The position vector for a particle moving on a helix is c(t)= (cost, sint, t^2). find the speed of the particle at time t = 4pi

We differentiate the position vector to obtain the velocity vector:
v(t) = (-sin(t), cos(t), 2t)
Now, substituting 4π
v(t) = 0, 1, 8π

We calculate the magnitude of the velocity to obtain the speed:
s = √(0² + 1² + (8π)²)
s = 25.2  

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