Question

a circle has a central angle measuring $\frac{3\pi}{4}$ radians that intersects an arc of length 45 in. what is the length of the radius of the circle? round your answer to the nearest tenth. use 3.14 for $\pi$.
2.4 in.
19.1 in.
105.6 in.
135.0 in.

Explanation:

Step1: Recall the arc - length formula

The formula for arc length \(s\) is \(s = r\theta\), where \(s\) is the arc length, \(r\) is the radius, and \(\theta\) is the central angle in radians.
We are given \(s = 45\) in and \(\theta=\frac{3\pi}{4}\). Substitute \(\pi = 3.14\), then \(\theta=\frac{3\times3.14}{4}=\frac{9.42}{4}=2.355\).

Step2: Solve for \(r\)

From \(s = r\theta\), we can rewrite it as \(r=\frac{s}{\theta}\).
Substitute \(s = 45\) and \(\theta = 2.355\) into the formula: \(r=\frac{45}{2.355}\approx19.1\).

Answer:

19.1 in.