Question

find the radius of this circle if the arc length is \frac{18\pi}{7} and \theta = \frac{6\pi}{7} radians. 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 of \(s\) and \(\theta\)

We are given that \(s = \frac{18\pi}{7}\) and \(\theta=\frac{6\pi}{7}\). Then \(r=\frac{\frac{18\pi}{7}}{\frac{6\pi}{7}}\).

Step4: Simplify the expression

When dividing by a fraction, we multiply by its reciprocal. So \(r=\frac{18\pi}{7}\times\frac{7}{6\pi}\). The \(\pi\) terms cancel out, and \(\frac{18}{6}=3\), \(\frac{7}{7} = 1\).

Answer:

\(3\)