To find the area of the sector [tex]\(AOB\)[/tex] given the values [tex]\(OA = 5\)[/tex] and [tex]\(\frac{\text{length of arc A}}{\text{circumference}} = \frac{1}{4}\)[/tex], we can perform the following steps:
1. Calculate the circumference of the circle:
- The formula for the circumference of a circle is given by [tex]\(C = 2\pi r\)[/tex].
- Here, [tex]\(r = OA = 5\)[/tex], and [tex]\(\pi = 3.14\)[/tex].
- Thus, [tex]\(C = 2 \times 3.14 \times 5 = 31.4\)[/tex].
2. Find the length of the arc [tex]\(A\)[/tex]:
- The arc length is a fraction of the circumference.
- Given the fraction is [tex]\(\frac{1}{4}\)[/tex], the length of arc [tex]\(A\)[/tex] is [tex]\(\frac{1}{4} \times 31.4 = 7.85\)[/tex].
3. Determine the angle of the sector [tex]\(AOB\)[/tex]:
- Because the arc represents [tex]\(\frac{1}{4}\)[/tex] of the circle, the angle of the sector in radians is [tex]\(\frac{1}{4} \times 2\pi\)[/tex].
- [tex]\(2\pi\)[/tex] is approximately [tex]\(2 \times 3.14 = 6.28\)[/tex].
- Thus, the angle of the sector is [tex]\(\frac{1}{4} \times 6.28 = 1.57\)[/tex] radians.
4. Calculate the area of sector [tex]\(AOB\)[/tex]:
- The formula for the area of a sector is [tex]\(\text{Area} = \frac{\theta}{2\pi} \times \pi r^2\)[/tex], where [tex]\(\theta\)[/tex] is the angle in radians.
- Here, [tex]\(\theta = 1.57\)[/tex] and [tex]\(r = 5\)[/tex].
- Substituting the values, we get:
[tex]\[
\text{Area} = \frac{1.57}{2 \times 3.14} \times 3.14 \times (5^2)
\][/tex]
- Simplifying this, we find:
[tex]\[
\text{Area} = \frac{1.57}{6.28} \times 78.5 = 0.25 \times 78.5 = 19.625
\][/tex]
Therefore, the area of sector [tex]\(AOB\)[/tex] is closest to option [tex]\(A\)[/tex], which is 19.6 square units. So, the correct answer is:
A. 19.6 square units
