To find the measure of the angle of a sector given its area and the radius of the circle, we use the formula for the area of a sector:
[tex]\[ \text{Area of sector} = \frac{\theta}{360} \times \pi r^2 \][/tex]
Where:
- [tex]\(\theta\)[/tex] is the angle in degrees.
- [tex]\(r\)[/tex] is the radius of the circle.
We are given:
- The area of the sector, [tex]\( \text{Area of sector} = 18.55 \)[/tex].
- The radius, [tex]\( r = 5 \)[/tex].
1. First, substitute the known values into the formula:
[tex]\[ 18.55 = \frac{\theta}{360} \times \pi \times 5^2 \][/tex]
2. Simplify the equation by calculating [tex]\( 5^2 \)[/tex]:
[tex]\[ 18.55 = \frac{\theta}{360} \times \pi \times 25 \][/tex]
3. Multiply [tex]\( \pi \)[/tex] and 25:
[tex]\[ 18.55 = \frac{\theta}{360} \times 25\pi \][/tex]
4. Isolate [tex]\( \theta \)[/tex] by multiplying both sides of the equation by 360:
[tex]\[ 18.55 \times 360 = \theta \times 25\pi \][/tex]
5. Calculate [tex]\( 18.55 \times 360 \)[/tex]:
[tex]\[ 6687 = \theta \times 25\pi \][/tex]
6. Divide both sides of the equation by [tex]\( 25\pi \)[/tex]:
[tex]\[ \theta = \frac{6687}{25\pi} \][/tex]
7. Calculate the value of [tex]\( \frac{6687}{25\pi} \)[/tex]. Using an approximate value for [tex]\( \pi \)[/tex] (3.14159):
[tex]\[ \theta = \frac{6687}{25 \times 3.14159} \][/tex]
[tex]\[ \theta \approx \frac{6687}{78.53975} \][/tex]
[tex]\[ \theta \approx 85.13^\circ \][/tex]
So, the measure of the angle of the sector is approximately [tex]\( 85^\circ \)[/tex].
The correct answer is:
A) 85°
