Question

for a circle with a radius of 5 meters, what is the length of an arc that subtends a 30° angle?
a. 10π meters
b. 5π meters
c. $\frac{5pi}{6}$ meters
d. $\frac{10pi}{3}$ meters

if a sector has a central angle of 120° and a radius of 3 meters, what is the area of this sector?
a. 2π square meters
b. 6π square meters
c. 3π square meters
d. 9π square meters

what is the area of a sector with a radius of 10 meters and a central angle of 45°?
a. $\frac{25pi}{2}$ square meters
b. 50π square meters
c. $\frac{10pi}{2}$ square meters
d. 25π square meters

Explanation:

Step1: Recall arc - length formula

The formula for the length of an arc $s$ of a circle is $s = r\theta$, where $r$ is the radius of the circle and $\theta$ is the central - angle in radians. First, convert the angle from degrees to radians. Given $\theta = 30^{\circ}$, and we know that to convert degrees to radians, we use the conversion factor $\theta_{rad}=\theta_{deg}\times\frac{\pi}{180}$. So, $\theta = 30\times\frac{\pi}{180}=\frac{\pi}{6}$ radians, and $r = 5$ meters. Then $s=r\theta=5\times\frac{\pi}{6}=\frac{5\pi}{6}$ meters.

Step2: Recall sector - area formula

The formula for the area of a sector $A$ of a circle is $A=\frac{\theta}{360}\times\pi r^{2}$, where $\theta$ is the central - angle in degrees and $r$ is the radius of the circle.
For the second question, $\theta = 120^{\circ}$ and $r = 3$ meters. Then $A=\frac{120}{360}\times\pi\times3^{2}=\frac{1}{3}\times\pi\times9 = 3\pi$ square meters.

Step3: Apply sector - area formula again

For the third question, $\theta = 45^{\circ}$ and $r = 10$ meters. Then $A=\frac{45}{360}\times\pi\times10^{2}=\frac{1}{8}\times\pi\times100=\frac{25\pi}{2}$ square meters.

Answer:

1. C. $\frac{5\pi}{6}$ meters
2. C. $3\pi$ square meters
3. A. $\frac{25\pi}{2}$ square meters