To determine the area of sector [tex]\(AOB\)[/tex], we need to follow these steps:
1. Calculate the circumference of the circle:
Given that the radius [tex]\(OA = 5\)[/tex] and [tex]\(\pi = 3.14\)[/tex], we use the formula for the circumference of a circle:
[tex]\[
\text{Circumference} = 2 \pi r
\][/tex]
Substituting the given values:
[tex]\[
\text{Circumference} = 2 \times 3.14 \times 5 = 31.4 \text{ units}
\][/tex]
2. Calculate the ratio for the given arc:
The problem states that the ratio of the length of arc [tex]\(\widehat{AB}\)[/tex] to the circumference is [tex]\(\frac{1}{4}\)[/tex].
3. Determine the area of the sector:
The area [tex]\(A\)[/tex] of the sector [tex]\(AOB\)[/tex] can be determined using the proportion of the arc length to the entire circumference. Because the ratio is [tex]\(\frac{1}{4}\)[/tex], the area of the sector will be [tex]\(\frac{1}{4}\)[/tex] of the circle's total area.
The total area of the circle is calculated using:
[tex]\[
\text{Area of the circle} = \pi r^2
\][/tex]
Substituting the given values:
[tex]\[
\text{Area of the circle} = 3.14 \times 5^2 = 3.14 \times 25 = 78.5 \text{ square units}
\][/tex]
Therefore, the area of sector [tex]\(AOB\)[/tex] is:
[tex]\[
\text{Area of sector } AOB = \frac{1}{4} \times \text{Area of the circle} = \frac{1}{4} \times 78.5 = 19.625 \text{ square units}
\][/tex]
Among the given options, the closest answer to 19.625 square units is:
A. 19.6 square units
Therefore, the correct answer is:
A. 19.6 square units
