To find the area of the sector of a circle with a given radius and central angle, we can follow these steps:
1. Convert the central angle from degrees to radians:
- The formula to convert degrees to radians is [tex]\(\theta_{\text{radians}} = \frac{\theta_{\text{degrees}}}{360} \times 2\pi\)[/tex].
- Given [tex]\(\theta = 315^\circ\)[/tex], we plug this value into the formula:
[tex]\[
\theta_{\text{radians}} = \frac{315}{360} \times 2\pi
\][/tex]
- The result of this conversion is approximately:
[tex]\[
\theta_{\text{radians}} \approx 5.497787143782138
\][/tex]
2. Use the formula for the area of a sector:
- The area of a sector (A) is given by the formula [tex]\(A = \frac{1}{2} r^2 \theta_{\text{radians}}\)[/tex].
- With [tex]\(r = 3.5 \)[/tex] kilometers, and [tex]\(\theta_{\text{radians}} \approx 5.497787143782138\)[/tex], we substitute these values into the formula:
[tex]\[
A = \frac{1}{2} \times (3.5)^2 \times 5.497787143782138
\][/tex]
- Calculating this, we get:
[tex]\[
A \approx 33.6739462556656 \text{ square kilometers}
\][/tex]
3. Round the result to two decimal places:
- The final area of the sector, rounded to two decimal places, is:
[tex]\[
A \approx 33.67 \text{ square kilometers}
\][/tex]
Hence, the area of the sector is:
[tex]\[
33.67 \text{ square kilometers}
\][/tex]
