Given the parametric equations for the position of an object, find the object’s velocity and speed at the give

Given the parametric equations for the position of an object, find the object’s velocity and speed at the give?
Given the parametric equations for the position of an object, find the object’s velocity and speed at the given times and describe its motion.

x=3cost+sin3t

y=3sint+cos3t A) t=0 B) t= (pi/2)

Velocity is the derivative of time, so we have

velocity_x = 3cos3t - 3sint
velocity_y = 3cost -3sin3t

A)
Just plug in t=0 and we get
velocity_x = 3
velocity_y = 3
speed can be found using the distance formula, speed = sqrt(velocity_x ^ 2 + velocity_y ^ 2)
speed = sqrt(9 + 9) = sqrt(18)
speed = 4.24
To describe the motion, I assume you mean the angle of motion. Since the x and y components of velocity are equal, the motion is at a 45 degree angle.

B)
Plug in t=pi/2 and get
velocity_x = -3
velocity_y = 3
The speed here will be the same as in A. You can check with the distance formula if you want.
speed = 4.24
The motion here is at a 135 degree angle, since the x is negative but the y is positive.

Hope this helps.