Question

find the radius of this circle if the arc length is \\( \frac { 10 \pi } { 3 } \\) and \\( \theta = \frac { 2 \pi } { 3 } \\) radians. s \\( \theta \\) r \\( r = \\) ?

Explanation:

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.

Step2: Solve the formula for \(r\)

From \(s = r\theta\), we can isolate \(r\) by dividing both sides of the equation by \(\theta\). So \(r=\frac{s}{\theta}\).

Step3: Substitute the given values

We are given that \(s = \frac{10\pi}{3}\) and \(\theta=\frac{2\pi}{3}\). Then \(r=\frac{\frac{10\pi}{3}}{\frac{2\pi}{3}}\).
When dividing by a fraction, we multiply by its reciprocal: \(r=\frac{10\pi}{3}\times\frac{3}{2\pi}\).
The \(\pi\) terms cancel out (\(\frac{\pi}{\pi} = 1\)), and \(\frac{10}{3}\times\frac{3}{2}=\frac{10\times3}{3\times2}\). The \(3\) terms cancel out (\(\frac{3}{3}=1\)), and \(\frac{10}{2}=5\).

Answer:

\(5\)