the height, in metres, of a nail in a water wheel above the surface of the water , as a function of time, can be modelled by the function h(t) = -4sin pie/4 (t-1)+2.5, where t is the time in seconds. During what periods of time is the nail below the water in the first 24s that the wheel is rotating?

Question

I have the answer i would just like to know the steps of how to solve for this sort of question.

Answer 1

h(t) = -4sin (π/4)(t-1)+2.5
The period of this function is 2π/(π/4) = 8
So we want to see where the sine curve falls below the x-axis for 3 periods of the curve, starting at t=0

0 = -4sin (π/4)(t-1)+2.5
4sin (π/4)(t-1) = 2.5
sin (π/4)(t-1) = .625

make sure your calculator is set to radians
(π/4)(t-1) = .67513 or (π/4)(t-1) = π - .67513

case1: (π/4)(t-1) = .67513
t-1 = .8596
t = 1.8596

case2: (π/4)(t-1) = π - .67513
(π/4)(t-1) = 2.46646..
t-1 = 3.1404
t = 4.1404

my Wolfram graph confirms this:
http://www.wolframalpha.com/input/?i=plot+h(t)+%3D+-4sin+((%CF%80%2F4)(t-1))+%2B2.5

so the nail is below the x-axis or 4.1404-1.8596 or
2.28 seconds for each cycle

my edited graph shows that this happens 3 times in your domain,
http://www.wolframalpha.com/input/?i=plot+h(t)+%3D+-4sin+((%CF%80%2F4)(t-1))+%2B2.5+for+t%3D0+to+24

So for your domain, the nail is underwater for a total of 3(2.28..) seconds or 6.842 seconds
or
the nail is below:
between 1.860 and 4.140 seconds
between 9.860 and 12.140 seconds and
between 17.860 and 20.140 seconds

depending on how you interpret your question.
(note that each interval for the last interpretation is 2.28 seconds for a total of 6.842 seconds)

again, my second Wolfram graph confirms this