[/tex] is the central angle in radians.
Given the following values:
- Central angle [tex]\( \theta = \frac{10 \pi}{7} \)[/tex] radians
- Radius [tex]\( r = 18.4 \)[/tex] meters
Since the central angle is given in terms of [tex]\(\pi\)[/tex] and the question suggests using [tex]\( \pi \approx 3.14 \)[/tex], we substitute [tex]\(\pi\)[/tex] with 3.14.
First, we need to convert the central angle [tex]\( \frac{10 \pi}{7} \)[/tex] to a numerical value:
[tex]\[ \theta = \frac{10 \times 3.14}{7} \][/tex]
Perform the multiplication and division:
[tex]\[ \theta = \frac{31.4}{7} \][/tex]
[tex]\[ \theta \approx 4.485714 \,\text{radians} \][/tex]
Now we substitute the radius [tex]\( r = 18.4 \)[/tex] meters and the obtained central angle [tex]\( \theta \approx 4.485714 \)[/tex] radians into the area formula:
[tex]\[ \text{Area} = \frac{1}{2} \times (18.4)^2 \times 4.485714 \][/tex]
First, calculate [tex]\( (18.4)^2 \)[/tex]:
[tex]\[ (18.4)^2 = 338.56 \][/tex]
Then multiply by [tex]\( 4.485714 \)[/tex]:
[tex]\[ 338.56 \times 4.485714 \approx 1518.68 \][/tex]
Finally, divide by 2:
[tex]\[ \text{Area} = \frac{1518.68}{2} = 759.34 \][/tex]
So, the area of the sector is approximately [tex]\( 759.34 \)[/tex] square meters when rounded to the nearest hundredth.
Therefore, the area of the sector is:
[tex]\[ \boxed{759.34} \, m^2 \][/tex]
