Select the correct answer.

Points \(A\) and \(B\) lie on a circle centered at point \(O\). If \(OA=5\) and \(\frac{\text{length of } \widehat{AB}}{\text{circumference}} = \frac{1}{4}\), what is the area of sector \(AOB\)? Use the value \(\pi = 3.14\), and choose the closest answer.

A. 19.6 square units
B. 39.3 square units
C. 7.85 square units
D. 15.7 square units



Answer :

To solve this problem, let's break it down step by step.

1. Identify the given information:
- The radius of the circle \( OA = 5 \) units.
- The fraction of the circumference represented by the arc length \( \frac{\text{length of } \widehat{AB}}{\text{circumference}} = \frac{1}{4} \).

2. Understand what is asked:
- We need to find the area of the sector \( AOB \).

3. Calculate the area of the whole circle:
- The formula for the area of a circle is \( \pi r^2 \).
- Given \( \pi = 3.14 \) and \( r = 5 \),
[tex]\[ \text{Area of the circle} = \pi \times r^2 = 3.14 \times 5^2 = 3.14 \times 25 = 78.5 \text{ square units} \][/tex]

4. Determine the fraction of the circle represented by the sector:
- The fraction given is \( \frac{1}{4} \).
- Therefore, the area of the sector \( AOB \) will be \( \frac{1}{4} \) of the area of the whole circle.

5. Calculate the area of the sector:
- Multiply the area of the circle by the fraction:
[tex]\[ \text{Area of the sector} = \frac{1}{4} \times 78.5 = 19.625 \text{ square units} \][/tex]

6. Choose the closest answer:
- Looking at the provided options, the closest one to 19.625 square units is:
[tex]\[ \boxed{19.6 \text{ square units}} \][/tex]

Therefore, the correct answer is [tex]\( \boxed{19.6 \text{ square units}} \)[/tex].

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