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Answer :
To solve the problem, we need to find the area of sector [tex]\(AOB\)[/tex] on a circle with center [tex]\(O\)[/tex], where the radius [tex]\(OA\)[/tex] is 5 units and the arc length of [tex]\(AB\)[/tex] is [tex]\(\frac{1}{4}\)[/tex] of the circle's circumference.
Here's a step-by-step guide to solving the problem:
1. Find the Circumference of the Circle:
The formula for the circumference [tex]\(C\)[/tex] of a circle is given by:
[tex]\[
C = 2\pi \times \text{radius}
\][/tex]
Given that [tex]\(\pi = 3.14\)[/tex] and the radius is 5:
[tex]\[
C = 2 \times 3.14 \times 5 = 31.4
\][/tex]
2. Determine the Arc Length of [tex]\(AB\)[/tex]:
We're told that the arc length of [tex]\(AB\)[/tex] is [tex]\(\frac{1}{4}\)[/tex] of the circle's circumference. So:
[tex]\[
\text{Arc length of } \overparen{AB} = \frac{1}{4} \times 31.4 = 7.85
\][/tex]
3. Find the Angle of Sector [tex]\(AOB\)[/tex]:
The angle [tex]\(\theta\)[/tex] of the sector in radians can be found using the formula:
[tex]\[
\theta = \frac{\text{Arc length of } \overparen{AB}}{\text{Circumference}} \times 2\pi
\][/tex]
Substituting the values:
[tex]\[
\theta = \frac{7.85}{31.4} \times 2 \times 3.14 = 1.57 \text{ radians}
\][/tex]
4. Calculate the Area of the Sector [tex]\(AOB\)[/tex]:
The area [tex]\(A\)[/tex] of a sector of a circle is given by:
[tex]\[
A = \frac{\theta}{2\pi} \times \pi \times \text{radius}^2
\][/tex]
Substituting the values:
[tex]\[
A = \frac{1.57}{2 \times 3.14} \times 3.14 \times 5^2 = 19.6 \text{ square units}
\][/tex]
The closest answer choice is A. 19.6 square units.
Here's a step-by-step guide to solving the problem:
1. Find the Circumference of the Circle:
The formula for the circumference [tex]\(C\)[/tex] of a circle is given by:
[tex]\[
C = 2\pi \times \text{radius}
\][/tex]
Given that [tex]\(\pi = 3.14\)[/tex] and the radius is 5:
[tex]\[
C = 2 \times 3.14 \times 5 = 31.4
\][/tex]
2. Determine the Arc Length of [tex]\(AB\)[/tex]:
We're told that the arc length of [tex]\(AB\)[/tex] is [tex]\(\frac{1}{4}\)[/tex] of the circle's circumference. So:
[tex]\[
\text{Arc length of } \overparen{AB} = \frac{1}{4} \times 31.4 = 7.85
\][/tex]
3. Find the Angle of Sector [tex]\(AOB\)[/tex]:
The angle [tex]\(\theta\)[/tex] of the sector in radians can be found using the formula:
[tex]\[
\theta = \frac{\text{Arc length of } \overparen{AB}}{\text{Circumference}} \times 2\pi
\][/tex]
Substituting the values:
[tex]\[
\theta = \frac{7.85}{31.4} \times 2 \times 3.14 = 1.57 \text{ radians}
\][/tex]
4. Calculate the Area of the Sector [tex]\(AOB\)[/tex]:
The area [tex]\(A\)[/tex] of a sector of a circle is given by:
[tex]\[
A = \frac{\theta}{2\pi} \times \pi \times \text{radius}^2
\][/tex]
Substituting the values:
[tex]\[
A = \frac{1.57}{2 \times 3.14} \times 3.14 \times 5^2 = 19.6 \text{ square units}
\][/tex]
The closest answer choice is A. 19.6 square units.
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