To find the area of a sector of a circle, we use the formula:
[tex]\[ \text{Area of a sector} = \frac{1}{2} r^2 \theta \][/tex]
where:
- [tex]\( r \)[/tex] is the radius of the circle
- [tex]\( \theta \)[/tex] is the central angle in radians
Given:
- The radius [tex]\( r = 8 \)[/tex]
- The central angle [tex]\( \theta = \frac{5 \pi}{3} \)[/tex]
We can substitute these values into the formula to get:
[tex]\[ \text{Area} = \frac{1}{2} \times 8^2 \times \frac{5 \pi}{3} \][/tex]
Now let's calculate step-by-step:
1. Calculate [tex]\( 8^2 \)[/tex]:
[tex]\[ 8^2 = 64 \][/tex]
2. Multiply by [tex]\(\frac{5 \pi}{3}\)[/tex]:
[tex]\[ 64 \times \frac{5 \pi}{3} = \frac{320 \pi }{3} \][/tex]
3. Finally, multiply by [tex]\(\frac{1}{2}\)[/tex]:
[tex]\[ \text{Area} = \frac{1}{2} \times \frac{320 \pi}{3} = \frac{160 \pi}{3} \][/tex]
So the area of the sector is:
[tex]\[ \text{Area} = \frac{160 \pi}{3} \text{ units}^2 \][/tex]
Thus, the correct answer is:
A. [tex]\(\frac{160 \pi}{3}\)[/tex] units [tex]\( ^2 \)[/tex]
