Answer :
To find the area of sector [tex]\(AOB\)[/tex] when points [tex]\(A\)[/tex] and [tex]\(B\)[/tex] lie on a circle centered at point [tex]\(O\)[/tex], with [tex]\(OA = 5\)[/tex] and the ratio of the length of arc [tex]\(AB\)[/tex] to the circumference of the circle is [tex]\(\frac{1}{4}\)[/tex], we can follow these steps:
1. Determine the Circumference of the Circle:
The formula for the circumference of a circle is:
[tex]\[
\text{Circumference} = 2 \pi \times \text{radius}
\][/tex]
Given the radius is 5, and [tex]\(\pi = 3.14\)[/tex]:
[tex]\[
\text{Circumference} = 2 \times 3.14 \times 5 = 31.4
\][/tex]
2. Understand the Arc Length Ratio:
The problem states that the length of arc [tex]\(AB\)[/tex] is [tex]\(\frac{1}{4}\)[/tex] of the total circumference. This means:
[tex]\[
\text{Arc Length Ratio} = \frac{\text{length of arc } AB}{\text{circumference}} = \frac{1}{4}
\][/tex]
3. Calculate the Area of Sector [tex]\(AOB\)[/tex]:
The area of the sector is proportional to the length of the arc. Since the arc [tex]\(AB\)[/tex] is [tex]\(\frac{1}{4}\)[/tex] of the circumference, the area of sector [tex]\(AOB\)[/tex] is [tex]\(\frac{1}{4}\)[/tex] of the total area of the circle.
The formula for the area of a circle is:
[tex]\[
\text{Area of Circle} = \pi \times \text{radius}^2
\][/tex]
Plugging in the values:
[tex]\[
\text{Area of Circle} = 3.14 \times 5^2 = 3.14 \times 25 = 78.5
\][/tex]
Therefore, the area of sector [tex]\(AOB\)[/tex] is:
[tex]\[
\text{Area of Sector } AOB = \frac{1}{4} \times 78.5 = 19.625
\][/tex]
After rounding to match the provided options, the area of sector [tex]\(AOB\)[/tex] is closest to:
A. 19.6 square units
Thus, the correct answer is option A, 19.6 square units.
1. Determine the Circumference of the Circle:
The formula for the circumference of a circle is:
[tex]\[
\text{Circumference} = 2 \pi \times \text{radius}
\][/tex]
Given the radius is 5, and [tex]\(\pi = 3.14\)[/tex]:
[tex]\[
\text{Circumference} = 2 \times 3.14 \times 5 = 31.4
\][/tex]
2. Understand the Arc Length Ratio:
The problem states that the length of arc [tex]\(AB\)[/tex] is [tex]\(\frac{1}{4}\)[/tex] of the total circumference. This means:
[tex]\[
\text{Arc Length Ratio} = \frac{\text{length of arc } AB}{\text{circumference}} = \frac{1}{4}
\][/tex]
3. Calculate the Area of Sector [tex]\(AOB\)[/tex]:
The area of the sector is proportional to the length of the arc. Since the arc [tex]\(AB\)[/tex] is [tex]\(\frac{1}{4}\)[/tex] of the circumference, the area of sector [tex]\(AOB\)[/tex] is [tex]\(\frac{1}{4}\)[/tex] of the total area of the circle.
The formula for the area of a circle is:
[tex]\[
\text{Area of Circle} = \pi \times \text{radius}^2
\][/tex]
Plugging in the values:
[tex]\[
\text{Area of Circle} = 3.14 \times 5^2 = 3.14 \times 25 = 78.5
\][/tex]
Therefore, the area of sector [tex]\(AOB\)[/tex] is:
[tex]\[
\text{Area of Sector } AOB = \frac{1}{4} \times 78.5 = 19.625
\][/tex]
After rounding to match the provided options, the area of sector [tex]\(AOB\)[/tex] is closest to:
A. 19.6 square units
Thus, the correct answer is option A, 19.6 square units.
