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Answer :
To solve the problem, let's go through it step-by-step:
1. Understand the Information Given:
- The circle is centered at point [tex]\( O \)[/tex] and has a radius [tex]\( OA = 5 \)[/tex].
- The fraction of the circumference represented by the arc [tex]\( \hat{AB} \)[/tex] is [tex]\( \frac{1}{4} \)[/tex].
2. Find the Circumference of the Circle:
- The formula for the circumference of a circle is [tex]\( 2 \pi \times \text{radius} \)[/tex].
- Using the value [tex]\(\pi = 3.14\)[/tex], the circumference is:
[tex]\[
2 \times 3.14 \times 5 = 31.4
\][/tex]
3. Determine the Arc Length [tex]\( \hat{AB} \)[/tex]:
- Since the arc length [tex]\( \hat{AB} \)[/tex] is [tex]\(\frac{1}{4}\)[/tex] of the total circumference, the length of [tex]\( \hat{AB} \)[/tex] is:
[tex]\[
\frac{1}{4} \times 31.4 = 7.85
\][/tex]
4. Calculate the Area of Sector [tex]\( AOB \)[/tex]:
- The area of the sector is proportional to the fraction of the circumference that the arc represents.
- Hence, the area of the sector [tex]\( AOB \)[/tex] is [tex]\(\frac{1}{4}\)[/tex] of the total area of the circle.
- The area of the entire circle is [tex]\(\pi \times \text{radius}^2\)[/tex]:
[tex]\[
3.14 \times 5^2 = 3.14 \times 25 = 78.5
\][/tex]
- Therefore, the area of sector [tex]\( AOB \)[/tex] is:
[tex]\[
\frac{1}{4} \times 78.5 = 19.625
\][/tex]
5. Round to the Nearest Given Option:
- The area of sector [tex]\( AOB \)[/tex] is approximately 19.625 square units.
- The closest option to 19.625 is 19.6 square units.
Therefore, the correct answer is A. 19.6 square units.
1. Understand the Information Given:
- The circle is centered at point [tex]\( O \)[/tex] and has a radius [tex]\( OA = 5 \)[/tex].
- The fraction of the circumference represented by the arc [tex]\( \hat{AB} \)[/tex] is [tex]\( \frac{1}{4} \)[/tex].
2. Find the Circumference of the Circle:
- The formula for the circumference of a circle is [tex]\( 2 \pi \times \text{radius} \)[/tex].
- Using the value [tex]\(\pi = 3.14\)[/tex], the circumference is:
[tex]\[
2 \times 3.14 \times 5 = 31.4
\][/tex]
3. Determine the Arc Length [tex]\( \hat{AB} \)[/tex]:
- Since the arc length [tex]\( \hat{AB} \)[/tex] is [tex]\(\frac{1}{4}\)[/tex] of the total circumference, the length of [tex]\( \hat{AB} \)[/tex] is:
[tex]\[
\frac{1}{4} \times 31.4 = 7.85
\][/tex]
4. Calculate the Area of Sector [tex]\( AOB \)[/tex]:
- The area of the sector is proportional to the fraction of the circumference that the arc represents.
- Hence, the area of the sector [tex]\( AOB \)[/tex] is [tex]\(\frac{1}{4}\)[/tex] of the total area of the circle.
- The area of the entire circle is [tex]\(\pi \times \text{radius}^2\)[/tex]:
[tex]\[
3.14 \times 5^2 = 3.14 \times 25 = 78.5
\][/tex]
- Therefore, the area of sector [tex]\( AOB \)[/tex] is:
[tex]\[
\frac{1}{4} \times 78.5 = 19.625
\][/tex]
5. Round to the Nearest Given Option:
- The area of sector [tex]\( AOB \)[/tex] is approximately 19.625 square units.
- The closest option to 19.625 is 19.6 square units.
Therefore, the correct answer is A. 19.6 square units.
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