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Answer :
To find the area of sector [tex]\(AOB\)[/tex] in the circle, let's go through the solution step by step:
1. Determine the radius and circumference of the circle:
- The radius [tex]\(OA\)[/tex] is given as 5 units.
- The formula for the circumference of a circle is [tex]\(2\pi r\)[/tex]. Plugging the given radius into the formula, the circumference is:
[tex]\[
\text{Circumference} = 2 \times 3.14 \times 5 = 31.4 \text{ units}
\][/tex]
2. Relate the arc length [tex]\(AB\)[/tex] to the circumference:
- We know that the ratio of the length of [tex]\(AB\)[/tex] to the circumference is [tex]\(\frac{1}{4}\)[/tex].
- Therefore, the length of arc [tex]\(AB\)[/tex] is:
[tex]\[
\text{Length of } AB = \frac{1}{4} \times 31.4 = 7.85 \text{ units}
\][/tex]
3. Determine the angle of sector [tex]\(AOB\)[/tex]:
- The angle corresponding to the arc [tex]\(AB\)[/tex] in a circle is proportional to the fraction of the arc length to the circumference.
- Since the arc [tex]\(AB\)[/tex] is [tex]\(\frac{1}{4}\)[/tex] of the circumference, the angle at the center is also [tex]\(\frac{1}{4}\)[/tex] of the full angle [tex]\(360^\circ\)[/tex]:
[tex]\[
\text{Angle of sector } AOB = 360^\circ \times \frac{1}{4} = 90^\circ
\][/tex]
4. Calculate the area of sector [tex]\(AOB\)[/tex]:
- The area of a sector is given by the formula [tex]\(\left(\frac{\text{angle}}{360^\circ}\right) \times \pi r^2\)[/tex].
- Substitute the values:
[tex]\[
\text{Area of sector } AOB = \left(\frac{90}{360}\right) \times 3.14 \times (5)^2
\][/tex]
- Simplifying:
[tex]\[
= \frac{1}{4} \times 3.14 \times 25 = 19.625 \text{ square units}
\][/tex]
5. Choose the closest answer:
- The closest answer to our calculated area, which is approximately [tex]\(19.625\)[/tex] square units, is option A. 19.6 square units.
Therefore, the correct choice for the area of sector [tex]\(AOB\)[/tex] is A. 19.6 square units.
1. Determine the radius and circumference of the circle:
- The radius [tex]\(OA\)[/tex] is given as 5 units.
- The formula for the circumference of a circle is [tex]\(2\pi r\)[/tex]. Plugging the given radius into the formula, the circumference is:
[tex]\[
\text{Circumference} = 2 \times 3.14 \times 5 = 31.4 \text{ units}
\][/tex]
2. Relate the arc length [tex]\(AB\)[/tex] to the circumference:
- We know that the ratio of the length of [tex]\(AB\)[/tex] to the circumference is [tex]\(\frac{1}{4}\)[/tex].
- Therefore, the length of arc [tex]\(AB\)[/tex] is:
[tex]\[
\text{Length of } AB = \frac{1}{4} \times 31.4 = 7.85 \text{ units}
\][/tex]
3. Determine the angle of sector [tex]\(AOB\)[/tex]:
- The angle corresponding to the arc [tex]\(AB\)[/tex] in a circle is proportional to the fraction of the arc length to the circumference.
- Since the arc [tex]\(AB\)[/tex] is [tex]\(\frac{1}{4}\)[/tex] of the circumference, the angle at the center is also [tex]\(\frac{1}{4}\)[/tex] of the full angle [tex]\(360^\circ\)[/tex]:
[tex]\[
\text{Angle of sector } AOB = 360^\circ \times \frac{1}{4} = 90^\circ
\][/tex]
4. Calculate the area of sector [tex]\(AOB\)[/tex]:
- The area of a sector is given by the formula [tex]\(\left(\frac{\text{angle}}{360^\circ}\right) \times \pi r^2\)[/tex].
- Substitute the values:
[tex]\[
\text{Area of sector } AOB = \left(\frac{90}{360}\right) \times 3.14 \times (5)^2
\][/tex]
- Simplifying:
[tex]\[
= \frac{1}{4} \times 3.14 \times 25 = 19.625 \text{ square units}
\][/tex]
5. Choose the closest answer:
- The closest answer to our calculated area, which is approximately [tex]\(19.625\)[/tex] square units, is option A. 19.6 square units.
Therefore, the correct choice for the area of sector [tex]\(AOB\)[/tex] is A. 19.6 square units.
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