Question

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

Explanation:

Step1: Recall the arc - length formula

The formula for the length of an arc \(s\) 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. We know that \(\theta\) (in radians) \(=\frac{\pi}{180}\times\theta\) (in degrees). Given \(\theta = 300^{\circ}\), then \(\theta=\frac{300\pi}{180}=\frac{5\pi}{3}\) radians.

Step2: Substitute the values into the arc - length formula

We are given \(r = 50\) meters. Using the formula \(s=r\theta\), substitute \(r = 50\) and \(\theta=\frac{5\pi}{3}\). Then \(s=50\times\frac{5\pi}{3}=\frac{250\pi}{3}\) meters.

Answer:

a. \(\frac{250\pi}{3}\) meters