Answered

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

B. [tex]50 \pi[/tex] units[tex]\(^2\)[/tex]

C. [tex]\frac{140 \pi}{3}[/tex] units[tex]\(^2\)[/tex]

D. [tex]\frac{320 \pi}{3}[/tex] units[tex]\(^2\)[/tex]



Answer :

GinnyAnswer
To find the area of a sector of a circle, we can 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, and
- [tex]\( \theta \)[/tex] is the central angle in radians.

Given:
- The radius [tex]\( r \)[/tex] is 8 units.
- The central angle [tex]\( \theta \)[/tex] is [tex]\(\frac{5 \pi}{3}\)[/tex] radians.

Substitute these values into the formula:

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

First, calculate [tex]\( 8^2 \)[/tex]:
[tex]\[ 8^2 = 64 \][/tex]

Now, substitute [tex]\( 64 \)[/tex] back into the formula:
[tex]\[ \text{Area of the sector} = \frac{1}{2} \times 64 \times \frac{5 \pi}{3} \][/tex]

Next, simplify the calculation step-by-step:

1. Multiply [tex]\( \frac{1}{2} \)[/tex] by [tex]\( 64 \)[/tex]:
[tex]\[ \frac{1}{2} \times 64 = 32 \][/tex]

2. Now multiply [tex]\( 32 \)[/tex] by [tex]\( \frac{5 \pi}{3} \)[/tex]:
[tex]\[ 32 \times \frac{5 \pi}{3} = \frac{32 \times 5 \pi}{3} = \frac{160 \pi}{3} \][/tex]

Thus, the area of the sector is:
[tex]\[ \frac{160 \pi}{3} \text{ units}^2 \][/tex]

Therefore, the correct answer is:
[tex]\[ \boxed{\frac{160 \pi}{3} \text{ units}^2} \][/tex]

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