Question

find the length of \\( \overparen{qp} \\) in terms of \\( \pi \\).

1. \\( \theta=\frac{\pi}{3}, r = 3 \\)
2. \\( \theta=\frac{\pi}{2}, r = 4 \\)
3. \\( \theta=60^{circ}, r = 18 \\)
4. \\( \theta=100^{circ}, r = 27 \\)

Explanation:

Step1: Recall the arc - length formula

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

Step2: Convert degree to radian (if necessary)

For problem 17: \(\theta=\frac{\pi}{6}\), \(r = 3\).
Substitute into the formula \(s=r\theta\).
\(s=3\times\frac{\pi}{6}\)
Simplify the expression: \(s=\frac{\pi}{2}\)
For problem 18: \(\theta=\frac{\pi}{2}\), \(r = 4\)
Substitute into the formula \(s=r\theta\).
\(s = 4\times\frac{\pi}{2}\)
Simplify the expression: \(s = 2\pi\)
For problem 19: First, convert \(\theta = 60^{\circ}\) to radians. \(\theta=60\times\frac{\pi}{180}=\frac{\pi}{3}\), \(r = 18\)
Substitute into the formula \(s=r\theta\).
\(s=18\times\frac{\pi}{3}\)
Simplify the expression: \(s = 6\pi\)
For problem 20: First, convert \(\theta = 100^{\circ}\) to radians. \(\theta=100\times\frac{\pi}{180}=\frac{5\pi}{9}\), \(r = 27\)
Substitute into the formula \(s=r\theta\).
\(s=27\times\frac{5\pi}{9}\)
Simplify the expression: \(s = 15\pi\)

Answer:

1. \(\frac{\pi}{2}\)
2. \(2\pi\)
3. \(6\pi\)
4. \(15\pi\)