Question

for #16 and 17, in circle o with radius of length r, central angle poq intercepts qp. find the length of qp in terms of π for each given degree measure of ∠poq and value of r.

1. θ = 175°, r = 20
2. θ = 90°, r = 50

Explanation:

Step1: Recall the arc - length formula

The formula for the length of an arc \(s\) is \(s=\frac{\theta}{360}\times2\pi r\), where \(\theta\) is the central angle in degrees and \(r\) is the radius of the circle.

Step2: Solve for problem 16

Given \(\theta = 175^{\circ}\) and \(r = 20\).
Substitute into the formula: \(s=\frac{175}{360}\times2\pi\times20\).
First, simplify \(\frac{175}{360}=\frac{35}{72}\).
Then \(s=\frac{35}{72}\times40\pi=\frac{350\pi}{9}\).

Step3: Solve for problem 17

Given \(\theta = 90^{\circ}\) and \(r = 50\).
Substitute into the formula: \(s=\frac{90}{360}\times2\pi\times50\).
Since \(\frac{90}{360}=\frac{1}{4}\), then \(s=\frac{1}{4}\times100\pi = 25\pi\).

Answer:

1. \(\frac{350\pi}{9}\)
2. \(25\pi\)