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