Answer :
To solve this problem, let's go through the steps:
1. Identify the Given Information:
- Points [tex]\( A \)[/tex] and [tex]\( B \)[/tex] lie on the circle.
- The radius [tex]\( OA = 5 \)[/tex] units.
- The length of the arc [tex]\(\hat{AB}\)[/tex] is [tex]\(\frac{1}{4}\)[/tex] of the circle's circumference.
2. Determine the Circle's Circumference:
- The formula for the circumference of a circle is [tex]\( C = 2\pi r \)[/tex].
- Using [tex]\(\pi = 3.14\)[/tex] and [tex]\( r = 5 \)[/tex], the circumference is:
[tex]\[
C = 2 \times 3.14 \times 5 = 31.4 \text{ units}
\][/tex]
3. Calculate the Full Circle's Area:
- The formula for the area of a circle is [tex]\( A = \pi r^2 \)[/tex].
- So, the full circle's area is:
[tex]\[
A = 3.14 \times 5^2 = 3.14 \times 25 = 78.5 \text{ square units}
\][/tex]
4. Find the Area of Sector [tex]\( AOB \)[/tex]:
- Since the arc length of [tex]\(\hat{AB}\)[/tex] is [tex]\(\frac{1}{4}\)[/tex] of the circle's circumference, the sector area is also [tex]\(\frac{1}{4}\)[/tex] of the full circle's area.
- Therefore, the area of sector [tex]\( AOB \)[/tex] is:
[tex]\[
\text{Sector Area} = \frac{1}{4} \times 78.5 = 19.625 \text{ square units}
\][/tex]
5. Select the Closest Answer:
- From the options given, 19.6 square units is the closest to our calculated area.
Thus, the area of sector [tex]\( AOB \)[/tex] is approximately [tex]\( 19.6 \)[/tex] square units, which corresponds to option A.
1. Identify the Given Information:
- Points [tex]\( A \)[/tex] and [tex]\( B \)[/tex] lie on the circle.
- The radius [tex]\( OA = 5 \)[/tex] units.
- The length of the arc [tex]\(\hat{AB}\)[/tex] is [tex]\(\frac{1}{4}\)[/tex] of the circle's circumference.
2. Determine the Circle's Circumference:
- The formula for the circumference of a circle is [tex]\( C = 2\pi r \)[/tex].
- Using [tex]\(\pi = 3.14\)[/tex] and [tex]\( r = 5 \)[/tex], the circumference is:
[tex]\[
C = 2 \times 3.14 \times 5 = 31.4 \text{ units}
\][/tex]
3. Calculate the Full Circle's Area:
- The formula for the area of a circle is [tex]\( A = \pi r^2 \)[/tex].
- So, the full circle's area is:
[tex]\[
A = 3.14 \times 5^2 = 3.14 \times 25 = 78.5 \text{ square units}
\][/tex]
4. Find the Area of Sector [tex]\( AOB \)[/tex]:
- Since the arc length of [tex]\(\hat{AB}\)[/tex] is [tex]\(\frac{1}{4}\)[/tex] of the circle's circumference, the sector area is also [tex]\(\frac{1}{4}\)[/tex] of the full circle's area.
- Therefore, the area of sector [tex]\( AOB \)[/tex] is:
[tex]\[
\text{Sector Area} = \frac{1}{4} \times 78.5 = 19.625 \text{ square units}
\][/tex]
5. Select the Closest Answer:
- From the options given, 19.6 square units is the closest to our calculated area.
Thus, the area of sector [tex]\( AOB \)[/tex] is approximately [tex]\( 19.6 \)[/tex] square units, which corresponds to option A.