To solve this problem, we need to determine the area of sector [tex]\(AOB\)[/tex] in a circle with a given radius and fractional arc length. Here is the detailed step-by-step solution:
1. Determine the circumference of the circle:
The circumference [tex]\(C\)[/tex] of a circle is given by the formula:
[tex]\[
C = 2 \pi r
\][/tex]
Given the radius [tex]\(r = 5\)[/tex] units and [tex]\(\pi = 3.14\)[/tex]:
[tex]\[
C = 2 \cdot 3.14 \cdot 5 = 31.4 \text{ units}
\][/tex]
2. Find the length of arc [tex]\(AB\)[/tex]:
The problem states that the length of arc [tex]\(AB\)[/tex] is [tex]\(\frac{1}{4}\)[/tex] of the circumference. Thus:
[tex]\[
\text{Length of arc } AB = \frac{1}{4} \cdot 31.4 = 7.85 \text{ units}
\][/tex]
3. Calculate the area of sector [tex]\(AOB\)[/tex]:
The area of a sector of a circle is given by the formula:
[tex]\[
\text{Area of sector } = \frac{1}{2} r s
\][/tex]
where [tex]\(r\)[/tex] is the radius and [tex]\(s\)[/tex] is the arc length.
Given [tex]\(r = 5\)[/tex] units and [tex]\(s = 7.85\)[/tex] units:
[tex]\[
\text{Area of sector } AOB = \frac{1}{2} \cdot 5 \cdot 7.85 = 19.625 \text{ square units}
\][/tex]
4. Choose the closest answer:
From the given options:
[tex]\[
\text{A. } 19.6 \text{ square units}
\][/tex]
The closest answer to 19.625 square units is indeed:
[tex]\[
\boxed{19.6} \text{ square units}
\][/tex]
Therefore, the correct answer is:
[tex]\[
\boxed{19.6}
\][/tex]
