Question

1. find the length of the circular arc and then the area of the sector outlined in the picture.

a) \\(\frac{32\pi}{3}\\), \\(64\pi\\) b) \\(64\pi, \frac{32\pi}{3}\\) c) \\(24\pi, 144\pi\\) d) \\(\frac{3\pi}{64}, \frac{\pi}{128}\\)
(image of a circle with center c, radius r = 12 cm, central angle \\(\theta = 160^\circ\\))
options: d, c, a, b

Explanation:

Step1: Convert angle to radians

First, convert \( \theta = 160^\circ \) to radians. The formula to convert degrees to radians is \( \theta_{\text{rad}}=\theta_{\text{deg}}\times\frac{\pi}{180} \). So, \( 160^\circ\times\frac{\pi}{180}=\frac{8\pi}{9} \)? Wait, no, wait. Wait, the radius \( r = 12 \) cm. Wait, maybe I made a mistake. Wait, let's recalculate: \( 160\times\frac{\pi}{180}=\frac{8\pi}{9} \)? Wait, no, \( 160\div20 = 8 \), \( 180\div20 = 9 \), so \( \frac{8\pi}{9} \)? Wait, but let's check the arc length formula: \( s = r\theta \) (where \( \theta \) is in radians) and sector area \( A=\frac{1}{2}r^2\theta \) ( \( \theta \) in radians). Wait, maybe I miscalculated the angle. Wait, no, let's check the options. Option A is \( \frac{32\pi}{3}, 64\pi \)? Wait, no, wait the radius is 12. Wait, let's do it again.

Wait, \( \theta = 160^\circ \). Convert to radians: \( 160\times\frac{\pi}{180}=\frac{8\pi}{9} \)? Wait, no, \( 160\div 20 = 8 \), \( 180\div20 = 9 \), so \( \frac{8\pi}{9} \). Then arc length \( s = r\theta = 12\times\frac{8\pi}{9}=\frac{96\pi}{9}=\frac{32\pi}{3} \). Then sector area \( A=\frac{1}{2}r^2\theta=\frac{1}{2}\times12^2\times\frac{8\pi}{9}=\frac{1}{2}\times144\times\frac{8\pi}{9}=72\times\frac{8\pi}{9}=8\times8\pi = 64\pi \). Oh! So that matches option A.

Step2: Verify arc length and sector area

Arc length formula: \( s = r\theta \) ( \( \theta \) in radians). \( r = 12 \), \( \theta = 160^\circ=\frac{8\pi}{9} \) radians. So \( s = 12\times\frac{8\pi}{9}=\frac{96\pi}{9}=\frac{32\pi}{3} \). Sector area formula: \( A=\frac{1}{2}r^2\theta \). \( r = 12 \), so \( r^2 = 144 \). Then \( A=\frac{1}{2}\times144\times\frac{8\pi}{9}=72\times\frac{8\pi}{9}=64\pi \). So the arc length is \( \frac{32\pi}{3} \) and sector area is \( 64\pi \), which is option A.

Answer:

A. \( \frac{32\pi}{3}, 64\pi \)