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Answer :
To find the area of sector [tex]\( AOB \)[/tex], we can follow these steps:
1. Identify the Given Information:
- The radius of the circle, [tex]\( OA \)[/tex], is [tex]\( 5 \)[/tex] units.
- The fraction of the circumference corresponding to arc [tex]\( \hat{AB} \)[/tex] is [tex]\( \frac{1}{4} \)[/tex].
2. Calculate the Total Area of the Circle:
- The formula for the area of a circle is [tex]\(\pi \times \text{radius}^2\)[/tex].
- Using the given radius and [tex]\(\pi = 3.14\)[/tex],
[tex]\[
\text{Area of the circle} = 3.14 \times 5^2 = 3.14 \times 25 = 78.5 \text{ square units}
\][/tex]
3. Determine the Area of Sector [tex]\( AOB \)[/tex]:
- Since the arc [tex]\(\hat{AB}\)[/tex] makes up [tex]\(\frac{1}{4}\)[/tex] of the circle's circumference, the area of the sector [tex]\( AOB \)[/tex] is also [tex]\(\frac{1}{4}\)[/tex] of the total area of the circle.
- Calculate the area of the sector:
[tex]\[
\text{Area of sector } AOB = \frac{1}{4} \times 78.5 = 19.625 \text{ square units}
\][/tex]
4. Select the Closest Answer:
- Comparing [tex]\( 19.625 \)[/tex] square units with the given options, the closest answer is:
- A. 19.6 square units
Therefore, the area of sector [tex]\( AOB \)[/tex] is approximately [tex]\( 19.6 \)[/tex] square units.
1. Identify the Given Information:
- The radius of the circle, [tex]\( OA \)[/tex], is [tex]\( 5 \)[/tex] units.
- The fraction of the circumference corresponding to arc [tex]\( \hat{AB} \)[/tex] is [tex]\( \frac{1}{4} \)[/tex].
2. Calculate the Total Area of the Circle:
- The formula for the area of a circle is [tex]\(\pi \times \text{radius}^2\)[/tex].
- Using the given radius and [tex]\(\pi = 3.14\)[/tex],
[tex]\[
\text{Area of the circle} = 3.14 \times 5^2 = 3.14 \times 25 = 78.5 \text{ square units}
\][/tex]
3. Determine the Area of Sector [tex]\( AOB \)[/tex]:
- Since the arc [tex]\(\hat{AB}\)[/tex] makes up [tex]\(\frac{1}{4}\)[/tex] of the circle's circumference, the area of the sector [tex]\( AOB \)[/tex] is also [tex]\(\frac{1}{4}\)[/tex] of the total area of the circle.
- Calculate the area of the sector:
[tex]\[
\text{Area of sector } AOB = \frac{1}{4} \times 78.5 = 19.625 \text{ square units}
\][/tex]
4. Select the Closest Answer:
- Comparing [tex]\( 19.625 \)[/tex] square units with the given options, the closest answer is:
- A. 19.6 square units
Therefore, the area of sector [tex]\( AOB \)[/tex] is approximately [tex]\( 19.6 \)[/tex] square units.
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