Answer :
To find the area of the sector [tex]\( AOB \)[/tex] of the circle, follow these steps:
1. Identify the Radius of the Circle:
The circle has a radius, [tex]\( OA = 5 \)[/tex].
2. Find the Circumference of the Circle:
The formula for the circumference of a circle is [tex]\( C = 2\pi r \)[/tex].
So, using [tex]\(\pi = 3.14\)[/tex] and [tex]\(r = 5\)[/tex], we have:
[tex]\[
C = 2 \times 3.14 \times 5 = 31.4
\][/tex]
3. Determine the Arc Length Ratio:
The ratio of the length of arc [tex]\( \widehat{AB} \)[/tex] to the circle’s circumference is [tex]\( \frac{1}{4} \)[/tex]. This means that the arc length of [tex]\( \widehat{AB} \)[/tex] is one-fourth of the circumference of the circle.
4. Calculate the Area of the Circle:
The formula for the area of a circle is [tex]\( A = \pi r^2 \)[/tex].
So, substituting in the values, we have:
[tex]\[
A = 3.14 \times 5^2 = 3.14 \times 25 = 78.5
\][/tex]
5. Calculate the Area of Sector [tex]\( AOB \)[/tex]:
Since the arc [tex]\( \widehat{AB} \)[/tex] corresponds to [tex]\(\frac{1}{4}\)[/tex] of the circle, the area of sector [tex]\( AOB \)[/tex] is also [tex]\(\frac{1}{4}\)[/tex] of the total area of the circle.
[tex]\[
\text{Area of sector } AOB = \frac{1}{4} \times 78.5 = 19.625
\][/tex]
After the calculations, we find that the area of sector [tex]\( AOB \)[/tex] is approximately [tex]\( 19.6 \)[/tex] square units. Therefore, the closest answer is:
- A. 19.6 square units
1. Identify the Radius of the Circle:
The circle has a radius, [tex]\( OA = 5 \)[/tex].
2. Find the Circumference of the Circle:
The formula for the circumference of a circle is [tex]\( C = 2\pi r \)[/tex].
So, using [tex]\(\pi = 3.14\)[/tex] and [tex]\(r = 5\)[/tex], we have:
[tex]\[
C = 2 \times 3.14 \times 5 = 31.4
\][/tex]
3. Determine the Arc Length Ratio:
The ratio of the length of arc [tex]\( \widehat{AB} \)[/tex] to the circle’s circumference is [tex]\( \frac{1}{4} \)[/tex]. This means that the arc length of [tex]\( \widehat{AB} \)[/tex] is one-fourth of the circumference of the circle.
4. Calculate the Area of the Circle:
The formula for the area of a circle is [tex]\( A = \pi r^2 \)[/tex].
So, substituting in the values, we have:
[tex]\[
A = 3.14 \times 5^2 = 3.14 \times 25 = 78.5
\][/tex]
5. Calculate the Area of Sector [tex]\( AOB \)[/tex]:
Since the arc [tex]\( \widehat{AB} \)[/tex] corresponds to [tex]\(\frac{1}{4}\)[/tex] of the circle, the area of sector [tex]\( AOB \)[/tex] is also [tex]\(\frac{1}{4}\)[/tex] of the total area of the circle.
[tex]\[
\text{Area of sector } AOB = \frac{1}{4} \times 78.5 = 19.625
\][/tex]
After the calculations, we find that the area of sector [tex]\( AOB \)[/tex] is approximately [tex]\( 19.6 \)[/tex] square units. Therefore, the closest answer is:
- A. 19.6 square units