1)
Consider P to be at the same vertical distance between the maximum and minimum height the blades can reach. Therefore, at t=0, P has to be at h=0 (the maximum height is 1 and the minimum one is -1).
Then, since the fan blades rotate counterclockwise, after t=0, h has to increase until it reaches a value of 1.
Finally, a revolution is completed after 1 second; therefore, at t=1, the h has to be equal to 0 (the same height as at t=0).
Thus, the graph that models the function is option D.
2)
Approximately, every year the Earth reaches its maximum distance from the sun and its minimum distance once a year, and that is a cycle that repeats almost without alteration. Furthermore, since the position of the Earth around the sun is given by an ellipse, the position of the Earth around the Sun can be expressed using trigonometric functions (because of polar coordinates).
Similarly, a rotating wheel will reach its maximum and minimum height eventually; therefore, it can be modeled using trigonometric functions which are periodic.
Finally, as for the area of a sheet of paper. Notice that the more we fold it, the less its area becomes, it does not reach its maximum anymore.
Given x inches, the number of centimeters is 2.54x, this is not a periodic function.
Due to the effects of gravity, the height of a pendulum can be expressed using a periodic function given that no energy is lost.
The trajectory of a tossed ball is given by a parabola, unless one tosses the ball again at the same initial height, option F cannot be modeled using a periodic function.
The answers to part 2 are A, B, and E.