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Answer :
To find the area of sector [tex]\(AOB\)[/tex], we need to follow these steps:
1. Determine the Radius:
- The radius of the circle is given as [tex]\(OA = 5\)[/tex] units.
2. Calculate the Area of the Whole Circle:
- The formula for the area of a circle is [tex]\(\pi \times \text{radius}^2\)[/tex].
- Substituting the given values, [tex]\( \pi = 3.14 \)[/tex] and [tex]\(\text{radius} = 5\)[/tex], the area of the circle is:
[tex]\[
\text{Area} = 3.14 \times 5^2 = 3.14 \times 25 = 78.5 \text{ square units}
\][/tex]
3. Determine the Fraction of the Circle:
- We know the arc [tex]\( \widehat{AB} \)[/tex] of the circle is [tex]\(\frac{1}{4}\)[/tex] of the entire circumference. Therefore, the sector [tex]\(AOB\)[/tex] also covers [tex]\(\frac{1}{4}\)[/tex] of the circle’s area.
4. Calculate the Area of Sector [tex]\(AOB\)[/tex]:
- The area of the sector is [tex]\(\frac{1}{4}\)[/tex] of the area of the entire circle.
- The calculation becomes:
[tex]\[
\text{Area of Sector } AOB = \frac{1}{4} \times 78.5 = 19.625 \text{ square units}
\][/tex]
5. Choose the Closest Answer:
- Among the given answer choices, the closest to 19.625 is [tex]\( \textbf{A. 19.6 square units} \)[/tex].
Therefore, the area of sector [tex]\(AOB\)[/tex] is approximately [tex]\( \text{19.6 square units} \)[/tex].
1. Determine the Radius:
- The radius of the circle is given as [tex]\(OA = 5\)[/tex] units.
2. Calculate the Area of the Whole Circle:
- The formula for the area of a circle is [tex]\(\pi \times \text{radius}^2\)[/tex].
- Substituting the given values, [tex]\( \pi = 3.14 \)[/tex] and [tex]\(\text{radius} = 5\)[/tex], the area of the circle is:
[tex]\[
\text{Area} = 3.14 \times 5^2 = 3.14 \times 25 = 78.5 \text{ square units}
\][/tex]
3. Determine the Fraction of the Circle:
- We know the arc [tex]\( \widehat{AB} \)[/tex] of the circle is [tex]\(\frac{1}{4}\)[/tex] of the entire circumference. Therefore, the sector [tex]\(AOB\)[/tex] also covers [tex]\(\frac{1}{4}\)[/tex] of the circle’s area.
4. Calculate the Area of Sector [tex]\(AOB\)[/tex]:
- The area of the sector is [tex]\(\frac{1}{4}\)[/tex] of the area of the entire circle.
- The calculation becomes:
[tex]\[
\text{Area of Sector } AOB = \frac{1}{4} \times 78.5 = 19.625 \text{ square units}
\][/tex]
5. Choose the Closest Answer:
- Among the given answer choices, the closest to 19.625 is [tex]\( \textbf{A. 19.6 square units} \)[/tex].
Therefore, the area of sector [tex]\(AOB\)[/tex] is approximately [tex]\( \text{19.6 square units} \)[/tex].
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