Question

find the radius of this circle if the arc length is $\frac{5\pi}{2}$ and $\theta=\frac{3\pi}{2}$ radians.

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 \(r\)

Given \(s=\frac{5\pi}{2}\) and \(\theta = \frac{3\pi}{2}\), from \(s = r\theta\), we can solve for \(r\) by \(r=\frac{s}{\theta}\).
Substitute the values: \(r=\frac{\frac{5\pi}{2}}{\frac{3\pi}{2}}\).
When dividing by a fraction, we multiply by its reciprocal: \(r=\frac{5\pi}{2}\times\frac{2}{3\pi}\).
The \(\pi\) and \(2\) terms cancel out: \(r = \frac{5}{3}\).

Answer:

\(\frac{5}{3}\)