To find the angular speed [tex]\( \omega \)[/tex] given a central angle [tex]\(\theta\)[/tex] and a time [tex]\( t \)[/tex], we can use the formula for angular speed:
[tex]\[ \omega = \frac{\theta}{t} \][/tex]
Given:
[tex]\[ \theta = 21\pi \, \text{radians} \][/tex]
[tex]\[ t = 6 \, \text{seconds} \][/tex]
Substitute the given values into the formula:
[tex]\[ \omega = \frac{21\pi}{6} \][/tex]
Simplify the fraction:
[tex]\[ \omega = \frac{21\pi}{6} = \frac{21}{6}\pi = 3.5\pi \][/tex]
Thus, the exact value for the angular speed [tex]\(\omega\)[/tex] is:
[tex]\[ \omega = 3.5\pi \,\text{radians/second} \][/tex]
So, [tex]\(\omega = \boxed{10.995574287564276} \, \text{radians/second}\)[/tex].
