Answer :
We are given that the circle has radius
[tex]$$r = 5,$$[/tex]
and the arc [tex]$\widehat{AB}$[/tex] represents a fraction
[tex]$$\frac{1}{4}$$[/tex]
of the total circumference.
1. First, compute the total circumference of the circle using the formula
[tex]$$C = 2\pi r.$$[/tex]
Substituting the given values, we get
[tex]$$C = 2 \times 3.14 \times 5 = 31.4.$$[/tex]
2. Since the arc is [tex]$\frac{1}{4}$[/tex] of the entire circumference, the corresponding central angle in radians is
[tex]$$\theta = \frac{1}{4} \times 2\pi.$$[/tex]
With [tex]$\pi = 3.14$[/tex], this becomes
[tex]$$\theta = \frac{1}{4} \times (2 \times 3.14) = 1.57 \text{ radians}.$$[/tex]
3. Next, the area of the sector [tex]$AOB$[/tex] is calculated using the formula
[tex]$$\text{Area} = \frac{1}{2}r^2\theta.$$[/tex]
Substitute the values we have:
[tex]$$\text{Area} = \frac{1}{2} \times 5^2 \times 1.57 = \frac{1}{2} \times 25 \times 1.57.$$[/tex]
Performing the multiplication:
[tex]$$\text{Area} = 12.5 \times 1.57 \approx 19.625.$$[/tex]
4. Rounding this value gives an area of approximately
[tex]$$19.6 \text{ square units}.$$[/tex]
Thus, the closest answer is:
A. 19.6 square units.
[tex]$$r = 5,$$[/tex]
and the arc [tex]$\widehat{AB}$[/tex] represents a fraction
[tex]$$\frac{1}{4}$$[/tex]
of the total circumference.
1. First, compute the total circumference of the circle using the formula
[tex]$$C = 2\pi r.$$[/tex]
Substituting the given values, we get
[tex]$$C = 2 \times 3.14 \times 5 = 31.4.$$[/tex]
2. Since the arc is [tex]$\frac{1}{4}$[/tex] of the entire circumference, the corresponding central angle in radians is
[tex]$$\theta = \frac{1}{4} \times 2\pi.$$[/tex]
With [tex]$\pi = 3.14$[/tex], this becomes
[tex]$$\theta = \frac{1}{4} \times (2 \times 3.14) = 1.57 \text{ radians}.$$[/tex]
3. Next, the area of the sector [tex]$AOB$[/tex] is calculated using the formula
[tex]$$\text{Area} = \frac{1}{2}r^2\theta.$$[/tex]
Substitute the values we have:
[tex]$$\text{Area} = \frac{1}{2} \times 5^2 \times 1.57 = \frac{1}{2} \times 25 \times 1.57.$$[/tex]
Performing the multiplication:
[tex]$$\text{Area} = 12.5 \times 1.57 \approx 19.625.$$[/tex]
4. Rounding this value gives an area of approximately
[tex]$$19.6 \text{ square units}.$$[/tex]
Thus, the closest answer is:
A. 19.6 square units.