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 [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 :

To determine the area of sector [tex]\(AOB\)[/tex], we need to follow these steps:

1. Calculate the circumference of the circle:
Given that the radius [tex]\(OA = 5\)[/tex] and [tex]\(\pi = 3.14\)[/tex], we use the formula for the circumference of a circle:
[tex]\[ \text{Circumference} = 2 \pi r \][/tex]
Substituting the given values:
[tex]\[ \text{Circumference} = 2 \times 3.14 \times 5 = 31.4 \text{ units} \][/tex]

2. Calculate the ratio for the given arc:
The problem states that the ratio of the length of arc [tex]\(\widehat{AB}\)[/tex] to the circumference is [tex]\(\frac{1}{4}\)[/tex].

3. Determine the area of the sector:
The area [tex]\(A\)[/tex] of the sector [tex]\(AOB\)[/tex] can be determined using the proportion of the arc length to the entire circumference. Because the ratio is [tex]\(\frac{1}{4}\)[/tex], the area of the sector will be [tex]\(\frac{1}{4}\)[/tex] of the circle's total area.

The total area of the circle is calculated using:
[tex]\[ \text{Area of the circle} = \pi r^2 \][/tex]
Substituting the given values:
[tex]\[ \text{Area of the circle} = 3.14 \times 5^2 = 3.14 \times 25 = 78.5 \text{ square units} \][/tex]

Therefore, the area of sector [tex]\(AOB\)[/tex] is:
[tex]\[ \text{Area of sector } AOB = \frac{1}{4} \times \text{Area of the circle} = \frac{1}{4} \times 78.5 = 19.625 \text{ square units} \][/tex]

Among the given options, the closest answer to 19.625 square units is:
A. 19.6 square units

Therefore, the correct answer is:
A. 19.6 square units