Answer :
To solve this problem, we need to find the area of a sector of a circle. The circle has a radius of 5 units, and we know that the arc from point A to point B is [tex]\(\frac{1}{4}\)[/tex] of the circle's circumference.
First, let's calculate the area of the whole circle using the formula:
[tex]\[
\text{Area of circle} = \pi \times \text{radius}^2
\][/tex]
Plugging in the values given:
[tex]\[
\text{Area of circle} = 3.14 \times 5^2 = 3.14 \times 25 = 78.5 \text{ square units}
\][/tex]
Next, the problem states that the arc length of [tex]\( \hat{AB} \)[/tex] constitutes [tex]\(\frac{1}{4}\)[/tex] of the circle's circumference. This tells us that the sector also represents [tex]\(\frac{1}{4}\)[/tex] of the total area of the circle.
So, we find the area of sector [tex]\(AOB\)[/tex] by taking [tex]\(\frac{1}{4}\)[/tex] of the total area of the circle:
[tex]\[
\text{Area of sector } AOB = \frac{1}{4} \times 78.5 = 19.625 \text{ square units}
\][/tex]
Among the options provided, we need to choose the closest value to the calculated area:
The closest is:
A. 19.6 square units
Therefore, the area of sector [tex]\(AOB\)[/tex] is approximately 19.6 square units.
First, let's calculate the area of the whole circle using the formula:
[tex]\[
\text{Area of circle} = \pi \times \text{radius}^2
\][/tex]
Plugging in the values given:
[tex]\[
\text{Area of circle} = 3.14 \times 5^2 = 3.14 \times 25 = 78.5 \text{ square units}
\][/tex]
Next, the problem states that the arc length of [tex]\( \hat{AB} \)[/tex] constitutes [tex]\(\frac{1}{4}\)[/tex] of the circle's circumference. This tells us that the sector also represents [tex]\(\frac{1}{4}\)[/tex] of the total area of the circle.
So, we find the area of sector [tex]\(AOB\)[/tex] by taking [tex]\(\frac{1}{4}\)[/tex] of the total area of the circle:
[tex]\[
\text{Area of sector } AOB = \frac{1}{4} \times 78.5 = 19.625 \text{ square units}
\][/tex]
Among the options provided, we need to choose the closest value to the calculated area:
The closest is:
A. 19.6 square units
Therefore, the area of sector [tex]\(AOB\)[/tex] is approximately 19.6 square units.
