The position of an object moving along a line is given by #p(t) = 2t - tsin(( pi )/4t) #. What is the speed of the object at #t = 7 #?

Question

The position of an object moving along a line is given by #p(t) = 2t - tsin((  pi )/4t)  #.  What is the speed of the object at #t = 7 #?

Sebastian Acosta · Accepted Answer

#v(7)=-1.117# #p(t)=2t-t sin (pi/4 t)" the equation of object's position"# #v(t)=d/(d t)p(t)=d/(d t) (2t-t sin(pi/4 t))# #v(t)=2-[sin(pi/4 t)+t*pi/4 cos(pi/4t)]# #v(7)=2-[sin(pi/4*7)+7*pi/4cos(pi/4*7)]# #v(7)=2-[-0.707+7*pi/4 *0.707]# #v(7)=2-[-0.707+3.887]# #v(7)=2-3.117# #v(7)=-1.117#

Layla Ahmed · Answer

undefined To find the speed of the object at $ t = 7 $, we need to find the derivative of the position function $ p(t) $ with respect to time $ t $, and then evaluate it at $ t = 7 $. The derivative of $ p(t) $ gives us the velocity function $ v(t) $, which represents the speed of the object at any given time $ t $.

So, first, let's find the derivative of $ p(t) $ with respect to $ t $:

$$ p'(t) = 2 - \left(1 - \frac{\pi}{4}\right) \cos\left(\frac{\pi}{4t}\right) + \frac{\pi}{4} \sin\left(\frac{\pi}{4t}\right) $$

Then, we evaluate $ p'(t) $ at $ t = 7 $ to find the speed of the object at that time:

$$ v(7) = 2 - \left(1 - \frac{\pi}{4}\right) \cos\left(\frac{\pi}{28}\right) + \frac{\pi}{4} \sin\left(\frac{\pi}{28}\right) $$