Answer :
To find the area of sector [tex]\(AOB\)[/tex] given the parameters of the circle and the fraction of the circumference, we can follow these steps:
1. Determine the radius of the circle:
[tex]\[
r = OA = 5 \text{ units}
\][/tex]
2. Calculate the circumference of the circle:
Since the circumference [tex]\(C\)[/tex] of a circle is given by the formula:
[tex]\[
C = 2 \pi r
\][/tex]
Substituting the given values:
[tex]\[
C = 2 \times 3.14 \times 5 = 31.4 \text{ units}
\][/tex]
3. Determine the fraction of the circumference:
It is given that:
[tex]\[
\frac{\text{length of } \hat{AB}}{\text{circumference}} = \frac{2}{4} = \frac{1}{2}
\][/tex]
4. Calculate the length of the arc [tex]\(AB\)[/tex]:
The length of the arc [tex]\(L_{AB}\)[/tex] can be found by multiplying the fraction by the total circumference:
[tex]\[
L_{AB} = \frac{1}{2} \times 31.4 = 15.7 \text{ units}
\][/tex]
5. Determine the area of the entire circle:
The area [tex]\(A\)[/tex] of a circle is given by the formula:
[tex]\[
A = \pi r^2
\][/tex]
Substituting the given values:
[tex]\[
A = 3.14 \times 5^2 = 3.14 \times 25 = 78.5 \text{ square units}
\][/tex]
6. Calculate the area of the sector [tex]\(AOB\)[/tex]:
The area of the sector [tex]\(AOB\)[/tex] can be given as a proportion of the total area of the circle, corresponding to the fraction of the arc length over the circumference:
[tex]\[
\text{Area of sector } AOB = \left( \frac{L_{AB}}{C} \right) \times A
\][/tex]
Substituting the values:
[tex]\[
\text{Area of sector } AOB = \left( \frac{15.7}{31.4} \right) \times 78.5
\][/tex]
Since [tex]\(\frac{15.7}{31.4} = \frac{1}{2}\)[/tex]:
[tex]\[
\text{Area of sector } AOB = \frac{1}{2} \times 78.5 = 39.25 \text{ square units}
\][/tex]
Given the options for the closest answer:
A. 19.6 square units
B. 39.3 square units
C. 7.85 square units
D. 15.7 square units
The closest answer to [tex]\(39.25\)[/tex] square units is:
[tex]\[
\boxed{39.3}
\][/tex]
1. Determine the radius of the circle:
[tex]\[
r = OA = 5 \text{ units}
\][/tex]
2. Calculate the circumference of the circle:
Since the circumference [tex]\(C\)[/tex] of a circle is given by the formula:
[tex]\[
C = 2 \pi r
\][/tex]
Substituting the given values:
[tex]\[
C = 2 \times 3.14 \times 5 = 31.4 \text{ units}
\][/tex]
3. Determine the fraction of the circumference:
It is given that:
[tex]\[
\frac{\text{length of } \hat{AB}}{\text{circumference}} = \frac{2}{4} = \frac{1}{2}
\][/tex]
4. Calculate the length of the arc [tex]\(AB\)[/tex]:
The length of the arc [tex]\(L_{AB}\)[/tex] can be found by multiplying the fraction by the total circumference:
[tex]\[
L_{AB} = \frac{1}{2} \times 31.4 = 15.7 \text{ units}
\][/tex]
5. Determine the area of the entire circle:
The area [tex]\(A\)[/tex] of a circle is given by the formula:
[tex]\[
A = \pi r^2
\][/tex]
Substituting the given values:
[tex]\[
A = 3.14 \times 5^2 = 3.14 \times 25 = 78.5 \text{ square units}
\][/tex]
6. Calculate the area of the sector [tex]\(AOB\)[/tex]:
The area of the sector [tex]\(AOB\)[/tex] can be given as a proportion of the total area of the circle, corresponding to the fraction of the arc length over the circumference:
[tex]\[
\text{Area of sector } AOB = \left( \frac{L_{AB}}{C} \right) \times A
\][/tex]
Substituting the values:
[tex]\[
\text{Area of sector } AOB = \left( \frac{15.7}{31.4} \right) \times 78.5
\][/tex]
Since [tex]\(\frac{15.7}{31.4} = \frac{1}{2}\)[/tex]:
[tex]\[
\text{Area of sector } AOB = \frac{1}{2} \times 78.5 = 39.25 \text{ square units}
\][/tex]
Given the options for the closest answer:
A. 19.6 square units
B. 39.3 square units
C. 7.85 square units
D. 15.7 square units
The closest answer to [tex]\(39.25\)[/tex] square units is:
[tex]\[
\boxed{39.3}
\][/tex]
