To solve the problem of finding the area of sector [tex]\( AOB \)[/tex], we need to follow these steps:
1. Calculate the radius of the circle:
Given [tex]\( OA = 5 \)[/tex], the circle's radius [tex]\( r \)[/tex] is 5 units.
2. Determine the value of [tex]\(\pi\)[/tex]:
For this problem, [tex]\(\pi = 3.14\)[/tex].
3. Calculate the circumference of the circle:
The formula for the circumference [tex]\( C \)[/tex] of a circle is:
[tex]\[
C = 2 \pi r
\][/tex]
Plugging in the values:
[tex]\[
C = 2 \times 3.14 \times 5 = 31.4 \text{ units}
\][/tex]
4. Find the length of the arc [tex]\( \hat{AB} \)[/tex]:
According to the problem, the length of [tex]\( \hat{AB} \)[/tex] as a fraction of the circumference is [tex]\(\frac{1}{4}\)[/tex]. Therefore:
[tex]\[
\text{Length of } \hat{AB} = \frac{1}{4} \times 31.4 = 7.85 \text{ units}
\][/tex]
5. Calculate the area of the circle:
The formula for the area [tex]\( A \)[/tex] of a circle is:
[tex]\[
A = \pi r^2
\][/tex]
Substituting the known values:
[tex]\[
A = 3.14 \times (5^2) = 3.14 \times 25 = 78.5 \text{ square units}
\][/tex]
6. Determine the area of sector [tex]\( AOB \)[/tex]:
The area of sector [tex]\( AOB \)[/tex] is proportional to the arc length, which in this case is [tex]\(\frac{1}{4}\)[/tex] of the circle's area. Therefore:
[tex]\[
\text{Area of sector } AOB = \frac{\text{Length of } \hat{AB}}{\text{Circumference}} \times A = \frac{7.85}{31.4} \times 78.5 = \frac{1}{4} \times 78.5 = 19.625 \text{ square units}
\][/tex]
The final area of sector [tex]\( AOB \)[/tex] is approximately 19.625 square units. Therefore, the answer closest to this value is:
A. 19.6 square units
