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].
