Question

what is the arc length of the subtending arc for an angle of 72 degrees on a circle of radius 4?
a. 8π
b. \\(\frac{4π}{5}\\)
c. \\(\frac{16π}{5}\\)
d. \\(\frac{8π}{5}\\)

Explanation:

Step1: Recall arc length formula

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

Step2: Substitute values

Given \( \theta = 72^\circ \) and \( r = 4 \). Substitute into the formula: \( s=\frac{72^\circ}{360^\circ}\times2\pi\times4 \).

Step3: Simplify the expression

First, simplify \( \frac{72}{360}=\frac{1}{5} \). Then, \( \frac{1}{5}\times2\pi\times4=\frac{8\pi}{5} \)? Wait, no, wait: Wait, \( \frac{72}{360}=\frac{1}{5} \), then \( \frac{1}{5}\times2\pi\times4=\frac{8\pi}{5} \)? Wait, no, let's recalculate. Wait, \( 72\div360 = \frac{1}{5} \), then \( 2\pi r=2\pi\times4 = 8\pi \), so \( \frac{1}{5}\times8\pi=\frac{8\pi}{5} \)? Wait, but let's check the options. Wait, option D is \( \frac{8\pi}{5} \)? Wait, no, wait the options: Option D is \( \frac{8\pi}{5} \)? Wait, let's re - check the calculation. Wait, \( \theta = 72^\circ \), \( r = 4 \). The formula can also be written as \( s = r\theta \) when \( \theta \) is in radians. Let's convert \( 72^\circ \) to radians. \( 72^\circ\times\frac{\pi}{180^\circ}=\frac{2\pi}{5} \) radians. Then arc length \( s=r\theta=4\times\frac{2\pi}{5}=\frac{8\pi}{5} \). Wait, but let's check the options again. Wait, the options are: A. \( 8\pi \), B. \( \frac{4\pi}{5} \), C. \( \frac{16\pi}{5} \), D. \( \frac{8\pi}{5} \). Wait, so the correct calculation gives \( \frac{8\pi}{5} \), which is option D? Wait, no, wait I think I made a mistake earlier. Wait, let's do it with the degree formula again. \( s=\frac{\theta}{360}\times2\pi r \). \( \theta = 72 \), \( r = 4 \). So \( \frac{72}{360}\times2\pi\times4=\frac{1}{5}\times8\pi=\frac{8\pi}{5} \). Yes, that's correct.

Answer:

D. \(\frac{8\pi}{5}\)