To find the area of a sector in a circle, you can use the formula:
[tex]\[ A = \frac{1}{2} r^2 \theta \][/tex]
where:
- [tex]\( r \)[/tex] is the radius of the circle,
- [tex]\( \theta \)[/tex] is the central angle of the sector in radians.
Given:
- The radius [tex]\( r = 9 \)[/tex] meters,
- The central angle [tex]\( \theta = 120^\circ \)[/tex].
First, we need to convert the angle from degrees to radians. The conversion factor between degrees and radians is given by:
[tex]\[ \theta_{\text{radians}} = \theta_{\text{degrees}} \times \frac{\pi}{180} \][/tex]
So for [tex]\( \theta = 120^\circ \)[/tex]:
[tex]\[ \theta_{\text{radians}} = 120^\circ \times \frac{\pi}{180} = \frac{2\pi}{3} \][/tex]
Now, substitute the radius and the angle in radians into the area formula:
[tex]\[ A = \frac{1}{2} \times 9^2 \times \frac{2\pi}{3} \][/tex]
[tex]\[ A = \frac{1}{2} \times 81 \times \frac{2\pi}{3} \][/tex]
[tex]\[ A = \frac{1}{2} \times 81 \times \frac{2\cdot\pi}{3} \][/tex]
[tex]\[ A = \frac{81}{1} \times \frac{\pi}{3} \][/tex]
[tex]\[ A = 81 \times \frac{\pi}{3} \][/tex]
[tex]\[ A = \frac{81 \pi}{3} \][/tex]
[tex]\[ A = 27 \pi \text{ square meters} \][/tex]
Therefore, the area of each sector is:
[tex]\[ 27 \pi \text{ square meters} \][/tex]
So, the correct answer is:
C. [tex]\( A = 27 \pi \text{ m}^2 \)[/tex]
