Answer :
To find the area of sector [tex]\( AOB \)[/tex] in the circle with center [tex]\( O \)[/tex], we follow these steps:
1. Understand the problem:
- Point [tex]\( O \)[/tex] is the center of the circle.
- The radius [tex]\( OA = 5 \)[/tex] units.
- The ratio of the arc length [tex]\(\widehat{AB}\)[/tex] to the circumference is [tex]\(\frac{1}{4}\)[/tex].
- We need to find the area of the sector [tex]\( AOB \)[/tex].
2. Calculate the circumference of the circle:
- The formula for the circumference of a circle is [tex]\( 2\pi r \)[/tex].
- With [tex]\( r = 5 \)[/tex], the circumference is:
[tex]\[ 2 \times 3.14 \times 5 = 31.4 \text{ units} \][/tex]
3. Determine the arc length of [tex]\(\widehat{AB}\)[/tex]:
- The arc length [tex]\(\widehat{AB}\)[/tex] is [tex]\(\frac{1}{4}\)[/tex] of the total circumference.
- So, the arc length is:
[tex]\[ \frac{1}{4} \times 31.4 = 7.85 \text{ units} \][/tex]
4. Calculate the area of sector [tex]\( AOB \)[/tex]:
- The area of a sector is given by the formula:
[tex]\(\text{Area of sector} = \frac{\text{arc length}}{\text{circumference}} \times (\pi \times r^2)\)[/tex]
- Substituting the known values:
[tex]\[\text{Area of sector} = \frac{1}{4} \times 3.14 \times 5^2\][/tex]
[tex]\[= \frac{1}{4} \times 3.14 \times 25\][/tex]
[tex]\[= \frac{78.5}{4}\][/tex]
[tex]\[= 19.625 \][/tex]
5. Choose the closest answer:
- The closest answer to 19.625 square units from the given options is [tex]\(19.6\)[/tex] square units.
Therefore, the area of sector [tex]\( AOB \)[/tex] is approximately [tex]\( \boxed{19.6} \)[/tex] square units.
1. Understand the problem:
- Point [tex]\( O \)[/tex] is the center of the circle.
- The radius [tex]\( OA = 5 \)[/tex] units.
- The ratio of the arc length [tex]\(\widehat{AB}\)[/tex] to the circumference is [tex]\(\frac{1}{4}\)[/tex].
- We need to find the area of the sector [tex]\( AOB \)[/tex].
2. Calculate the circumference of the circle:
- The formula for the circumference of a circle is [tex]\( 2\pi r \)[/tex].
- With [tex]\( r = 5 \)[/tex], the circumference is:
[tex]\[ 2 \times 3.14 \times 5 = 31.4 \text{ units} \][/tex]
3. Determine the arc length of [tex]\(\widehat{AB}\)[/tex]:
- The arc length [tex]\(\widehat{AB}\)[/tex] is [tex]\(\frac{1}{4}\)[/tex] of the total circumference.
- So, the arc length is:
[tex]\[ \frac{1}{4} \times 31.4 = 7.85 \text{ units} \][/tex]
4. Calculate the area of sector [tex]\( AOB \)[/tex]:
- The area of a sector is given by the formula:
[tex]\(\text{Area of sector} = \frac{\text{arc length}}{\text{circumference}} \times (\pi \times r^2)\)[/tex]
- Substituting the known values:
[tex]\[\text{Area of sector} = \frac{1}{4} \times 3.14 \times 5^2\][/tex]
[tex]\[= \frac{1}{4} \times 3.14 \times 25\][/tex]
[tex]\[= \frac{78.5}{4}\][/tex]
[tex]\[= 19.625 \][/tex]
5. Choose the closest answer:
- The closest answer to 19.625 square units from the given options is [tex]\(19.6\)[/tex] square units.
Therefore, the area of sector [tex]\( AOB \)[/tex] is approximately [tex]\( \boxed{19.6} \)[/tex] square units.
