Let's start by understanding the problem:
1. Circle and Sector: We are given a circle with a radius of 6 cm. A sector of this circle has a given perimeter of [tex]\(2(6 + 5\pi)\)[/tex] cm.
2. Perimeter of the Sector: The perimeter of a sector is the sum of the lengths of two radii and the length of the arc. We can represent this relationship as:
[tex]\[
\text{Perimeter of sector} = 2 \times \text{radius} + \text{arc length}
\][/tex]
3. Given Data: Using the given perimeter, we can set up the equation:
[tex]\[
2 \times 6 + \text{arc length} = 2(6 + 5\pi)
\][/tex]
4. Solving for Arc Length: To find the arc length:
[tex]\[
\text{arc length} = 2(6 + 5\pi) - 2 \times 6
\][/tex]
5. Plugging in the numbers:
[tex]\[
\text{arc length} = 2(6 + 5\pi) - 12 = 12 + 10\pi - 12 = 10\pi
\][/tex]
Therefore, the arc length is approximately 31.41592653589793 cm.
6. Angle of the Sector: The arc length [tex]\( l \)[/tex] can also be described using the angle [tex]\( \theta \)[/tex] in radians with the relationship:
[tex]\[
\text{arc length} = \theta \times \text{radius}
\][/tex]
Solving for [tex]\( \theta\)[/tex]:
[tex]\[
\theta = \frac{\text{arc length}}{\text{radius}}
\][/tex]
Plugging in the values:
[tex]\[
\theta = \frac{31.41592653589793}{6} \approx 5.235987755982989 \text{ radians}
\][/tex]
7. Area of the Sector: The area [tex]\( A \)[/tex] of a sector of a circle can be determined using the formula:
[tex]\[
A = \frac{1}{2} \times \theta \times \text{radius}^2
\][/tex]
Substituting our values in:
[tex]\[
A = \frac{1}{2} \times 5.235987755982989 \times 6^2
\][/tex]
8. Performing the multiplication:
[tex]\[
A = \frac{1}{2} \times 5.235987755982989 \times 36 \approx 94.2477796076938 \text{ square centimeters}
\][/tex]
So, the area of the sector is approximately 94.2477796076938 cm[tex]\(^2\)[/tex].
