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\\).\
\\(\bigcirc\\) 2.4 in.\
\\(\bigcirc\\) 19.1 in.\
\\(\bigcirc\\) 105.6 in.\
\\(\bigcirc\\) 135.0 in.

Explanation:

Step1: Recall arc length formula

The formula for arc length \( s \) in radians is \( s = r\theta \), where \( r \) is the radius and \( \theta \) is the central angle in radians. We need to solve for \( r \), so rearrange the formula to \( r=\frac{s}{\theta} \).

Step2: Substitute values

Given \( s = 45 \) in and \( \theta=\frac{3\pi}{4} \). First, calculate \( \theta \) with \( \pi = 3.14 \): \( \theta=\frac{3\times3.14}{4}=\frac{9.42}{4} = 2.355 \) radians.

Step3: Calculate radius

Now substitute \( s = 45 \) and \( \theta = 2.355 \) into \( r=\frac{s}{\theta} \): \( r=\frac{45}{2.355}\approx19.1 \) in.

Answer:

B. 19.1 in.