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Answer :
To find the area of sector [tex]\(AOB\)[/tex], follow these steps:
1. Identify the Given Information:
- The circle's center is at point [tex]\(O\)[/tex] with a radius [tex]\(O A = 5\)[/tex] units.
- The arc length ratio [tex]\(\frac{\text{length of } \hat{A B}}{\text{circumference}}\)[/tex] 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].
- Plug in the values: [tex]\(C = 2 \times 3.14 \times 5\)[/tex].
- This gives a circumference of [tex]\(31.4\)[/tex] units.
3. Determine the Angle of the Sector [tex]\(AOB\)[/tex]:
- Since the arc length of [tex]\( \hat{AB} \)[/tex] corresponds to [tex]\(\frac{1}{4}\)[/tex] of the circumference, the sector angle is also [tex]\(\frac{1}{4}\)[/tex] of a full circle.
- A full circle is [tex]\(360^\circ\)[/tex], so the sector angle is [tex]\(\frac{1}{4} \times 360 = 90^\circ\)[/tex].
4. Calculate the Area of Sector [tex]\(AOB\)[/tex]:
- The formula for the area of a sector is [tex]\(\text{Area} = \frac{\text{angle of sector}}{360} \times \pi r^2\)[/tex].
- Substitute the known values: [tex]\(\text{Area} = \frac{90}{360} \times 3.14 \times (5)^2\)[/tex].
- This simplifies to [tex]\(\text{Area} = \frac{1}{4} \times 3.14 \times 25\)[/tex].
- Calculate the area: [tex]\(\text{Area} = \frac{1}{4} \times 78.5 = 19.625\)[/tex] square units.
5. Select the Closest Answer:
- The area of the sector is approximately [tex]\(19.625\)[/tex] square units.
- Among the given options, [tex]\( \boxed{19.6} \)[/tex] square units is the closest answer.
So, the correct answer is A. 19.6 square units.
1. Identify the Given Information:
- The circle's center is at point [tex]\(O\)[/tex] with a radius [tex]\(O A = 5\)[/tex] units.
- The arc length ratio [tex]\(\frac{\text{length of } \hat{A B}}{\text{circumference}}\)[/tex] 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].
- Plug in the values: [tex]\(C = 2 \times 3.14 \times 5\)[/tex].
- This gives a circumference of [tex]\(31.4\)[/tex] units.
3. Determine the Angle of the Sector [tex]\(AOB\)[/tex]:
- Since the arc length of [tex]\( \hat{AB} \)[/tex] corresponds to [tex]\(\frac{1}{4}\)[/tex] of the circumference, the sector angle is also [tex]\(\frac{1}{4}\)[/tex] of a full circle.
- A full circle is [tex]\(360^\circ\)[/tex], so the sector angle is [tex]\(\frac{1}{4} \times 360 = 90^\circ\)[/tex].
4. Calculate the Area of Sector [tex]\(AOB\)[/tex]:
- The formula for the area of a sector is [tex]\(\text{Area} = \frac{\text{angle of sector}}{360} \times \pi r^2\)[/tex].
- Substitute the known values: [tex]\(\text{Area} = \frac{90}{360} \times 3.14 \times (5)^2\)[/tex].
- This simplifies to [tex]\(\text{Area} = \frac{1}{4} \times 3.14 \times 25\)[/tex].
- Calculate the area: [tex]\(\text{Area} = \frac{1}{4} \times 78.5 = 19.625\)[/tex] square units.
5. Select the Closest Answer:
- The area of the sector is approximately [tex]\(19.625\)[/tex] square units.
- Among the given options, [tex]\( \boxed{19.6} \)[/tex] square units is the closest answer.
So, the correct answer is A. 19.6 square units.
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