# The position of an object moving along a line is given by p(t) = 5t - cos(( pi )/3t) + 2 . What is the speed of the object at t = 13 ?

May 31, 2018

$v \left(13\right) = 5 + \frac{\pi}{2 \sqrt{3}} \text{ distance per unit time}$

or

$v \left(13\right) = 5.9 \text{ distance per unit time}$

#### Explanation:

The position function is given as

$p \left(t\right) = 5 t - \cos \left(\frac{\pi}{3} t\right) + 2$

We differentiate to obtain a velocity function

$v \left(t\right) = 5 + \frac{\pi}{3} \sin \left(\frac{\pi}{3} t\right)$

Substitute $t = 13$ to find the speed at this time

$v \left(13\right) = 5 + \frac{\pi}{3} \sin \left(\frac{\pi}{3} \left(13\right)\right)$

which can be simplified to

$v \left(13\right) = 5 + \frac{\pi}{2 \sqrt{3}} \text{ distance per unit time}$

or

$v \left(13\right) = 5.9 \text{ distance per unit time}$