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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{1}{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. 15.7 square units

To find the area of sector [tex]\(AOB\)[/tex], let's go step-by-step:

1. Identify the radius and arc length ratio:
- The radius [tex]\(OA\)[/tex] of the circle is given as 5 units.
- The ratio of the length of arc [tex]\(\hat{AB}\)[/tex] to the circumference is [tex]\(\frac{1}{4}\)[/tex].

2. Calculate the circumference of the circle:
- The formula for the circumference [tex]\(C\)[/tex] of a circle is [tex]\(C = 2\pi r\)[/tex].
- By substituting the given radius, [tex]\(C = 2 \times 3.14 \times 5 = 31.4\)[/tex] units.

3. Determine the length of arc [tex]\(\hat{AB}\)[/tex]:
- Since arc [tex]\(\hat{AB}\)[/tex] is [tex]\(\frac{1}{4}\)[/tex] of the circumference, calculate the length of arc [tex]\(\hat{AB}\)[/tex] as follows:
- Length of arc [tex]\(\hat{AB}\)[/tex] = [tex]\(\frac{1}{4} \times 31.4 = 7.85\)[/tex] units.

4. Calculate the area of sector [tex]\(AOB\)[/tex]:
- The formula for the area of a sector is [tex]\(\text{Area of sector} = \frac{\text{arc length}}{\text{circumference}} \times \pi r^2\)[/tex].
- Plugging in the values, we get:
- [tex]\(\text{Area of sector} = \frac{7.85}{31.4} \times 3.14 \times 5^2 = 19.6\)[/tex] square units.

Based on the calculations, the area of sector [tex]\(AOB\)[/tex] is closest to option A. 19.6 square units.