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Answer :
To solve this problem, we need to find the area of the sector [tex]\(AOB\)[/tex] of a circle. Here are the steps:
1. Identify the provided values:
- The radius of the circle, [tex]\(OA\)[/tex], is 5 units.
- The fraction of the circle's circumference that the arc [tex]\(\hat{AB}\)[/tex] takes up 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], where [tex]\(r\)[/tex] is the radius.
- Substituting the radius:
[tex]\[
C = 2 \times 3.14 \times 5 = 31.4 \text{ units}
\][/tex]
3. Determine the length of arc [tex]\(AB\)[/tex]:
- Since arc [tex]\(AB\)[/tex] is [tex]\(\frac{1}{4}\)[/tex] of the entire circumference:
[tex]\[
\text{Length of } \hat{AB} = \frac{1}{4} \times 31.4 = 7.85 \text{ units}
\][/tex]
4. Calculate the area of the entire circle:
- The formula for the area, [tex]\(A\)[/tex], of a circle is [tex]\(A = \pi r^2\)[/tex].
- Substituting the radius:
[tex]\[
A = 3.14 \times (5)^2 = 78.5 \text{ square units}
\][/tex]
5. Calculate the area of sector [tex]\(AOB\)[/tex]:
- The area of a sector is proportional to the length of its arc compared to the total circumference.
- So, the area of sector [tex]\(AOB\)[/tex] is the same fraction of the circle's area as the arc length is of the circumference:
[tex]\[
\text{Sector area } AOB = \frac{7.85}{31.4} \times 78.5
\][/tex]
[tex]\[
\text{Sector area } AOB = 19.625 \text{ square units}
\][/tex]
6. Choose the closest answer from the options:
- Comparing the sector area [tex]\(19.625\)[/tex] with the given options, the closest answer is:
- Option A: 19.6 square units
So, the correct answer is A. 19.6 square units.
1. Identify the provided values:
- The radius of the circle, [tex]\(OA\)[/tex], is 5 units.
- The fraction of the circle's circumference that the arc [tex]\(\hat{AB}\)[/tex] takes up 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], where [tex]\(r\)[/tex] is the radius.
- Substituting the radius:
[tex]\[
C = 2 \times 3.14 \times 5 = 31.4 \text{ units}
\][/tex]
3. Determine the length of arc [tex]\(AB\)[/tex]:
- Since arc [tex]\(AB\)[/tex] is [tex]\(\frac{1}{4}\)[/tex] of the entire circumference:
[tex]\[
\text{Length of } \hat{AB} = \frac{1}{4} \times 31.4 = 7.85 \text{ units}
\][/tex]
4. Calculate the area of the entire circle:
- The formula for the area, [tex]\(A\)[/tex], of a circle is [tex]\(A = \pi r^2\)[/tex].
- Substituting the radius:
[tex]\[
A = 3.14 \times (5)^2 = 78.5 \text{ square units}
\][/tex]
5. Calculate the area of sector [tex]\(AOB\)[/tex]:
- The area of a sector is proportional to the length of its arc compared to the total circumference.
- So, the area of sector [tex]\(AOB\)[/tex] is the same fraction of the circle's area as the arc length is of the circumference:
[tex]\[
\text{Sector area } AOB = \frac{7.85}{31.4} \times 78.5
\][/tex]
[tex]\[
\text{Sector area } AOB = 19.625 \text{ square units}
\][/tex]
6. Choose the closest answer from the options:
- Comparing the sector area [tex]\(19.625\)[/tex] with the given options, the closest answer is:
- Option A: 19.6 square units
So, the correct answer is A. 19.6 square units.
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