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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]$\frac{\text{length of } \hat{AB}}{\text{circumference}} = \frac{2}{4}$[/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 find the area of sector [tex]\(AOB\)[/tex] given the parameters of the circle and the fraction of the circumference, we can follow these steps:

1. Determine the radius of the circle:
[tex]\[ r = OA = 5 \text{ units} \][/tex]

2. Calculate the circumference of the circle:
Since the circumference [tex]\(C\)[/tex] of a circle is given by the formula:
[tex]\[ C = 2 \pi r \][/tex]
Substituting the given values:
[tex]\[ C = 2 \times 3.14 \times 5 = 31.4 \text{ units} \][/tex]

3. Determine the fraction of the circumference:
It is given that:
[tex]\[ \frac{\text{length of } \hat{AB}}{\text{circumference}} = \frac{2}{4} = \frac{1}{2} \][/tex]

4. Calculate the length of the arc [tex]\(AB\)[/tex]:
The length of the arc [tex]\(L_{AB}\)[/tex] can be found by multiplying the fraction by the total circumference:
[tex]\[ L_{AB} = \frac{1}{2} \times 31.4 = 15.7 \text{ units} \][/tex]

5. Determine the area of the entire circle:
The area [tex]\(A\)[/tex] of a circle is given by the formula:
[tex]\[ A = \pi r^2 \][/tex]
Substituting the given values:
[tex]\[ A = 3.14 \times 5^2 = 3.14 \times 25 = 78.5 \text{ square units} \][/tex]

6. Calculate the area of the sector [tex]\(AOB\)[/tex]:
The area of the sector [tex]\(AOB\)[/tex] can be given as a proportion of the total area of the circle, corresponding to the fraction of the arc length over the circumference:
[tex]\[ \text{Area of sector } AOB = \left( \frac{L_{AB}}{C} \right) \times A \][/tex]
Substituting the values:
[tex]\[ \text{Area of sector } AOB = \left( \frac{15.7}{31.4} \right) \times 78.5 \][/tex]
Since [tex]\(\frac{15.7}{31.4} = \frac{1}{2}\)[/tex]:
[tex]\[ \text{Area of sector } AOB = \frac{1}{2} \times 78.5 = 39.25 \text{ square units} \][/tex]

Given the options for the closest answer:
A. 19.6 square units
B. 39.3 square units
C. 7.85 square units
D. 15.7 square units

The closest answer to [tex]\(39.25\)[/tex] square units is:
[tex]\[ \boxed{39.3} \][/tex]

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