To find the area of a sector with a given radius [tex]\( r \)[/tex] and central angle [tex]\( \theta \)[/tex] in radians, we can use the formula for the area of a sector:
[tex]\[ \text{Area} = \frac{1}{2} r^2 \theta \][/tex]
where [tex]\( r \)[/tex] is the radius and [tex]\( \theta \)[/tex] is the central angle in radians.
Given:
- Radius, [tex]\( r = 8 \)[/tex]
- Central angle, [tex]\( \theta = \frac{5\pi}{3} \)[/tex] radians
Let's follow these steps to find the area of the sector:
1. Substitute the values into the formula:
[tex]\[
\text{Area} = \frac{1}{2} \cdot 8^2 \cdot \frac{5\pi}{3}
\][/tex]
2. Calculate the values in the formula:
- First, calculate [tex]\( 8^2 \)[/tex]:
[tex]\[
8^2 = 64
\][/tex]
- Next, substitute this value back into the formula:
[tex]\[
\text{Area} = \frac{1}{2} \cdot 64 \cdot \frac{5\pi}{3}
\][/tex]
3. Simplify the multiplication:
- Multiply [tex]\( \frac{1}{2} \)[/tex] and [tex]\( 64 \)[/tex]:
[tex]\[
\frac{1}{2} \cdot 64 = 32
\][/tex]
- Now the formula becomes:
[tex]\[
\text{Area} = 32 \cdot \frac{5\pi}{3}
\][/tex]
4. Complete the calculation:
- Multiply [tex]\( 32 \)[/tex] and [tex]\( \frac{5\pi}{3} \)[/tex]:
[tex]\[
32 \cdot \frac{5\pi}{3} = \frac{160\pi}{3}
\][/tex]
Therefore, the area of the sector is [tex]\(\frac{160\pi}{3}\)[/tex] square units.
Based on the given options, the correct answer is:
D. [tex]\(\frac{160 \pi}{3}\)[/tex] units[tex]\(^2\)[/tex]
