# The position of an object moving along a line is given by p(t) = 7t - cos(( pi )/3t) + 2 . What is the speed of the object at t = 8 ?

$7.907 \frac{m}{s}$
Speed is the magnitude of velocity. Velocity is the change in position. $p ' \left(t\right) = v \left(t\right)$
$p \left(t\right) = 7 t - \cos \left(\frac{\pi}{3} t\right) + 2 \implies p ' \left(t\right) = v \left(t\right) = 7 + \frac{\pi}{3} \sin \left(\frac{\pi}{3} t\right)$
at $t = 8$ we have $v \left(8\right) = 7 + \frac{\pi}{3} \sin \left(\frac{\pi}{3} \left(8\right)\right) = 7 + \frac{\pi}{3} \sin \left(\frac{2 \pi}{3}\right) = 7 + \frac{\pi}{3} \left(\frac{\sqrt{3}}{2}\right) = 7 + \frac{\sqrt{3} \pi}{6} \approx 7.907 \frac{m}{s}$