Answer :
To solve this problem, we need to find the area of sector [tex]\(AOB\)[/tex] in a circle with a radius of 5 units, where the arc length [tex]\(\widehat{AB}\)[/tex] is [tex]\(\frac{1}{4}\)[/tex] of the circle's circumference. Let's break this down step-by-step:
1. Find the Circumference of the Circle:
The formula for the circumference [tex]\( C \)[/tex] of a circle is:
[tex]\[
C = 2 \pi r
\][/tex]
Given [tex]\(\pi = 3.14\)[/tex] and the radius [tex]\( r = 5 \)[/tex],
[tex]\[
C = 2 \times 3.14 \times 5 = 31.4
\][/tex]
2. Calculate the Arc Length [tex]\(\widehat{AB}\)[/tex]:
Since the arc length [tex]\(\widehat{AB}\)[/tex] is [tex]\(\frac{1}{4}\)[/tex] of the circumference,
[tex]\[
\text{Arc Length } = \frac{1}{4} \times 31.4 = 7.85
\][/tex]
3. Determine the Area of the Whole Circle:
The formula for the area [tex]\( A \)[/tex] of a circle is:
[tex]\[
A = \pi r^2
\][/tex]
[tex]\[
A = 3.14 \times 5^2 = 3.14 \times 25 = 78.5
\][/tex]
4. Calculate the Area of the Sector [tex]\(AOB\)[/tex]:
The area of sector [tex]\(AOB\)[/tex] is a fraction of the total area of the circle, similar to the arc's fraction of the circumference. Since the arc [tex]\(\widehat{AB}\)[/tex] represents [tex]\(\frac{1}{4}\)[/tex] of the circumference, the sector will also represent [tex]\(\frac{1}{4}\)[/tex] of the circle's area:
[tex]\[
\text{Area of Sector } = \frac{1}{4} \times 78.5 = 19.625
\][/tex]
Comparing this with the provided options, we select the closest answer:
A. 19.6 square units
Thus, the area of sector [tex]\(AOB\)[/tex] is approximately 19.6 square units.
1. Find the Circumference of the Circle:
The formula for the circumference [tex]\( C \)[/tex] of a circle is:
[tex]\[
C = 2 \pi r
\][/tex]
Given [tex]\(\pi = 3.14\)[/tex] and the radius [tex]\( r = 5 \)[/tex],
[tex]\[
C = 2 \times 3.14 \times 5 = 31.4
\][/tex]
2. Calculate the Arc Length [tex]\(\widehat{AB}\)[/tex]:
Since the arc length [tex]\(\widehat{AB}\)[/tex] is [tex]\(\frac{1}{4}\)[/tex] of the circumference,
[tex]\[
\text{Arc Length } = \frac{1}{4} \times 31.4 = 7.85
\][/tex]
3. Determine the Area of the Whole Circle:
The formula for the area [tex]\( A \)[/tex] of a circle is:
[tex]\[
A = \pi r^2
\][/tex]
[tex]\[
A = 3.14 \times 5^2 = 3.14 \times 25 = 78.5
\][/tex]
4. Calculate the Area of the Sector [tex]\(AOB\)[/tex]:
The area of sector [tex]\(AOB\)[/tex] is a fraction of the total area of the circle, similar to the arc's fraction of the circumference. Since the arc [tex]\(\widehat{AB}\)[/tex] represents [tex]\(\frac{1}{4}\)[/tex] of the circumference, the sector will also represent [tex]\(\frac{1}{4}\)[/tex] of the circle's area:
[tex]\[
\text{Area of Sector } = \frac{1}{4} \times 78.5 = 19.625
\][/tex]
Comparing this with the provided options, we select the closest answer:
A. 19.6 square units
Thus, the area of sector [tex]\(AOB\)[/tex] is approximately 19.6 square units.
