Answer :
To solve the problem of finding the area of sector [tex]\( AOB \)[/tex], we need to follow these steps:
1. Identify the radius and given information:
- The radius of the circle, [tex]\( OA \)[/tex], is given as 5 units.
- The ratio of the length of arc [tex]\( AB \)[/tex] to the circumference of the circle is given as [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].
- Substituting the radius [tex]\( r = 5 \)[/tex] and using [tex]\(\pi \approx 3.14\)[/tex], we get:
[tex]\[
C = 2 \times 3.14 \times 5 = 31.4 \text{ units}
\][/tex]
3. Determine the arc length of [tex]\( AB \)[/tex]:
- Since the arc length of [tex]\( AB \)[/tex] is [tex]\(\frac{1}{4}\)[/tex] of the circumference, we calculate:
[tex]\[
\text{Arc length of } AB = \frac{1}{4} \times 31.4 = 7.85 \text{ units}
\][/tex]
4. Calculate the area of sector [tex]\( AOB \)[/tex]:
- The area of sector [tex]\( AOB \)[/tex] can be calculated using the formula:
[tex]\[
\text{Area of sector} = \left(\frac{\text{Arc length of } AB}{\text{Circumference}}\right) \times (\pi r^2)
\][/tex]
- Substituting values:
[tex]\[
\text{Area of sector} = \left(\frac{7.85}{31.4}\right) \times (3.14 \times 5^2)
\][/tex]
5. Perform the calculations:
- First calculate the fraction [tex]\(\frac{7.85}{31.4}\)[/tex], which is [tex]\(\frac{1}{4}\)[/tex] (as given).
- Then compute [tex]\((3.14 \times 5^2) = 78.5\)[/tex].
- Finally, calculate the sector area:
[tex]\[
\text{Area of sector} = \frac{1}{4} \times 78.5 = 19.625
\][/tex]
6. Choose the closest answer:
The closest answer to 19.625 from the given options is:
A. 19.6 square units
So, the area of sector [tex]\( AOB \)[/tex] is approximately 19.6 square units.
1. Identify the radius and given information:
- The radius of the circle, [tex]\( OA \)[/tex], is given as 5 units.
- The ratio of the length of arc [tex]\( AB \)[/tex] to the circumference of the circle is given as [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].
- Substituting the radius [tex]\( r = 5 \)[/tex] and using [tex]\(\pi \approx 3.14\)[/tex], we get:
[tex]\[
C = 2 \times 3.14 \times 5 = 31.4 \text{ units}
\][/tex]
3. Determine the arc length of [tex]\( AB \)[/tex]:
- Since the arc length of [tex]\( AB \)[/tex] is [tex]\(\frac{1}{4}\)[/tex] of the circumference, we calculate:
[tex]\[
\text{Arc length of } AB = \frac{1}{4} \times 31.4 = 7.85 \text{ units}
\][/tex]
4. Calculate the area of sector [tex]\( AOB \)[/tex]:
- The area of sector [tex]\( AOB \)[/tex] can be calculated using the formula:
[tex]\[
\text{Area of sector} = \left(\frac{\text{Arc length of } AB}{\text{Circumference}}\right) \times (\pi r^2)
\][/tex]
- Substituting values:
[tex]\[
\text{Area of sector} = \left(\frac{7.85}{31.4}\right) \times (3.14 \times 5^2)
\][/tex]
5. Perform the calculations:
- First calculate the fraction [tex]\(\frac{7.85}{31.4}\)[/tex], which is [tex]\(\frac{1}{4}\)[/tex] (as given).
- Then compute [tex]\((3.14 \times 5^2) = 78.5\)[/tex].
- Finally, calculate the sector area:
[tex]\[
\text{Area of sector} = \frac{1}{4} \times 78.5 = 19.625
\][/tex]
6. Choose the closest answer:
The closest answer to 19.625 from the given options is:
A. 19.6 square units
So, the area of sector [tex]\( AOB \)[/tex] is approximately 19.6 square units.
