To find the area of sector [tex]\(AOB\)[/tex] in a circle with center [tex]\(O\)[/tex], where the radius [tex]\(OA = 5\)[/tex] and the ratio of the length of arc [tex]\(\widehat{AB}\)[/tex] to the circumference of the circle is [tex]\(\frac{1}{4}\)[/tex], we can follow these steps:
1. Identify the radius of the circle:
Since [tex]\(OA\)[/tex] is the radius and given that [tex]\(OA = 5\)[/tex], the radius [tex]\(r = 5\)[/tex] units.
2. Understand the given ratio:
The length of arc [tex]\(\widehat{AB}\)[/tex] is given as [tex]\(\frac{1}{4}\)[/tex] of the total circumference of the circle.
3. Find the total circumference of the circle:
The circumference [tex]\(C\)[/tex] of a circle is given by the formula:
[tex]\[
C = 2 \pi r
\][/tex]
Plugging in the given values:
[tex]\[
C = 2 \times 3.14 \times 5 = 31.4 \text{ units}
\][/tex]
4. Calculate the area of the complete circle:
The area [tex]\(A\)[/tex] of a circle is given by:
[tex]\[
A = \pi r^2
\][/tex]
Plugging in the given values:
[tex]\[
A = 3.14 \times (5)^2 = 78.5 \text{ square units}
\][/tex]
5. Determine the area of sector [tex]\(AOB\)[/tex]:
Since the arc [tex]\(\widehat{AB}\)[/tex] spans [tex]\(\frac{1}{4}\)[/tex] of the circle's circumference, the sector also encompasses [tex]\(\frac{1}{4}\)[/tex] of the circle's area. Therefore, the area of sector [tex]\(AOB\)[/tex] is:
[tex]\[
\text{Area of sector } AOB = \frac{1}{4} \times \text{Total Area} = \frac{1}{4} \times 78.5 = 19.625 \text{ square units}
\][/tex]
The closest answer to 19.625 square units, based on the options provided, is [tex]\( 19.6 \)[/tex] square units.
Thus, the correct answer is:
A. [tex]\( \text{19.6 square units} \)[/tex]
