Question

find the radius of the circle containing a 20° arc whose length is 15π m. the radius of the circle is □ (simplify your answer. type ar using π as needed.)

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 \(\theta( \text{in radians})=\frac{\pi}{180}\times\theta(\text{in degrees})\). Given \(\theta = 20^{\circ}\), then \(\theta=\frac{\pi}{180}\times20=\frac{\pi}{9}\) radians. And \(s = 15\pi\) m.

Step2: Solve for \(r\)

Substitute \(s = 15\pi\) and \(\theta=\frac{\pi}{9}\) into the arc - length formula \(s = r\theta\). We get \(15\pi=r\times\frac{\pi}{9}\).
To solve for \(r\), multiply both sides of the equation by \(\frac{9}{\pi}\).
\(r=\frac{15\pi\times9}{\pi}\)

Answer:

\(135\) m