To determine the area of a circular sector, we can use the formula:
[tex]\[ A = \frac{1}{2} r^2 \theta \][/tex]
where:
- [tex]\( A \)[/tex] is the area of the sector
- [tex]\( r \)[/tex] is the radius of the circle (in meters)
- [tex]\( \theta \)[/tex] is the central angle of the sector in radians
Given the values:
- Radius [tex]\( r = 34 \)[/tex] meters
- Central angle [tex]\( \theta = \frac{7 \pi}{8} \)[/tex] radians
Following the steps, let's substitute the given values into the formula:
1. Calculate [tex]\( r^2 \)[/tex]:
[tex]\[ r^2 = 34^2 = 1156 \][/tex]
2. Substitute [tex]\( r^2 \)[/tex] and [tex]\( \theta \)[/tex] into the formula:
[tex]\[ A = \frac{1}{2} \times 1156 \times \frac{7 \pi}{8} \][/tex]
3. Multiply the constants:
[tex]\[ A = \frac{1}{2} \times 1156 \times \frac{7}{8} \times \pi \][/tex]
First, compute the product:
[tex]\[ \frac{1}{2} \times 1156 = 578 \][/tex]
Next:
[tex]\[ 578 \times \frac{7}{8} = 506.5 \][/tex]
4. Incorporate [tex]\(\pi\)[/tex] in the multiplication:
[tex]\[ A = 506.5 \times \pi \][/tex]
Using the approximate value [tex]\(\pi \approx 3.141592653589793\)[/tex],
5. Finally, compute:
[tex]\[ A \approx 506.5 \times 3.141592653589793 \][/tex]
[tex]\[ A \approx 1588.8604845530378 \][/tex]
Rounding the result to three significant digits:
[tex]\[ A \approx 1588.86 \, \text{m}^2 \][/tex]
So, the area of the circular sector is approximately [tex]\( 1588.86 \)[/tex] square meters when rounded to three significant digits.
