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Answer :
We are given that:
- The radius of the circle is [tex]$5$[/tex].
- The ratio of the length of arc [tex]$\widehat{AB}$[/tex] to the entire circumference is [tex]$\frac{1}{4}$[/tex].
- We use [tex]$\pi = 3.14$[/tex].
Step 1. Calculate the area of the entire circle.
The area of a circle is given by:
[tex]$$
\text{Area}_{\text{circle}} = \pi r^2
$$[/tex]
Substituting the given values:
[tex]$$
\text{Area}_{\text{circle}} = 3.14 \times 5^2 = 3.14 \times 25 = 78.5 \text{ square units}
$$[/tex]
Step 2. Determine the area of sector [tex]$AOB$[/tex].
Since the sector represents [tex]$\frac{1}{4}$[/tex] of the entire circle, its area is:
[tex]$$
\text{Area}_{\text{sector}} = \frac{1}{4} \times 78.5 = 19.625 \text{ square units}
$$[/tex]
Rounding [tex]$19.625$[/tex] to a value close to the provided choices, the area of sector [tex]$AOB$[/tex] is approximately [tex]$19.6$[/tex] square units.
Final Answer:
[tex]$\boxed{19.6 \text{ square units}}$[/tex]
- The radius of the circle is [tex]$5$[/tex].
- The ratio of the length of arc [tex]$\widehat{AB}$[/tex] to the entire circumference is [tex]$\frac{1}{4}$[/tex].
- We use [tex]$\pi = 3.14$[/tex].
Step 1. Calculate the area of the entire circle.
The area of a circle is given by:
[tex]$$
\text{Area}_{\text{circle}} = \pi r^2
$$[/tex]
Substituting the given values:
[tex]$$
\text{Area}_{\text{circle}} = 3.14 \times 5^2 = 3.14 \times 25 = 78.5 \text{ square units}
$$[/tex]
Step 2. Determine the area of sector [tex]$AOB$[/tex].
Since the sector represents [tex]$\frac{1}{4}$[/tex] of the entire circle, its area is:
[tex]$$
\text{Area}_{\text{sector}} = \frac{1}{4} \times 78.5 = 19.625 \text{ square units}
$$[/tex]
Rounding [tex]$19.625$[/tex] to a value close to the provided choices, the area of sector [tex]$AOB$[/tex] is approximately [tex]$19.6$[/tex] square units.
Final Answer:
[tex]$\boxed{19.6 \text{ square units}}$[/tex]
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