# The position of an object moving along a line is given by #p(t) = sin(t- pi /4) +1 #. What is the speed of the object at #t = pi/3 #?

The speed is

The derivative of position is the speed.

Consequently,

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To find the speed of the object at ( t = \frac{\pi}{3} ), we need to find the derivative of the position function with respect to time ( t ), then evaluate it at ( t = \frac{\pi}{3} ).

The derivative of the position function ( p(t) ) with respect to time ( t ) is given by:

[ v(t) = \frac{d}{dt}[ \sin(t - \frac{\pi}{4}) + 1] ]

[ v(t) = \cos(t - \frac{\pi}{4}) ]

Now, to find the speed at ( t = \frac{\pi}{3} ), we substitute ( t = \frac{\pi}{3} ) into the velocity function:

[ v(\frac{\pi}{3}) = \cos(\frac{\pi}{3} - \frac{\pi}{4}) ]

[ v(\frac{\pi}{3}) = \cos(\frac{\pi}{12}) ]

[ v(\frac{\pi}{3}) \approx 0.9659 ]

Therefore, the speed of the object at ( t = \frac{\pi}{3} ) is approximately 0.9659.

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