Answer :
We are given that [tex]\(O A = 5\)[/tex] and that the arc [tex]\(\widehat{A B}\)[/tex] represents [tex]\(\frac{1}{4}\)[/tex] of the whole circumference of the circle. The steps to find the area of sector [tex]\(AOB\)[/tex] are as follows:
1. Calculate the area of the entire circle using the formula:
[tex]$$
\text{Area} = \pi r^2
$$[/tex]
Substituting [tex]\(r = 5\)[/tex] and [tex]\(\pi = 3.14\)[/tex]:
[tex]$$
\text{Area} = 3.14 \times (5)^2 = 3.14 \times 25 = 78.5 \text{ square units}
$$[/tex]
2. Since the arc [tex]\(\widehat{A B}\)[/tex] is [tex]\(\frac{1}{4}\)[/tex] of the entire circle's circumference, the central angle corresponding to this arc also represents [tex]\(\frac{1}{4}\)[/tex] of the full circle. Therefore, the area of sector [tex]\(AOB\)[/tex] is:
[tex]$$
\text{Area of sector } AOB = \frac{1}{4} \times \text{Area of circle} = \frac{1}{4} \times 78.5 = 19.625 \text{ square units}
$$[/tex]
3. Rounding [tex]\(19.625\)[/tex] to the closest option, we have approximately [tex]\(19.6\)[/tex] square units.
Thus, the area of sector [tex]\(AOB\)[/tex] is approximately [tex]\(19.6\)[/tex] square units, which corresponds to answer choice:
A. 19.6 square units.
1. Calculate the area of the entire circle using the formula:
[tex]$$
\text{Area} = \pi r^2
$$[/tex]
Substituting [tex]\(r = 5\)[/tex] and [tex]\(\pi = 3.14\)[/tex]:
[tex]$$
\text{Area} = 3.14 \times (5)^2 = 3.14 \times 25 = 78.5 \text{ square units}
$$[/tex]
2. Since the arc [tex]\(\widehat{A B}\)[/tex] is [tex]\(\frac{1}{4}\)[/tex] of the entire circle's circumference, the central angle corresponding to this arc also represents [tex]\(\frac{1}{4}\)[/tex] of the full circle. Therefore, the area of sector [tex]\(AOB\)[/tex] is:
[tex]$$
\text{Area of sector } AOB = \frac{1}{4} \times \text{Area of circle} = \frac{1}{4} \times 78.5 = 19.625 \text{ square units}
$$[/tex]
3. Rounding [tex]\(19.625\)[/tex] to the closest option, we have approximately [tex]\(19.6\)[/tex] square units.
Thus, the area of sector [tex]\(AOB\)[/tex] is approximately [tex]\(19.6\)[/tex] square units, which corresponds to answer choice:
A. 19.6 square units.
