What is the area of a sector with a central angle of [tex]\frac{4 \pi}{5}[/tex] radians and a radius of [tex]11 \, \text{cm}[/tex]?

Use [tex]3.14[/tex] for [tex]\pi[/tex] and round your final answer to the nearest hundredth.

Enter your answer as a decimal in the box.

[tex]\square \, \text{cm}^2[/tex]



Answer :

To determine the area of a sector with a central angle of [tex]\(\frac{4\pi}{5}\)[/tex] radians and a radius of 11 cm, follow these steps:

1. Understand the Formula:
The area [tex]\(A\)[/tex] of a sector of a circle is given by the formula:
[tex]\[ A = \frac{1}{2} r^2 \theta \][/tex]
where [tex]\(r\)[/tex] is the radius and [tex]\(\theta\)[/tex] is the central angle in radians.

2. Substitute the Given Values:
Here, the central angle [tex]\(\theta = \frac{4\pi}{5}\)[/tex].
Given [tex]\(\pi \approx 3.14\)[/tex], we can calculate:
[tex]\[ \theta = \frac{4 \times 3.14}{5} = \frac{12.56}{5} = 2.512 \text{ radians} \][/tex]

The radius [tex]\(r = 11 \text{ cm}\)[/tex].

3. Apply the Formula:
Now, we substitute [tex]\(r\)[/tex] and [tex]\(\theta\)[/tex] into the area formula:
[tex]\[ A = \frac{1}{2} \times 11^2 \times 2.512 \][/tex]

First, calculate [tex]\(11^2\)[/tex]:
[tex]\[ 11^2 = 121 \][/tex]

Then calculate the product:
[tex]\[ 121 \times 2.512 = 304.952 \][/tex]

Now, multiply by [tex]\(\frac{1}{2}\)[/tex]:
[tex]\[ \frac{1}{2} \times 304.952 = 152.476 \][/tex]

4. Round the Result to the Nearest Hundredth:
The area of the sector rounded to the nearest hundredth is:
[tex]\[ 152.48 \text{ cm}^2 \][/tex]

Thus, the area of the sector is [tex]\(\boxed{152.48} \text{ cm}^2\)[/tex].