High School

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------------------------------------------------ Select the correct answer.

Points [tex] A [/tex] and [tex] B [/tex] lie on a circle centered at point [tex] O [/tex]. If [tex] OA = 5 [/tex] and [tex]\frac{\text{length of } \hat{AB}}{\text{circumference}} = \frac{2}{4}[/tex], what is the area of sector [tex] AOB [/tex]? Use the value [tex]\pi = 3.14[/tex], and choose the closest answer.

A. 19.6 square units
B. 39.3 square units
C. 7.85 square units
D. 15.77 square units

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.