Answer :
The problem provides a circle centered at point [tex]$O$[/tex] with radius [tex]$r = 5$[/tex] and states that the arc [tex]$\widehat{AB}$[/tex] represents [tex]$\frac{1}{4}$[/tex] of the circle's circumference.
Step 1. Calculate the area of the full circle
The area of a circle is given by the formula:
[tex]$$
A = \pi r^2.
$$[/tex]
Substitute [tex]$r = 5$[/tex] and [tex]$\pi = 3.14$[/tex]:
[tex]$$
A = 3.14 \times 5^2 = 3.14 \times 25 = 78.5 \text{ square units}.
$$[/tex]
Step 2. Determine the area of sector [tex]$AOB$[/tex]
Since the arc covers [tex]$\frac{1}{4}$[/tex] of the circle's circumference, the area of the sector is [tex]$\frac{1}{4}$[/tex] of the area of the circle:
[tex]$$
\text{Area of sector} = \frac{1}{4} \times 78.5 = 19.625 \text{ square units}.
$$[/tex]
Step 3. Round the result and select the closest answer
The result, [tex]$19.625$[/tex] square units, is closest to [tex]$19.6$[/tex] square units.
Thus, the correct answer is:
A. 19.6 square units.
Step 1. Calculate the area of the full circle
The area of a circle is given by the formula:
[tex]$$
A = \pi r^2.
$$[/tex]
Substitute [tex]$r = 5$[/tex] and [tex]$\pi = 3.14$[/tex]:
[tex]$$
A = 3.14 \times 5^2 = 3.14 \times 25 = 78.5 \text{ square units}.
$$[/tex]
Step 2. Determine the area of sector [tex]$AOB$[/tex]
Since the arc covers [tex]$\frac{1}{4}$[/tex] of the circle's circumference, the area of the sector is [tex]$\frac{1}{4}$[/tex] of the area of the circle:
[tex]$$
\text{Area of sector} = \frac{1}{4} \times 78.5 = 19.625 \text{ square units}.
$$[/tex]
Step 3. Round the result and select the closest answer
The result, [tex]$19.625$[/tex] square units, is closest to [tex]$19.6$[/tex] square units.
Thus, the correct answer is:
A. 19.6 square units.
