A horse on a merry-go-round moves in such a way that its height (in metres) above the floor is h(t) = 0.8 sin[/2 (t

A horse on a merry-go-round moves in such a way that its height (in metres) above the floor is h(t) = 0.8 sin[π/2 (t − 1)] + 1.8 where t ≥ 0 is time in seconds.

(a) Using the formula for h(t), find its period. Sketch the graph of h(t) for 0 ≤ t ≤ 12.

(b) Find the time at which the horse reaches its maximum height for the 15th time, showing the reasoning.

(c) Find h′(t) and evaluate h′(3.5) exactly. Is the horse moving up or down at t = 3.5 and why?

(d) Find the smallest value of t ≥ 0 for which h ′ (t) is a maximum, showing the working.