Answer :
To solve this problem, we need to find the area of the sector [tex]\(AOB\)[/tex] in the circle.
1. Understand the Given Information:
- The circle is centered at point [tex]\(O\)[/tex].
- The radius [tex]\(OA\)[/tex] is 5 units.
- The length of arc [tex]\(\overset{\frown}{AB}\)[/tex] is [tex]\(\frac{1}{4}\)[/tex] of the circle's circumference.
2. Calculate the Circumference of the Circle:
- The formula for the circumference [tex]\(C\)[/tex] of a circle is [tex]\(C = 2\pi \times \text{radius}\)[/tex].
- Using [tex]\(\pi = 3.14\)[/tex] and radius = 5, we get:
[tex]\[
C = 2 \times 3.14 \times 5 = 31.4 \text{ units}
\][/tex]
3. Find the Length of Arc [tex]\(AB\)[/tex]:
- Arc [tex]\(AB\)[/tex] is [tex]\(\frac{1}{4}\)[/tex] of the circumference.
- So, the length of arc [tex]\(AB\)[/tex] is:
[tex]\[
\text{Arc length} = \frac{1}{4} \times 31.4 = 7.85 \text{ units}
\][/tex]
4. Determine the Angle of Sector [tex]\(AOB\)[/tex]:
- The fraction of the circle's circumference that the arc represents also equals the fraction of [tex]\(360^\circ\)[/tex] that the angle represents.
- Therefore, the angle [tex]\(\theta\)[/tex] is:
[tex]\[
\theta = \frac{1}{4} \times 360^\circ = 90^\circ
\][/tex]
5. Calculate the Area of Sector [tex]\(AOB\)[/tex]:
- The formula for the area of a sector of a circle is given by:
[tex]\[
\text{Area of sector} = \left(\frac{\theta}{360}\right) \times \pi \times \text{radius}^2
\][/tex]
- Substituting the known values:
[tex]\[
\text{Area of sector} = \left(\frac{90}{360}\right) \times 3.14 \times 5^2
\][/tex]
- Simplifying further:
[tex]\[
\text{Area of sector} = \left(\frac{1}{4}\right) \times 3.14 \times 25 = \frac{78.5}{4} = 19.625 \text{ square units}
\][/tex]
6. Choose the Closest Answer:
- The closest answer to the calculated area of 19.625 square units is option A: 19.6 square units.
Therefore, the area of sector [tex]\(AOB\)[/tex] is approximately 19.6 square units.
1. Understand the Given Information:
- The circle is centered at point [tex]\(O\)[/tex].
- The radius [tex]\(OA\)[/tex] is 5 units.
- The length of arc [tex]\(\overset{\frown}{AB}\)[/tex] is [tex]\(\frac{1}{4}\)[/tex] of the circle's circumference.
2. Calculate the Circumference of the Circle:
- The formula for the circumference [tex]\(C\)[/tex] of a circle is [tex]\(C = 2\pi \times \text{radius}\)[/tex].
- Using [tex]\(\pi = 3.14\)[/tex] and radius = 5, we get:
[tex]\[
C = 2 \times 3.14 \times 5 = 31.4 \text{ units}
\][/tex]
3. Find the Length of Arc [tex]\(AB\)[/tex]:
- Arc [tex]\(AB\)[/tex] is [tex]\(\frac{1}{4}\)[/tex] of the circumference.
- So, the length of arc [tex]\(AB\)[/tex] is:
[tex]\[
\text{Arc length} = \frac{1}{4} \times 31.4 = 7.85 \text{ units}
\][/tex]
4. Determine the Angle of Sector [tex]\(AOB\)[/tex]:
- The fraction of the circle's circumference that the arc represents also equals the fraction of [tex]\(360^\circ\)[/tex] that the angle represents.
- Therefore, the angle [tex]\(\theta\)[/tex] is:
[tex]\[
\theta = \frac{1}{4} \times 360^\circ = 90^\circ
\][/tex]
5. Calculate the Area of Sector [tex]\(AOB\)[/tex]:
- The formula for the area of a sector of a circle is given by:
[tex]\[
\text{Area of sector} = \left(\frac{\theta}{360}\right) \times \pi \times \text{radius}^2
\][/tex]
- Substituting the known values:
[tex]\[
\text{Area of sector} = \left(\frac{90}{360}\right) \times 3.14 \times 5^2
\][/tex]
- Simplifying further:
[tex]\[
\text{Area of sector} = \left(\frac{1}{4}\right) \times 3.14 \times 25 = \frac{78.5}{4} = 19.625 \text{ square units}
\][/tex]
6. Choose the Closest Answer:
- The closest answer to the calculated area of 19.625 square units is option A: 19.6 square units.
Therefore, the area of sector [tex]\(AOB\)[/tex] is approximately 19.6 square units.