# If the position of a car is given by s=(80/3)[t + (3/pi)sin(pi*t/6)]. Find its velocity.

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We are given the position if the car as s = (80/3)*[t + (3/pi) sin (pi*t/6)]

Now this is a function in terms of the time t.

The velocity of a car is the instantaneous change in position, which is given by the derivative of s with respect to t

Now ds/dt = [(80/3)*t]' + [(80/3) (3/pi) * sin (pi*t/6)]'

=> ds/dt = [(80/3)*t]' + [(80/pi) * sin (pi*t/6)]'

=> ds/dt = (80/3) + (80/pi) *(pi/6)* cos (pi*t/6)

=> ds/dt = (80/3) + (40/3) cos (pi*t/6)

**Therefore the velocity of the car is (80/3) + (40/3) cos (pi*t/6)**

The velocity is the derivative of distance, with respect to time:

v = ds/dt

We'll calculate the derivative:

v = d/dt[(80/3)t + (80/pi)sin (pi*t/6)]

We'll remove the brackets:

v = (80/3)(d/dt t) + (80/pi)(d/dt[sin (pi*t/6)]

v = (80/3) + (80/pi)(pi/6)[cos (pi*t/6)]

v = (80/3) + (40/3)[cos (pi*t/6)]

We'll factorize by 40/3 and we'll get:

**v = (40/3)[2 + cos (pi*t/6)]**