Answer :
To solve this problem, we need to find the area of the sector [tex]\(AOB\)[/tex] in a circle.
Here are the steps to follow:
1. Determine the given information:
- The radius of the circle [tex]\(OA\)[/tex] is [tex]\(5\)[/tex] units.
- The length of arc [tex]\(AB\)[/tex] divided by the circumference of the circle is [tex]\(\frac{1}{4}\)[/tex].
- Use [tex]\(\pi = 3.14\)[/tex].
2. Calculate the circumference of the circle:
The formula for the circumference of a circle is:
[tex]\[
\text{Circumference} = 2 \pi \times \text{radius}
\][/tex]
Plugging in the radius:
[tex]\[
\text{Circumference} = 2 \times 3.14 \times 5 = 31.4 \text{ units}
\][/tex]
3. Calculate the length of arc [tex]\(AB\)[/tex]:
Since the ratio of the length of arc [tex]\(AB\)[/tex] to the circumference is [tex]\(\frac{1}{4}\)[/tex], we can find the length of arc by multiplying the circumference by this ratio:
[tex]\[
\text{Length of arc } AB = \frac{1}{4} \times 31.4 = 7.85 \text{ units}
\][/tex]
4. Calculate the angle of sector [tex]\(AOB\)[/tex] in radians:
The angle [tex]\(\theta\)[/tex] in radians for a specific arc length is determined by the formula:
[tex]\[
\theta = \frac{\text{Arc Length}}{\text{Radius}}
\][/tex]
Thus:
[tex]\[
\theta = \frac{7.85}{5} = 1.57 \text{ radians}
\][/tex]
5. Calculate the area of sector [tex]\(AOB\)[/tex]:
The area of a sector is given by the formula:
[tex]\[
\text{Area of sector} = \frac{1}{2} \times \text{radius}^2 \times \theta
\][/tex]
Substitute the values we found:
[tex]\[
\text{Area of sector} = \frac{1}{2} \times 5^2 \times 1.57 = 19.625 \text{ square units}
\][/tex]
The closest answer to this calculation is:
- A. 19.6 square units
Therefore, the area of sector [tex]\(AOB\)[/tex] is approximately 19.6 square units.
Here are the steps to follow:
1. Determine the given information:
- The radius of the circle [tex]\(OA\)[/tex] is [tex]\(5\)[/tex] units.
- The length of arc [tex]\(AB\)[/tex] divided by the circumference of the circle is [tex]\(\frac{1}{4}\)[/tex].
- Use [tex]\(\pi = 3.14\)[/tex].
2. Calculate the circumference of the circle:
The formula for the circumference of a circle is:
[tex]\[
\text{Circumference} = 2 \pi \times \text{radius}
\][/tex]
Plugging in the radius:
[tex]\[
\text{Circumference} = 2 \times 3.14 \times 5 = 31.4 \text{ units}
\][/tex]
3. Calculate the length of arc [tex]\(AB\)[/tex]:
Since the ratio of the length of arc [tex]\(AB\)[/tex] to the circumference is [tex]\(\frac{1}{4}\)[/tex], we can find the length of arc by multiplying the circumference by this ratio:
[tex]\[
\text{Length of arc } AB = \frac{1}{4} \times 31.4 = 7.85 \text{ units}
\][/tex]
4. Calculate the angle of sector [tex]\(AOB\)[/tex] in radians:
The angle [tex]\(\theta\)[/tex] in radians for a specific arc length is determined by the formula:
[tex]\[
\theta = \frac{\text{Arc Length}}{\text{Radius}}
\][/tex]
Thus:
[tex]\[
\theta = \frac{7.85}{5} = 1.57 \text{ radians}
\][/tex]
5. Calculate the area of sector [tex]\(AOB\)[/tex]:
The area of a sector is given by the formula:
[tex]\[
\text{Area of sector} = \frac{1}{2} \times \text{radius}^2 \times \theta
\][/tex]
Substitute the values we found:
[tex]\[
\text{Area of sector} = \frac{1}{2} \times 5^2 \times 1.57 = 19.625 \text{ square units}
\][/tex]
The closest answer to this calculation is:
- A. 19.6 square units
Therefore, the area of sector [tex]\(AOB\)[/tex] is approximately 19.6 square units.
