Question

a circle has a radius of 6. an arc in this circle has a central angle of 48°. what is the length of the arc? either enter an exact answer in terms of π or use 3.14 for π and enter your answer as a decimal.

Explanation:

Step1: Recall arc length formula

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

Step2: Substitute given values

Given \( r = 6 \) and \( \theta=48^{\circ} \), substitute into the formula:
\( L=\frac{48^{\circ}}{360^{\circ}}\times2\pi\times6 \)
First, simplify \( \frac{48}{360}=\frac{4}{30}=\frac{2}{15} \), and \( 2\times6 = 12 \). So \( L=\frac{2}{15}\times12\pi=\frac{24\pi}{15}=\frac{8\pi}{5} \).
If we use \( \pi = 3.14 \), then \( L=\frac{8\times3.14}{5}=\frac{25.12}{5} = 5.024 \).

Answer:

In terms of \( \pi \), the arc length is \( \frac{8}{5}\pi \) (or \( 1.6\pi \)); as a decimal (using \( \pi = 3.14 \)), it is \( 5.024 \).