Question

if the circumference of a circle is 50 meters, what is the radius?
a. 25 meters
b. \\( \frac { 50 } { 2 } \\) meters
c. 10 meters
d. \\( \frac { 100 } { pi } \\) meters
if a sector has a central angle of \\( 120 ^ { \circ } \\) and a radius of 3 meters, what is the area of this sector?
a. 6\\( \pi \\) square meters
b. 3\\( \pi \\) square meters
c. 9\\( \pi \\) square meters
d. 2\\( \pi \\) square meters
if a sector has an arc length of \\( 8 \pi \\) meters and a radius of 8 meters, what is the central angle in radians?
a. \\( \frac { \pi } { 2 } \\) radians
b. 2\\( \pi \\) radians
c. 2 radians
d. \\( \pi \\) radians
if a circle has a radius of 7 meters and a central angle of \\( 60 ^ { \circ } \\), what is the length of the arc?
a. \\( \frac { 7 \pi } { 3 } \\) meters
b. 14\\( \pi \\) meters
c. 7 meters
d. \\( \frac { \pi } { 3 } \\) meters

Explanation:

First Question

Step1: Recall the circumference formula

The formula for the circumference of a circle is \(C = 2\pi r\). Given \(C=50\), we can solve for \(r\).

Step2: Solve for \(r\)

From \(C = 2\pi r\), we get \(r=\frac{C}{2\pi}\). Substituting \(C = 50\), \(r=\frac{50}{2\pi}=\frac{25}{\pi}\)

Second Question

Step1: Recall the sector - area formula

The formula for the area of a sector is \(A=\frac{\theta}{360}\times\pi r^{2}\), where \(\theta\) is the central angle in degrees and \(r\) is the radius. Given \(\theta = 120^{\circ}\) and \(r = 3\)

Step2: Substitute values into the formula

\(A=\frac{120}{360}\times\pi\times3^{2}=\frac{1}{3}\times\pi\times9 = 3\pi\)

Third Question

Step1: Recall the arc - length formula

The formula for arc length is \(s=r\theta\), where \(s\) is the arc length, \(r\) is the radius, and \(\theta\) is the central angle in radians. Given \(s = 8\pi\) and \(r = 8\)

Step2: Solve for \(\theta\)

From \(s=r\theta\), we get \(\theta=\frac{s}{r}\). Substituting \(s = 8\pi\) and \(r = 8\), \(\theta=\pi\)

Fourth Question

Step1: Convert the angle to radians

First, convert \(\theta=60^{\circ}\) to radians. \(\theta = 60\times\frac{\pi}{180}=\frac{\pi}{3}\) radians. Given \(r = 7\)

Step2: Use the arc - length formula \(s=r\theta\)

Substitute \(r = 7\) and \(\theta=\frac{\pi}{3}\) into \(s=r\theta\). \(s=7\times\frac{\pi}{3}=\frac{7\pi}{3}\)

Answer:

1. B. \(\frac{25}{\pi}\) meters
2. B. \(3\pi\) square meters
3. D. \(\pi\) radians
4. A. \(\frac{7\pi}{3}\) meters