# The position of an object moving along a line is given by p(t) = 4t - sin(( pi )/4t) . What is the speed of the object at t = 1 ?

Sep 27, 2017

$4 - \frac{\pi}{4 \sqrt{2}}$

#### Explanation:

Position of an object $p \left(t\right) = 4 t - \sin \left(\frac{\pi}{4} t\right)$

Speed is defined as the rate of change in position of object with respect to time.
To find the speed we need to differentiate $p \left(t\right)$ with respect to $t$.
speed $= \frac{d}{\mathrm{dt}} \left(p \left(t\right)\right)$
$v = \frac{d}{\mathrm{dt}} \left(p \left(t\right)\right) = \frac{d}{\mathrm{dt}} \left(4 t - \sin \left(\frac{\pi}{4} t\right)\right)$
$v = 4 - \left(\frac{\pi}{4}\right) \cos \left(\frac{\pi}{4} t\right)$

At time $t = 1$
${v}_{t = 1} = 4 - \left(\frac{\pi}{4}\right) \cos \left(\frac{\pi}{4} \left(1\right)\right) = 4 - \frac{\pi}{4} \left(\frac{1}{\sqrt{2}}\right) = 4 - \frac{\pi}{4 \sqrt{2}}$