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Answer :
To find the area of sector [tex]\(AOB\)[/tex], let's go step-by-step:
1. Identify the radius and arc length ratio:
- The radius [tex]\(OA\)[/tex] of the circle is given as 5 units.
- The ratio of the length of arc [tex]\(\hat{AB}\)[/tex] to the circumference is [tex]\(\frac{1}{4}\)[/tex].
2. Calculate the circumference of the circle:
- The formula for the circumference [tex]\(C\)[/tex] of a circle is [tex]\(C = 2\pi r\)[/tex].
- By substituting the given radius, [tex]\(C = 2 \times 3.14 \times 5 = 31.4\)[/tex] units.
3. Determine the length of arc [tex]\(\hat{AB}\)[/tex]:
- Since arc [tex]\(\hat{AB}\)[/tex] is [tex]\(\frac{1}{4}\)[/tex] of the circumference, calculate the length of arc [tex]\(\hat{AB}\)[/tex] as follows:
- Length of arc [tex]\(\hat{AB}\)[/tex] = [tex]\(\frac{1}{4} \times 31.4 = 7.85\)[/tex] units.
4. Calculate the area of sector [tex]\(AOB\)[/tex]:
- The formula for the area of a sector is [tex]\(\text{Area of sector} = \frac{\text{arc length}}{\text{circumference}} \times \pi r^2\)[/tex].
- Plugging in the values, we get:
- [tex]\(\text{Area of sector} = \frac{7.85}{31.4} \times 3.14 \times 5^2 = 19.6\)[/tex] square units.
Based on the calculations, the area of sector [tex]\(AOB\)[/tex] is closest to option A. 19.6 square units.
1. Identify the radius and arc length ratio:
- The radius [tex]\(OA\)[/tex] of the circle is given as 5 units.
- The ratio of the length of arc [tex]\(\hat{AB}\)[/tex] to the circumference is [tex]\(\frac{1}{4}\)[/tex].
2. Calculate the circumference of the circle:
- The formula for the circumference [tex]\(C\)[/tex] of a circle is [tex]\(C = 2\pi r\)[/tex].
- By substituting the given radius, [tex]\(C = 2 \times 3.14 \times 5 = 31.4\)[/tex] units.
3. Determine the length of arc [tex]\(\hat{AB}\)[/tex]:
- Since arc [tex]\(\hat{AB}\)[/tex] is [tex]\(\frac{1}{4}\)[/tex] of the circumference, calculate the length of arc [tex]\(\hat{AB}\)[/tex] as follows:
- Length of arc [tex]\(\hat{AB}\)[/tex] = [tex]\(\frac{1}{4} \times 31.4 = 7.85\)[/tex] units.
4. Calculate the area of sector [tex]\(AOB\)[/tex]:
- The formula for the area of a sector is [tex]\(\text{Area of sector} = \frac{\text{arc length}}{\text{circumference}} \times \pi r^2\)[/tex].
- Plugging in the values, we get:
- [tex]\(\text{Area of sector} = \frac{7.85}{31.4} \times 3.14 \times 5^2 = 19.6\)[/tex] square units.
Based on the calculations, the area of sector [tex]\(AOB\)[/tex] is closest to option A. 19.6 square units.
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