a) From the question, we are interested in the height of a specific carriage as it spins around, obviously, at different TIMES, the HEIGHT of the carriage above ground will be different.
Seeing time ticks away regardless of what happens to the carriage (or the radius. or the height of the lowest carriage for that matter), so it doesn't depend on anything that you're given... so TIME is the independent variable.
The height of the carriage changes at different times, so we can say that height depends on time, so here, HEIGHT is the dependent variable.
b) The carriage takes 1 minute to make 1 revolution, to break it down finely, you could consider the position of the carriage every 10 seconds? Or every 5 seconds? Or every second!!!
I've decided to do every 5 seconds to make less calculations (but you can obviously do more), the distances can be easily found using a right angled triangle (as shown in red in the attached diagram), and some basic trigonometry, bearing in mind the distance from the corner of the right-angled triangle to the ground as well!
I've shown you 12 of them!!! (And there are repeats as you can see here!)
c) If we were to move the circle such that the centre is at the origin, recalculating these distances. After plotting the distances, you will notice that they fall onto the standard sine curve .
Seeing the centre of the circle is at (0,5) rather than the origin, it is STILL a sine curve, but with some shifting!
In fact, the distances should trace the graph