Answer :
To solve this question, we need to find the area of the sector [tex]\(AOB\)[/tex] in the circle. We're given the following information:
- The radius [tex]\(OA = 5\)[/tex].
- The fraction of the circumference that corresponds to the arc [tex]\(\widehat{B}\)[/tex] is [tex]\(\frac{1}{4}\)[/tex].
- We also have the value of [tex]\(\pi = 3.14\)[/tex].
### Steps to solve the problem:
1. Calculate the Area of the Entire Circle:
The formula for the area of a circle is:
[tex]\[
\text{Area} = \pi \times \text{radius}^2
\][/tex]
Plugging in the values, the area of the circle is:
[tex]\[
\text{Area} = 3.14 \times 5^2 = 3.14 \times 25 = 78.5 \text{ square units}
\][/tex]
2. Find the Area of the Sector:
We're given that the arc associated with the sector is [tex]\(\frac{1}{4}\)[/tex] of the circle's circumference. The area of a sector is proportional to the arc length of that sector. This means the area of the sector [tex]\(AOB\)[/tex] will also be [tex]\(\frac{1}{4}\)[/tex] of the entire circle's area.
[tex]\[
\text{Sector Area} = \frac{1}{4} \times 78.5 = 19.625 \text{ square units}
\][/tex]
The closest answer from the given options is:
A. 19.6 square units
Therefore, the correct choice is A. 19.6 square units.
- The radius [tex]\(OA = 5\)[/tex].
- The fraction of the circumference that corresponds to the arc [tex]\(\widehat{B}\)[/tex] is [tex]\(\frac{1}{4}\)[/tex].
- We also have the value of [tex]\(\pi = 3.14\)[/tex].
### Steps to solve the problem:
1. Calculate the Area of the Entire Circle:
The formula for the area of a circle is:
[tex]\[
\text{Area} = \pi \times \text{radius}^2
\][/tex]
Plugging in the values, the area of the circle is:
[tex]\[
\text{Area} = 3.14 \times 5^2 = 3.14 \times 25 = 78.5 \text{ square units}
\][/tex]
2. Find the Area of the Sector:
We're given that the arc associated with the sector is [tex]\(\frac{1}{4}\)[/tex] of the circle's circumference. The area of a sector is proportional to the arc length of that sector. This means the area of the sector [tex]\(AOB\)[/tex] will also be [tex]\(\frac{1}{4}\)[/tex] of the entire circle's area.
[tex]\[
\text{Sector Area} = \frac{1}{4} \times 78.5 = 19.625 \text{ square units}
\][/tex]
The closest answer from the given options is:
A. 19.6 square units
Therefore, the correct choice is A. 19.6 square units.
