Answer :
Let's solve the problem step-by-step.
We are given:
- Point [tex]\(O\)[/tex] is the center of the circle.
- Points [tex]\(A\)[/tex] and [tex]\(B\)[/tex] lie on the circle.
- The radius, [tex]\(OA = 5\)[/tex].
- The ratio of the length of arc [tex]\( \hat{AB} \)[/tex] to the circumference of the circle is [tex]\(\frac{1}{4}\)[/tex].
The goal is to find the area of sector [tex]\(AOB\)[/tex].
1. Calculate the circumference of the circle:
[tex]\[
\text{Circumference} = 2\pi \times \text{radius} = 2 \times 3.14 \times 5 = 31.4
\][/tex]
2. Calculate the area of the circle:
[tex]\[
\text{Area of the circle} = \pi \times \text{radius}^2 = 3.14 \times 5^2 = 3.14 \times 25 = 78.5 \text{ square units}
\][/tex]
3. Determine the area of the sector:
Since the length of arc [tex]\( \hat{AB} \)[/tex] is [tex]\(\frac{1}{4}\)[/tex] of the circumference, the sector [tex]\(AOB\)[/tex] will also be [tex]\(\frac{1}{4}\)[/tex] of the area of the circle.
[tex]\[
\text{Area of sector AOB} = \frac{1}{4} \times \text{Area of the circle} = \frac{1}{4} \times 78.5 = 19.625 \text{ square units}
\][/tex]
4. Round to the nearest option:
[tex]\[
19.625 \text{ square units} \approx 19.6 \text{ square units}
\][/tex]
Among the given choices, the closest answer is:
A. 19.6 square units
So, the area of sector [tex]\(AOB\)[/tex] is [tex]\(\boxed{19.6 \text{ square units}}\)[/tex].
We are given:
- Point [tex]\(O\)[/tex] is the center of the circle.
- Points [tex]\(A\)[/tex] and [tex]\(B\)[/tex] lie on the circle.
- The radius, [tex]\(OA = 5\)[/tex].
- The ratio of the length of arc [tex]\( \hat{AB} \)[/tex] to the circumference of the circle is [tex]\(\frac{1}{4}\)[/tex].
The goal is to find the area of sector [tex]\(AOB\)[/tex].
1. Calculate the circumference of the circle:
[tex]\[
\text{Circumference} = 2\pi \times \text{radius} = 2 \times 3.14 \times 5 = 31.4
\][/tex]
2. Calculate the area of the circle:
[tex]\[
\text{Area of the circle} = \pi \times \text{radius}^2 = 3.14 \times 5^2 = 3.14 \times 25 = 78.5 \text{ square units}
\][/tex]
3. Determine the area of the sector:
Since the length of arc [tex]\( \hat{AB} \)[/tex] is [tex]\(\frac{1}{4}\)[/tex] of the circumference, the sector [tex]\(AOB\)[/tex] will also be [tex]\(\frac{1}{4}\)[/tex] of the area of the circle.
[tex]\[
\text{Area of sector AOB} = \frac{1}{4} \times \text{Area of the circle} = \frac{1}{4} \times 78.5 = 19.625 \text{ square units}
\][/tex]
4. Round to the nearest option:
[tex]\[
19.625 \text{ square units} \approx 19.6 \text{ square units}
\][/tex]
Among the given choices, the closest answer is:
A. 19.6 square units
So, the area of sector [tex]\(AOB\)[/tex] is [tex]\(\boxed{19.6 \text{ square units}}\)[/tex].
