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Select the correct answer.

Points [tex]$A$[/tex] and [tex]$B$[/tex] lie on a circle centered at point [tex]$O$[/tex]. If [tex]$OA = 5$[/tex] and [tex]$\frac{\text{length of } \hat{AB}}{\text{circumference}} = \frac{1}{4}$[/tex], what is the area of sector [tex]$AOB$[/tex]? Use the value [tex]$\pi = 3.14$[/tex], 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 :

GinnyAnswer
To solve this problem, we need to determine the area of sector [tex]\(AOB\)[/tex] in a circle with a given radius and fractional arc length. Here is the detailed step-by-step solution:

1. Determine the circumference of the circle:

The circumference [tex]\(C\)[/tex] of a circle is given by the formula:
[tex]\[ C = 2 \pi r \][/tex]

Given the radius [tex]\(r = 5\)[/tex] units and [tex]\(\pi = 3.14\)[/tex]:
[tex]\[ C = 2 \cdot 3.14 \cdot 5 = 31.4 \text{ units} \][/tex]

2. Find the length of arc [tex]\(AB\)[/tex]:

The problem states that the length of arc [tex]\(AB\)[/tex] is [tex]\(\frac{1}{4}\)[/tex] of the circumference. Thus:
[tex]\[ \text{Length of arc } AB = \frac{1}{4} \cdot 31.4 = 7.85 \text{ units} \][/tex]

3. Calculate the area of sector [tex]\(AOB\)[/tex]:

The area of a sector of a circle is given by the formula:
[tex]\[ \text{Area of sector } = \frac{1}{2} r s \][/tex]
where [tex]\(r\)[/tex] is the radius and [tex]\(s\)[/tex] is the arc length.

Given [tex]\(r = 5\)[/tex] units and [tex]\(s = 7.85\)[/tex] units:
[tex]\[ \text{Area of sector } AOB = \frac{1}{2} \cdot 5 \cdot 7.85 = 19.625 \text{ square units} \][/tex]

4. Choose the closest answer:

From the given options:
[tex]\[ \text{A. } 19.6 \text{ square units} \][/tex]

The closest answer to 19.625 square units is indeed:
[tex]\[ \boxed{19.6} \text{ square units} \][/tex]

Therefore, the correct answer is:
[tex]\[ \boxed{19.6} \][/tex]

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