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 the 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 the given values

We know that \( r = 6 \) and \( \theta=48^{\circ} \). Substituting these values into the formula, we get \( L=\frac{48^{\circ}}{360^{\circ}}\times2\pi\times6 \).

Step3: Simplify the expression

First, simplify \( \frac{48}{360}=\frac{4}{30}=\frac{2}{15} \). Then, \( 2\times6 = 12 \). So the expression becomes \( L=\frac{2}{15}\times12\pi \). Simplify \( \frac{2\times12}{15}\pi=\frac{24}{15}\pi=\frac{8}{5}\pi \). If we use \( \pi = 3.14 \), then \( L=\frac{8}{5}\times3.14 = 1.6\times3.14 = 5.024 \).

Answer:

The exact answer in terms of \( \pi \) is \( \frac{8}{5}\pi \) (or \( 1.6\pi \)) and the decimal answer is \( 5.024 \). If we take the exact form: \( \boldsymbol{\frac{8}{5}\pi} \) (or decimal: \( \boldsymbol{5.024} \))