Question

a circular garden has a radius of 12 meters. if you want to fence a sector with a central angle of 90°, what is the length of the fence along the arc?
a. 12π meters
b. 6π meters
c. 12 meters
d. 24 meters

a runner completes a 300° arc on a circular track of radius 50 meters. how long is the arc?
a. \frac{250π}{3} meters
b. \frac{300π}{3} meters
c. \frac{150π}{3} meters
d. 75π meters

which of the following describes the radius of a circle?
a. the line segment through the center connecting two points on the circle
b. the perimeter of the circle
c. the line segment from the center to any point on the circle
d. the line segment connecting two points on the circle

what is the radius of a circle if an arc of 90° has a length of 5π meters?
a. 10 meters
b. 5π meters
c. 20 meters
d. 5 meters

a circular track has a radius of 60 meters. what is the circumference of the track?
a. 240π meters
b. 60π meters
c. 120π meters
d. 130π meters

which of the following formulas is used to calculate the length of an arc in radians?
a. l = \frac{1}{2}θr²
b. l = 2πθ
c. l = θr²
d. l = θr

Explanation:

Step1: Recall the arc - length formula

The formula for the length of an arc \(L\) is \(L=\frac{\theta}{360}\times2\pi r\) (when \(\theta\) is in degrees), where \(\theta\) is the central angle and \(r\) is the radius of the circle.

Step2: Solve the first problem

For the circular garden with \(r = 12\) meters and \(\theta=90^{\circ}\), substitute into the formula:
\(L=\frac{90}{360}\times2\pi\times12\)
\(=\frac{1}{4}\times24\pi\)
\( = 6\pi\) meters.

Step3: Solve the second problem

For the runner on a circular track with \(r = 50\) meters and \(\theta = 300^{\circ}\), substitute into the formula:
\(L=\frac{300}{360}\times2\pi\times50\)
\(=\frac{5}{6}\times100\pi\)
\(=\frac{250\pi}{3}\) meters.

Step4: Define the radius

The radius of a circle is the line segment from the center to any point on the circle.

Step5: Solve for the radius in the fourth problem

Given \(L = 5\pi\) meters and \(\theta=90^{\circ}\), use \(L=\frac{\theta}{360}\times2\pi r\).
\(5\pi=\frac{90}{360}\times2\pi r\)
\(5\pi=\frac{1}{4}\times2\pi r\)
\(5\pi=\frac{\pi r}{2}\)
Multiply both sides by \(2\): \(10\pi=\pi r\), so \(r = 10\) meters.

Step6: Calculate the circumference

The formula for the circumference \(C\) of a circle is \(C = 2\pi r\). For \(r=60\) meters, \(C=2\pi\times60=120\pi\) meters.

Step7: Recall the arc - length formula in radians

The formula for the length of an arc \(L\) when the angle \(\theta\) is in radians is \(L = r\theta\).

Answer:

1. B. \(6\pi\) meters
2. A. \(\frac{250\pi}{3}\) meters
3. C. The line segment from the center to any point on the circle
4. A. \(10\) meters
5. A. \(120\pi\) meters
6. D. \(L = r\theta\)