Answer :
To solve this problem, we need to find the area of sector [tex]\(AOB\)[/tex] of the circle.
1. Understand the Given Information:
- The circle has a radius [tex]\(OA = 5\)[/tex].
- The fraction of the circumference that arc [tex]\(\widehat{AB}\)[/tex] occupies is [tex]\(\frac{1}{4}\)[/tex].
2. Calculate the Circumference of the Circle:
- The formula for the circumference of a circle is [tex]\(C = 2\pi r\)[/tex].
- With [tex]\(r = 5\)[/tex] and using [tex]\(\pi = 3.14\)[/tex], the circumference is:
[tex]\[
C = 2 \times 3.14 \times 5 = 31.4
\][/tex]
3. Determine the Length of Arc [tex]\(\widehat{AB}\)[/tex]:
- Since the arc length is [tex]\(\frac{1}{4}\)[/tex] of the total circumference, calculate:
[tex]\[
\text{Arc Length} = \frac{1}{4} \times 31.4 = 7.85
\][/tex]
4. Calculate the Area of Sector [tex]\(AOB\)[/tex]:
- The area of a sector is calculated as the fraction of the circle's area that is covered by the arc. It is given by:
[tex]\[
\text{Area of Sector} = \left(\frac{\text{Arc Length}}{\text{Circumference}}\right) \times \text{Area of Circle}
\][/tex]
- The area of the circle is [tex]\(\pi r^2\)[/tex], so substitute the values:
[tex]\[
\text{Area of Circle} = 3.14 \times 5^2 = 3.14 \times 25 = 78.5
\][/tex]
- Substituting in the values for the area of the sector:
[tex]\[
\text{Area of Sector} = \left(\frac{7.85}{31.4}\right) \times 78.5 = \frac{1}{4} \times 78.5 = 19.625
\][/tex]
5. Round the Area to the Closest Option:
- Rounding [tex]\(19.625\)[/tex] to the nearest option gives us [tex]\(19.6\)[/tex].
So, the closest answer is:
A. 19.6 square units
1. Understand the Given Information:
- The circle has a radius [tex]\(OA = 5\)[/tex].
- The fraction of the circumference that arc [tex]\(\widehat{AB}\)[/tex] occupies is [tex]\(\frac{1}{4}\)[/tex].
2. Calculate the Circumference of the Circle:
- The formula for the circumference of a circle is [tex]\(C = 2\pi r\)[/tex].
- With [tex]\(r = 5\)[/tex] and using [tex]\(\pi = 3.14\)[/tex], the circumference is:
[tex]\[
C = 2 \times 3.14 \times 5 = 31.4
\][/tex]
3. Determine the Length of Arc [tex]\(\widehat{AB}\)[/tex]:
- Since the arc length is [tex]\(\frac{1}{4}\)[/tex] of the total circumference, calculate:
[tex]\[
\text{Arc Length} = \frac{1}{4} \times 31.4 = 7.85
\][/tex]
4. Calculate the Area of Sector [tex]\(AOB\)[/tex]:
- The area of a sector is calculated as the fraction of the circle's area that is covered by the arc. It is given by:
[tex]\[
\text{Area of Sector} = \left(\frac{\text{Arc Length}}{\text{Circumference}}\right) \times \text{Area of Circle}
\][/tex]
- The area of the circle is [tex]\(\pi r^2\)[/tex], so substitute the values:
[tex]\[
\text{Area of Circle} = 3.14 \times 5^2 = 3.14 \times 25 = 78.5
\][/tex]
- Substituting in the values for the area of the sector:
[tex]\[
\text{Area of Sector} = \left(\frac{7.85}{31.4}\right) \times 78.5 = \frac{1}{4} \times 78.5 = 19.625
\][/tex]
5. Round the Area to the Closest Option:
- Rounding [tex]\(19.625\)[/tex] to the nearest option gives us [tex]\(19.6\)[/tex].
So, the closest answer is:
A. 19.6 square units
