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Answer :
To solve this problem, we're finding the area of sector [tex]\(AOB\)[/tex] of a circle with center [tex]\(O\)[/tex].
1. Understand the given information:
- Radius of the circle ([tex]\(OA\)[/tex]) is 5 units.
- The fraction of the arc length of [tex]\(\hat{AB}\)[/tex] to the entire circumference is [tex]\(\frac{2}{4}\)[/tex] or [tex]\(\frac{1}{2}\)[/tex].
2. Calculate the circumference of the circle:
- The formula for the circumference ([tex]\(C\)[/tex]) of a circle is [tex]\(2 \pi \times \text{radius}\)[/tex].
- With [tex]\(\pi = 3.14\)[/tex] and radius = 5, we get:
[tex]\[
C = 2 \times 3.14 \times 5 = 31.4
\][/tex]
3. Find the arc length of [tex]\(\hat{AB}\)[/tex]:
- The arc length [tex]\(\hat{AB}\)[/tex] is [tex]\(\frac{1}{2}\)[/tex] of the circumference.
- Therefore, the arc length is:
[tex]\[
\frac{1}{2} \times 31.4 = 15.7
\][/tex]
4. Calculate the area of sector [tex]\(AOB\)[/tex]:
- The area of the sector can be found using the fraction of the circumference the arc represents, multiplied by the total area of the circle.
- First, compute the total area of the circle: [tex]\(\pi \times \text{radius}^2 = 3.14 \times 5^2 = 78.5\)[/tex].
- Since the arc length is [tex]\(\frac{1}{2}\)[/tex] of the circumference, the area of sector [tex]\(AOB\)[/tex] is:
[tex]\[
\frac{1}{2} \times 78.5 = 39.25
\][/tex]
The answer closest to 39.25 from the provided options is:
B. 39.3 square units.
1. Understand the given information:
- Radius of the circle ([tex]\(OA\)[/tex]) is 5 units.
- The fraction of the arc length of [tex]\(\hat{AB}\)[/tex] to the entire circumference is [tex]\(\frac{2}{4}\)[/tex] or [tex]\(\frac{1}{2}\)[/tex].
2. Calculate the circumference of the circle:
- The formula for the circumference ([tex]\(C\)[/tex]) of a circle is [tex]\(2 \pi \times \text{radius}\)[/tex].
- With [tex]\(\pi = 3.14\)[/tex] and radius = 5, we get:
[tex]\[
C = 2 \times 3.14 \times 5 = 31.4
\][/tex]
3. Find the arc length of [tex]\(\hat{AB}\)[/tex]:
- The arc length [tex]\(\hat{AB}\)[/tex] is [tex]\(\frac{1}{2}\)[/tex] of the circumference.
- Therefore, the arc length is:
[tex]\[
\frac{1}{2} \times 31.4 = 15.7
\][/tex]
4. Calculate the area of sector [tex]\(AOB\)[/tex]:
- The area of the sector can be found using the fraction of the circumference the arc represents, multiplied by the total area of the circle.
- First, compute the total area of the circle: [tex]\(\pi \times \text{radius}^2 = 3.14 \times 5^2 = 78.5\)[/tex].
- Since the arc length is [tex]\(\frac{1}{2}\)[/tex] of the circumference, the area of sector [tex]\(AOB\)[/tex] is:
[tex]\[
\frac{1}{2} \times 78.5 = 39.25
\][/tex]
The answer closest to 39.25 from the provided options is:
B. 39.3 square units.
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