The position of an object moving along a line is given by #p(t) = cos(t- pi /3) +2 #. What is the speed of the object at #t = (2pi) /4 #?

#v(t) =(dp)/(dt) =d/(dt) cos(t-pi/3)+2 =-sin(t-pi/3)#

To find the speed of the object at ( t = \frac{2\pi}{4} ), we need to differentiate the position function ( p(t) ) with respect to time ( t ) to get the velocity function ( v(t) ). Then, we can evaluate the velocity function at ( t = \frac{2\pi}{4} ) to find the speed.

Given: [ p(t) = \cos(t - \frac{\pi}{3}) + 2 ]

We differentiate ( p(t) ) with respect to ( t ) to find ( v(t) ): [ v(t) = \frac{d}{dt}[\cos(t - \frac{\pi}{3}) + 2] ]

Using the chain rule and the derivative of cosine function: [ v(t) = -\sin(t - \frac{\pi}{3}) ]

Now, we evaluate ( v(t) ) at ( t = \frac{2\pi}{4} ): [ v(\frac{2\pi}{4}) = -\sin(\frac{2\pi}{4} - \frac{\pi}{3}) ] [ v(\frac{2\pi}{4}) = -\sin(\frac{\pi}{6}) ]

Recall that ( \sin(\frac{\pi}{6}) = \frac{1}{2} ), so: [ v(\frac{2\pi}{4}) = -\frac{1}{2} ]

Therefore, the speed of the object at ( t = \frac{2\pi}{4} ) is ( \frac{1}{2} ) units per time.