To solve this problem, let's break it down step by step.
1. Identify the given information:
- The radius of the circle \( OA = 5 \) units.
- The fraction of the circumference represented by the arc length \( \frac{\text{length of } \widehat{AB}}{\text{circumference}} = \frac{1}{4} \).
2. Understand what is asked:
- We need to find the area of the sector \( AOB \).
3. Calculate the area of the whole circle:
- The formula for the area of a circle is \( \pi r^2 \).
- Given \( \pi = 3.14 \) and \( r = 5 \),
[tex]\[
\text{Area of the circle} = \pi \times r^2 = 3.14 \times 5^2 = 3.14 \times 25 = 78.5 \text{ square units}
\][/tex]
4. Determine the fraction of the circle represented by the sector:
- The fraction given is \( \frac{1}{4} \).
- Therefore, the area of the sector \( AOB \) will be \( \frac{1}{4} \) of the area of the whole circle.
5. Calculate the area of the sector:
- Multiply the area of the circle by the fraction:
[tex]\[
\text{Area of the sector} = \frac{1}{4} \times 78.5 = 19.625 \text{ square units}
\][/tex]
6. Choose the closest answer:
- Looking at the provided options, the closest one to 19.625 square units is:
[tex]\[
\boxed{19.6 \text{ square units}}
\][/tex]
Therefore, the correct answer is [tex]\( \boxed{19.6 \text{ square units}} \)[/tex].
