Question

find the exact length of the arc intercepted by a central angle of θ on a circle of radius r. show work

1. θ = 220°, r = 6 cm

a) \frac{22}{3}π cm ≈ 23.0 cm
b) \frac{11}{9}π cm ≈ 3.8 cm
c) \frac{11}{3}π cm ≈ 11.5 cm
d) 1320π cm ≈ 4146.9 cm

Explanation:

Step1: Convert degree to radian

The formula to convert degrees to radians is \(\theta_{rad}=\theta_{deg}\times\frac{\pi}{180}\).
For \(\theta = 220^{\circ}\), \(\theta_{rad}=220\times\frac{\pi}{180}=\frac{11\pi}{9}\) radians.

Step2: Use the arc - length formula

The arc - length formula is \(s = r\theta\) (where \(s\) is the arc length, \(r\) is the radius, and \(\theta\) is the central angle in radians).
Given \(r = 6\mathrm{cm}\) and \(\theta=\frac{11\pi}{9}\), then \(s=6\times\frac{11\pi}{9}\).
Simplify \(6\times\frac{11\pi}{9}=\frac{66\pi}{9}=\frac{22\pi}{3}\mathrm{cm}\approx\frac{22\times3.14}{3}\approx23.0\mathrm{cm}\).

Answer:

A. \(\frac{22}{3}\pi\mathrm{cm}\approx23.0\mathrm{cm}\)