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}\\)
\\(\theta = 160^\circ\\)
\\(r = 12\\) cm
options: b, a, c, d

Explanation:

Step1: Convert angle to radians

First, convert \( \theta = 160^\circ \) to radians. The formula for converting 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: \( 160\div20 = 8 \), \( 180\div20 = 9 \)? Wait, no, 160 and 180 have a common factor of 20? Wait, 160 = 20×8, 180=20×9. So \( 160^\circ=\frac{8\pi}{9} \) radians? Wait, no, wait, maybe I made a mistake. Wait, the arc length formula is \( s = r\theta \) (when \( \theta \) is in radians), and the sector area formula is \( A=\frac{1}{2}r^2\theta \) (when \( \theta \) is in radians). Alternatively, we can use the proportion: arc length is a fraction of the circumference, and sector area is a fraction of the circle's area. The fraction is \( \frac{\theta}{360^\circ} \), where \( \theta \) is in degrees.

So, circumference \( C = 2\pi r \), so arc length \( s=\frac{\theta}{360^\circ}\times2\pi r \). Sector area \( A=\frac{\theta}{360^\circ}\times\pi r^2 \).

Given \( r = 12 \) cm, \( \theta = 160^\circ \).

First, calculate arc length:

\( s=\frac{160}{360}\times2\pi\times12 \). Simplify \( \frac{160}{360}=\frac{4}{9} \)? Wait, no, 160 and 360: divide numerator and denominator by 40: \( \frac{4}{9} \)? Wait, 160÷40=4, 360÷40=9. So \( s=\frac{4}{9}\times24\pi \)? Wait, 2π×12=24π. Then \( \frac{4}{9}\times24\pi=\frac{96\pi}{9}=\frac{32\pi}{3} \). Ah, there we go. So arc length \( s=\frac{32\pi}{3} \) cm.

Now, sector area: \( A=\frac{160}{360}\times\pi r^2 \). \( r = 12 \), so \( r^2 = 144 \). So \( \frac{160}{360}\times\pi\times144 \). Simplify \( \frac{160}{360}=\frac{4}{9} \), so \( \frac{4}{9}\times144\pi = 64\pi \). So arc length is \( \frac{32\pi}{3} \), sector area is \( 64\pi \), which matches option a.

Wait, let's recheck:

Arc length: fraction of circumference. Circumference is \( 2\pi r = 24\pi \). The angle is 160 degrees, so the fraction is \( \frac{160}{360}=\frac{4}{9} \)? Wait, no, 160/360 = 4/9? Wait, 160 divided by 40 is 4, 360 divided by 40 is 9. So 160/360 = 4/9. Then arc length is \( \frac{4}{9}\times24\pi = \frac{96\pi}{9} = \frac{32\pi}{3} \). Correct.

Sector area: fraction of the circle's area. Circle area is \( \pi r^2 = 144\pi \). Fraction is \( \frac{160}{360}=\frac{4}{9} \). So sector area is \( \frac{4}{9}\times144\pi = 64\pi \). So that's option a: \( \frac{32\pi}{3}, 64\pi \).

Step2: Verify the calculations

Arc length: \( s = \frac{\theta}{360^\circ} \times 2\pi r \). \( \theta = 160^\circ \), \( r = 12 \). So \( s = \frac{160}{360} \times 2\pi \times 12 \). Simplify \( \frac{160}{360} = \frac{4}{9} \), \( 2\times12 = 24 \), so \( s = \frac{4}{9} \times 24\pi = \frac{96\pi}{9} = \frac{32\pi}{3} \). Correct.

Sector area: \( A = \frac{\theta}{360^\circ} \times \pi r^2 \). \( \theta = 160^\circ \), \( r = 12 \), so \( r^2 = 144 \). \( A = \frac{160}{360} \times \pi \times 144 \). Simplify \( \frac{160}{360} = \frac{4}{9} \), \( \frac{4}{9} \times 144\pi = 64\pi \). Correct.

So the arc length is \( \frac{32\pi}{3} \) and the sector area is \( 64\pi \), which corresponds to option a.

Answer:

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