Answer :
Alright, let's tackle the problem step-by-step!
1. Understand the Problem:
- We have a circle with center at [tex]\( O \)[/tex] and radius [tex]\( OA = 5 \)[/tex].
- Points [tex]\( A \)[/tex] and [tex]\( B \)[/tex] lie on this circle.
- The arc that spans from [tex]\( A \)[/tex] to [tex]\( B \)[/tex] has a length which is one-fourth of the circle's circumference.
- We need to find the area of the sector [tex]\( AOB \)[/tex].
2. Calculate the Circumference:
- The formula for the circumference of a circle is [tex]\( 2 \pi r \)[/tex].
- Here, [tex]\( r = 5 \)[/tex] and [tex]\( \pi = 3.14 \)[/tex].
- So, the circumference is:
[tex]\[
2 \times 3.14 \times 5 = 31.4
\][/tex]
3. Find the Length of Arc [tex]\( AB \)[/tex]:
- It’s given that the length of arc [tex]\( AB \)[/tex] is one-fourth of the circumference.
- Thus, the length of [tex]\( AB \)[/tex] is:
[tex]\[
\frac{1}{4} \times 31.4 = 7.85
\][/tex]
4. Calculate the Sector Angle:
- The length of an arc is related to the sector angle by the formula:
[tex]\[
\text{Sector Angle (in radians)} = \frac{\text{Arc Length}}{\text{Circumference}} \times 2\pi
\][/tex]
- Plugging in the values we have:
[tex]\[
\frac{7.85}{31.4} \times 2 \times 3.14 = 1.57 \, \text{radians}
\][/tex]
5. Determine the Area of Sector [tex]\( AOB \)[/tex]:
- The area of a sector is given by:
[tex]\[
\text{Area} = \frac{1}{2} \times r^2 \times \text{Sector Angle}
\][/tex]
- Use [tex]\( r = 5 \)[/tex] and the sector angle [tex]\( 1.57 \)[/tex] radians:
[tex]\[
\frac{1}{2} \times 5^2 \times 1.57 = 19.625
\][/tex]
6. Choose the Closest Answer:
- The calculated area is [tex]\( 19.625 \)[/tex] square units.
- The closest answer from the given options is:
- A. 19.6 square units
Thus, the area of sector [tex]\( AOB \)[/tex] is approximately 19.6 square units.
1. Understand the Problem:
- We have a circle with center at [tex]\( O \)[/tex] and radius [tex]\( OA = 5 \)[/tex].
- Points [tex]\( A \)[/tex] and [tex]\( B \)[/tex] lie on this circle.
- The arc that spans from [tex]\( A \)[/tex] to [tex]\( B \)[/tex] has a length which is one-fourth of the circle's circumference.
- We need to find the area of the sector [tex]\( AOB \)[/tex].
2. Calculate the Circumference:
- The formula for the circumference of a circle is [tex]\( 2 \pi r \)[/tex].
- Here, [tex]\( r = 5 \)[/tex] and [tex]\( \pi = 3.14 \)[/tex].
- So, the circumference is:
[tex]\[
2 \times 3.14 \times 5 = 31.4
\][/tex]
3. Find the Length of Arc [tex]\( AB \)[/tex]:
- It’s given that the length of arc [tex]\( AB \)[/tex] is one-fourth of the circumference.
- Thus, the length of [tex]\( AB \)[/tex] is:
[tex]\[
\frac{1}{4} \times 31.4 = 7.85
\][/tex]
4. Calculate the Sector Angle:
- The length of an arc is related to the sector angle by the formula:
[tex]\[
\text{Sector Angle (in radians)} = \frac{\text{Arc Length}}{\text{Circumference}} \times 2\pi
\][/tex]
- Plugging in the values we have:
[tex]\[
\frac{7.85}{31.4} \times 2 \times 3.14 = 1.57 \, \text{radians}
\][/tex]
5. Determine the Area of Sector [tex]\( AOB \)[/tex]:
- The area of a sector is given by:
[tex]\[
\text{Area} = \frac{1}{2} \times r^2 \times \text{Sector Angle}
\][/tex]
- Use [tex]\( r = 5 \)[/tex] and the sector angle [tex]\( 1.57 \)[/tex] radians:
[tex]\[
\frac{1}{2} \times 5^2 \times 1.57 = 19.625
\][/tex]
6. Choose the Closest Answer:
- The calculated area is [tex]\( 19.625 \)[/tex] square units.
- The closest answer from the given options is:
- A. 19.6 square units
Thus, the area of sector [tex]\( AOB \)[/tex] is approximately 19.6 square units.
