Answer :
To solve this problem, we need to find the area of sector [tex]\(AOB\)[/tex] in a circle.
Here's how we can approach it:
1. Identify the Given Information:
- Radius of the circle, [tex]\(OA = 5\)[/tex].
- Relationship between the arc length of [tex]\(\widehat{AB}\)[/tex] and the diameter is given by [tex]\(\frac{\text{length of } \widehat{AB}}{\text{diameter}} = \frac{1}{4}\)[/tex].
2. Understand What’s Needed:
- We need to find the area of sector [tex]\(AOB\)[/tex].
3. Calculate the Circumference and Diameter of the Circle:
- Diameter of the circle is [tex]\(2 \times \text{radius} = 2 \times 5 = 10\)[/tex].
- Circumference is calculated as [tex]\(2 \pi \times \text{radius} = 2 \times 3.14 \times 5 = 31.4\)[/tex].
4. Find the Arc Length of [tex]\(\widehat{AB}\)[/tex]:
- Since [tex]\(\frac{\text{length of } \widehat{AB}}{\text{diameter}} = \frac{1}{4}\)[/tex], the length of [tex]\(\widehat{AB}\)[/tex] is [tex]\(\frac{1}{4} \times 10 = 2.5\)[/tex].
5. Calculate the Angle of the Sector in Degrees:
- The sector angle can be found using the proportion of the arc length to the full circumference.
- [tex]\(\frac{\text{arc length}}{\text{circumference}} = \frac{2.5}{31.4}\)[/tex].
- This ratio is equivalent to [tex]\(\frac{angle}{360^\circ}\)[/tex].
- Solving for the angle, we use [tex]\(\frac{2.5}{31.4} = \frac{angle}{360^\circ}\)[/tex], which simplifies to an angle of approximately [tex]\(90^\circ\)[/tex].
6. Calculate the Area of the Sector:
- The area of a sector is given by the formula [tex]\(\frac{\text{angle}}{360^\circ} \times \pi r^2\)[/tex].
- Substituting the known values, we get [tex]\(\frac{90^\circ}{360^\circ} \times 3.14 \times 5^2 = \frac{1}{4} \times 3.14 \times 25 = 19.625\)[/tex].
Thus, the area of sector [tex]\(AOB\)[/tex] is approximately [tex]\(19.625\)[/tex] square units. The closest answer choice is:
- A. 19.6 square units
Here's how we can approach it:
1. Identify the Given Information:
- Radius of the circle, [tex]\(OA = 5\)[/tex].
- Relationship between the arc length of [tex]\(\widehat{AB}\)[/tex] and the diameter is given by [tex]\(\frac{\text{length of } \widehat{AB}}{\text{diameter}} = \frac{1}{4}\)[/tex].
2. Understand What’s Needed:
- We need to find the area of sector [tex]\(AOB\)[/tex].
3. Calculate the Circumference and Diameter of the Circle:
- Diameter of the circle is [tex]\(2 \times \text{radius} = 2 \times 5 = 10\)[/tex].
- Circumference is calculated as [tex]\(2 \pi \times \text{radius} = 2 \times 3.14 \times 5 = 31.4\)[/tex].
4. Find the Arc Length of [tex]\(\widehat{AB}\)[/tex]:
- Since [tex]\(\frac{\text{length of } \widehat{AB}}{\text{diameter}} = \frac{1}{4}\)[/tex], the length of [tex]\(\widehat{AB}\)[/tex] is [tex]\(\frac{1}{4} \times 10 = 2.5\)[/tex].
5. Calculate the Angle of the Sector in Degrees:
- The sector angle can be found using the proportion of the arc length to the full circumference.
- [tex]\(\frac{\text{arc length}}{\text{circumference}} = \frac{2.5}{31.4}\)[/tex].
- This ratio is equivalent to [tex]\(\frac{angle}{360^\circ}\)[/tex].
- Solving for the angle, we use [tex]\(\frac{2.5}{31.4} = \frac{angle}{360^\circ}\)[/tex], which simplifies to an angle of approximately [tex]\(90^\circ\)[/tex].
6. Calculate the Area of the Sector:
- The area of a sector is given by the formula [tex]\(\frac{\text{angle}}{360^\circ} \times \pi r^2\)[/tex].
- Substituting the known values, we get [tex]\(\frac{90^\circ}{360^\circ} \times 3.14 \times 5^2 = \frac{1}{4} \times 3.14 \times 25 = 19.625\)[/tex].
Thus, the area of sector [tex]\(AOB\)[/tex] is approximately [tex]\(19.625\)[/tex] square units. The closest answer choice is:
- A. 19.6 square units
