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What is the area of the sector having a radius of 8 and a central angle of [tex]\frac{5 \pi}{3}[/tex] radians?

A. [tex]50 \pi[/tex] units [tex]^2[/tex]
B. [tex]\frac{320 \pi}{3}[/tex] units [tex]^2[/tex]
C. [tex]\frac{140 \pi}{3}[/tex] units [tex]^2[/tex]
D. [tex]\frac{160 \pi}{3}[/tex] units [tex]^2[/tex]


Sagot :

To find the area of a sector in a circle, you can use the formula:

[tex]\[ \text{Area of sector} = \frac{1}{2} \cdot r^2 \cdot \theta \][/tex]

where:
- [tex]\( r \)[/tex] is the radius of the circle
- [tex]\( \theta \)[/tex] is the central angle in radians

In this problem, we are given:
- Radius [tex]\( r = 8 \)[/tex]
- Central angle [tex]\( \theta = \frac{5 \pi}{3} \)[/tex]

Let's plug these values into the formula:

[tex]\[ \text{Area of sector} = \frac{1}{2} \cdot 8^2 \cdot \frac{5 \pi}{3} \][/tex]

First, calculate [tex]\( 8^2 \)[/tex]:

[tex]\[ 8^2 = 64 \][/tex]

Now, substitute this back into the formula:

[tex]\[ \text{Area of sector} = \frac{1}{2} \cdot 64 \cdot \frac{5 \pi}{3} \][/tex]

Next, simplify the term [tex]\( \frac{1}{2} \cdot 64 \)[/tex]:

[tex]\[ \frac{1}{2} \cdot 64 = 32 \][/tex]

Now, multiply this result by [tex]\( \frac{5 \pi}{3} \)[/tex]:

[tex]\[ 32 \cdot \frac{5 \pi}{3} = \frac{32 \cdot 5 \pi}{3} = \frac{160 \pi}{3} \][/tex]

Therefore, the area of the sector is:

[tex]\[ \frac{160 \pi}{3} \, \text{units}^2 \][/tex]

The correct answer is:

D. [tex]\(\frac{160 \pi}{3}\)[/tex] units[tex]\(^2\)[/tex]