Answer :
Sure! Let's solve this question step by step.
We're given that:
1. The radius [tex]\( OA = 5 \)[/tex].
2. The fraction of the circumference corresponding to arc [tex]\( AB \)[/tex] is [tex]\( \frac{1}{4} \)[/tex].
3. We need to find the area of sector [tex]\( AOB \)[/tex] using [tex]\( \pi = 3.14 \)[/tex].
### Step 1: Calculate the Circumference
The formula for the circumference [tex]\( C \)[/tex] of a circle is:
[tex]\[ C = 2 \pi r \][/tex]
Substitute [tex]\( r = 5 \)[/tex] and [tex]\( \pi = 3.14 \)[/tex]:
[tex]\[ C = 2 \times 3.14 \times 5 = 31.4 \][/tex]
### Step 2: Calculate the Length of Arc [tex]\( AB \)[/tex]
Since the length of arc [tex]\( AB \)[/tex] is [tex]\(\frac{1}{4}\)[/tex] of the circumference:
[tex]\[ \text{Arc length of } AB = \frac{1}{4} \times 31.4 = 7.85 \][/tex]
### Step 3: Calculate the Area of Sector [tex]\( AOB \)[/tex]
The area of a sector is given by:
[tex]\[ \text{Area of Sector } AOB = \frac{1}{2} \times r \times \text{arc length} \][/tex]
Substitute the values [tex]\( r = 5 \)[/tex] and the arc length of [tex]\( AB = 7.85 \)[/tex]:
[tex]\[ \text{Area of Sector } AOB = \frac{1}{2} \times 5 \times 7.85 = 19.625 \][/tex]
### Choosing the Closest Answer
The closest answer choice to 19.625 square units is:
[tex]\[ \text{A. } 19.6 \text{ square units} \][/tex]
So, the area of sector [tex]\( AOB \)[/tex] is approximately 19.6 square units. Therefore, the correct answer is A. 19.6 square units.
We're given that:
1. The radius [tex]\( OA = 5 \)[/tex].
2. The fraction of the circumference corresponding to arc [tex]\( AB \)[/tex] is [tex]\( \frac{1}{4} \)[/tex].
3. We need to find the area of sector [tex]\( AOB \)[/tex] using [tex]\( \pi = 3.14 \)[/tex].
### Step 1: Calculate the Circumference
The formula for the circumference [tex]\( C \)[/tex] of a circle is:
[tex]\[ C = 2 \pi r \][/tex]
Substitute [tex]\( r = 5 \)[/tex] and [tex]\( \pi = 3.14 \)[/tex]:
[tex]\[ C = 2 \times 3.14 \times 5 = 31.4 \][/tex]
### Step 2: Calculate the Length of Arc [tex]\( AB \)[/tex]
Since the length of arc [tex]\( AB \)[/tex] is [tex]\(\frac{1}{4}\)[/tex] of the circumference:
[tex]\[ \text{Arc length of } AB = \frac{1}{4} \times 31.4 = 7.85 \][/tex]
### Step 3: Calculate the Area of Sector [tex]\( AOB \)[/tex]
The area of a sector is given by:
[tex]\[ \text{Area of Sector } AOB = \frac{1}{2} \times r \times \text{arc length} \][/tex]
Substitute the values [tex]\( r = 5 \)[/tex] and the arc length of [tex]\( AB = 7.85 \)[/tex]:
[tex]\[ \text{Area of Sector } AOB = \frac{1}{2} \times 5 \times 7.85 = 19.625 \][/tex]
### Choosing the Closest Answer
The closest answer choice to 19.625 square units is:
[tex]\[ \text{A. } 19.6 \text{ square units} \][/tex]
So, the area of sector [tex]\( AOB \)[/tex] is approximately 19.6 square units. Therefore, the correct answer is A. 19.6 square units.
