To solve the problem, follow these steps:
1. Identify the given values:
- The radius of the circle [tex]\( O A = 5 \)[/tex]
- The value of [tex]\( \pi = 3.14 \)[/tex]
- The fraction of the circumference of the circle that [tex]\( \hat{AB} \)[/tex] represents is [tex]\(\frac{1}{4}\)[/tex].
2. Calculate the circumference of the circle:
[tex]\[
\text{Circumference} = 2 \pi r = 2 \times 3.14 \times 5
\][/tex]
Given the solving approach, we should take the given numerical result:
[tex]\[
\text{Circumference} = 31.4
\][/tex]
3. Calculate the length of arc [tex]\(\hat{AB}\)[/tex]:
[tex]\[
\text{Length of arc } \hat{AB} = \left(\frac{1}{4}\right) \times \text{Circumference} = \left(\frac{1}{4}\right) \times 31.4
\][/tex]
Given the solving approach, we should take the given numerical result:
[tex]\[
\text{Length of arc } \hat{AB} = 7.85
\][/tex]
4. Calculate the area of the sector [tex]\( \text{AOB} \)[/tex]:
[tex]\[
\text{Area of sector AOB} = \left(\frac{1}{4}\right) \times \pi \times r^2 = \left(\frac{1}{4}\right) \times 3.14 \times 5^2
\][/tex]
Given the solving approach, we should take the given numerical result:
[tex]\[
\text{Area of sector AOB} = 19.625
\][/tex]
Comparing the calculated area of sector [tex]\( \text{AOB} \)[/tex] with the possible answers provided:
- A. 19.6 square units
- B. 39.3 square units
- C. 7.85 square units
- D. 15.7 square units
The closest answer to 19.625 square units is:
A. 19.6 square units.
