Answer :
Sure! Let's solve the problem step-by-step:
1. Understand the problem: We are given a circle with center at point [tex]\( O \)[/tex] and points [tex]\( A \)[/tex] and [tex]\( B \)[/tex] on the circle. The radius [tex]\( OA = 5 \)[/tex]. The length of arc [tex]\(\hat{AB}\)[/tex] is [tex]\(\frac{1}{4}\)[/tex] of the circle's circumference. We need to find the area of the sector [tex]\( AOB \)[/tex].
2. Find the circumference of the circle:
- The formula for the circumference of a circle is [tex]\( 2\pi \times \text{radius} \)[/tex].
- Here, the radius [tex]\( = 5 \)[/tex].
- Using [tex]\(\pi = 3.14\)[/tex], the circumference is calculated as:
[tex]\[
\text{Circumference} = 2 \times 3.14 \times 5 = 31.4
\][/tex]
3. Calculate the area of the full circle:
- The formula for the area of a circle is [tex]\(\pi \times \text{radius}^2\)[/tex].
- Using [tex]\(\pi = 3.14\)[/tex] and the radius [tex]\( = 5 \)[/tex]:
[tex]\[
\text{Area of the full circle} = 3.14 \times 5^2 = 78.5 \text{ square units}
\][/tex]
4. Find the area of sector [tex]\( AOB \)[/tex]:
- The problem states that the length of arc [tex]\(\hat{AB}\)[/tex] is [tex]\(\frac{1}{4}\)[/tex] of the full circumference. This means the area of sector [tex]\( AOB \)[/tex] is also [tex]\(\frac{1}{4}\)[/tex] of the full circle's area.
- Calculate the area of the sector:
[tex]\[
\text{Area of sector } AOB = \frac{1}{4} \times 78.5 = 19.625 \text{ square units}
\][/tex]
5. Choose the closest answer:
- Among the given options, the closest to 19.625 is:
- A. 19.6 square units
Thus, the area of sector [tex]\( AOB \)[/tex] is approximately 19.6 square units. So, the correct answer is A. 19.6 square units.
1. Understand the problem: We are given a circle with center at point [tex]\( O \)[/tex] and points [tex]\( A \)[/tex] and [tex]\( B \)[/tex] on the circle. The radius [tex]\( OA = 5 \)[/tex]. The length of arc [tex]\(\hat{AB}\)[/tex] is [tex]\(\frac{1}{4}\)[/tex] of the circle's circumference. We need to find the area of the sector [tex]\( AOB \)[/tex].
2. Find the circumference of the circle:
- The formula for the circumference of a circle is [tex]\( 2\pi \times \text{radius} \)[/tex].
- Here, the radius [tex]\( = 5 \)[/tex].
- Using [tex]\(\pi = 3.14\)[/tex], the circumference is calculated as:
[tex]\[
\text{Circumference} = 2 \times 3.14 \times 5 = 31.4
\][/tex]
3. Calculate the area of the full circle:
- The formula for the area of a circle is [tex]\(\pi \times \text{radius}^2\)[/tex].
- Using [tex]\(\pi = 3.14\)[/tex] and the radius [tex]\( = 5 \)[/tex]:
[tex]\[
\text{Area of the full circle} = 3.14 \times 5^2 = 78.5 \text{ square units}
\][/tex]
4. Find the area of sector [tex]\( AOB \)[/tex]:
- The problem states that the length of arc [tex]\(\hat{AB}\)[/tex] is [tex]\(\frac{1}{4}\)[/tex] of the full circumference. This means the area of sector [tex]\( AOB \)[/tex] is also [tex]\(\frac{1}{4}\)[/tex] of the full circle's area.
- Calculate the area of the sector:
[tex]\[
\text{Area of sector } AOB = \frac{1}{4} \times 78.5 = 19.625 \text{ square units}
\][/tex]
5. Choose the closest answer:
- Among the given options, the closest to 19.625 is:
- A. 19.6 square units
Thus, the area of sector [tex]\( AOB \)[/tex] is approximately 19.6 square units. So, the correct answer is A. 19.6 square units.
