Achild is pushing a merry-go-round. the angle through which the merry-go-round has turned varies with time according to θ(t)=γt+βt3, where γ= 0.422 rad/s and β= 1.35×10−2 rad/s3.

I attached the missing part of the question.
Part A
Angular velocity is simply the rate at which angle changes, in other words, it is the first derivative of the angle function with respect to time:

Part B
Initial value is the value at t=0. To find initial value we simply plug t=0 into the equation.

Part C
To find these values we simply need to plug in t=5 in the equation.

Part D
Average angular velocity is total angular displacement divided by time:


The average angular velocity is:

It indicates no change in temperature and the process of matter changing its state