What is the angular displacement of the wheel between t = 5 s and t = 15 s?

The question seems incomplete. The complete text is:

a)What is the angular displacement of the wheel between t = 5 s and t = 15 s

b)What is the angular velocity of the wheel at 15 s

And it refers to the attached figure.

a) 25 rad

The graph shown represent the angular position of the wheel at different times.

Therefore, we can simply calculate the  angular displacement between two times by calculating the difference between the angular position at t2 and the angular position at t1.

At , the angular position from the graph is

At , the angular position from the graph is

Therefore, the angular displacement is

2) -5.0 rad/s

For a angular displacement vs time graph, the angular velocity at any time is simply equal to the slope of the curve at that time.

Here  we want to calculate the angular velocity at t = 15 s, so we have to calculate the slope at that time.

By noting that the slope is constant in the last part of the motion, we find that the slope between 10 s and 20 s is:

This slope is constant between 10 s and 20 s, so the angular velocity of the wheel at t = 15 s