Question

for #14 and 15, in circle o with radius of length r, central angle poq intercepts \\( \overparen{qp} \\). find the length of \\( \overparen{qp} \\) in terms of \\( \pi \\) for each given radian measure of \\( \angle poq \\) and value of r.

1. \\( \theta=\frac{4 \pi}{3}, r = 100 \\)
2. \\( \theta=\frac{9 \pi}{8}, r = 12 \\)

Explanation:

Step1: Recall the arc - length formula

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

Step2: Substitute the values for problem 14

Given \(\theta=\frac{4\pi}{3}\) and \(r = 100\). Substitute into the formula \(s=r\theta\):
\(s=100\times\frac{4\pi}{3}=\frac{400\pi}{3}\)

Step3: Substitute the values for problem 15

Given \(\theta=\frac{9\pi}{8}\) and \(r = 12\). Substitute into the formula \(s=r\theta\):
\(s=12\times\frac{9\pi}{8}=\frac{108\pi}{8}=\frac{27\pi}{2}\)

Answer:

1. \(\frac{400\pi}{3}\)
2. \(\frac{27\pi}{2}\)