Question

1. find the radius of the circle in the figure to the right.

the radius of the circle is
(type an integer or a simplified fraction.)

Explanation:

Step1: Recall the arc - length formula

The arc - length formula is \(s = r\theta\), where \(s\) is the arc length, \(r\) is the radius of the circle, and \(\theta\) is the central angle in radians.

Step2: Substitute the given values into the formula

We are given that \(s = 9\pi\) and \(\theta=\frac{4\pi}{3}\). Substituting these values into the formula \(s = r\theta\), we get \(9\pi=r\times\frac{4\pi}{3}\).

Step3: Solve for \(r\)

To solve for \(r\), we can use the following steps:
First, multiply both sides of the equation \(9\pi=r\times\frac{4\pi}{3}\) by \(\frac{3}{4\pi}\).

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Cancel out the \(\pi\) terms.

$$ r = \frac{27}{4} $$

Answer:

\(\frac{27}{4}\)