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], follow these steps:

1. Calculate the circumference of the circle:

The formula for the circumference of a circle is:
[tex]\[
\text{Circumference} = 2 \pi r
\][/tex]
Given [tex]\( r = OA = 5 \)[/tex] and [tex]\(\pi = 3.14\)[/tex], we have:
[tex]\[
\text{Circumference} = 2 \times 3.14 \times 5 = 31.4
\][/tex]

2. Find the length of arc [tex]\( \hat{AB} \)[/tex]:

We know that the length of arc [tex]\( \hat{AB} \)[/tex] is [tex]\(\frac{1}{4}\)[/tex] of the circumference of the circle. Thus:
[tex]\[
\text{Length of arc } \hat{AB} = \frac{1}{4} \times 31.4 = 7.85
\][/tex]

3. Calculate the angle [tex]\( \angle AOB \)[/tex]:

The formula to calculate the angle in degrees for a sector is:
[tex]\[
\text{Angle } \angle AOB = \left(\frac{\text{Length of arc } \hat{AB}}{\text{Circumference}}\right) \times 360
\][/tex]
Plugging in the known values:
[tex]\[
\angle AOB = \left(\frac{7.85}{31.4}\right) \times 360 = 90.0 \text{ degrees}
\][/tex]

4. Calculate the area of sector [tex]\( AOB \)[/tex]:

The formula for the area of a sector is:
[tex]\[
\text{Area of sector} = \left(\frac{\text{Angle}}{360}\right) \times \pi \times r^2
\][/tex]
Substituting the known values:
[tex]\[
\text{Area of sector} = \left(\frac{90}{360}\right) \times 3.14 \times 5^2
\][/tex]
[tex]\[
= \frac{1}{4} \times 3.14 \times 25
\][/tex]
[tex]\[
= 19.625
\][/tex]

The area of sector [tex]\( AOB \)[/tex] is approximately 19.6 square units, which matches closest with option A.