The height, h, in feet of a piece of cloth tied to a waterwheel in relation to sea level as a function of time, t, in seconds can be modeled by the equation h=8cos((pie/10)t) . what is the period of the function?

The period of any trig function is the time it takes for the argument (the angle)
to increase by 2π radians (360 degrees). 

So all we have to do is look at the angle-part of this function, start when the
angle is zero, and see how much 't' has to pass before the angle is  2π :

         cos(0.1 π t)  is the beginning.

         The end of the period is when    (0.1 π t)  =  2 π

                                                         0.1    t  =  2

                                                                 t  =  20 seconds.

The 8 at the beginning of the function ... multiplying the cosine ...
is just the amplitude of the function.  It doesn't affect the period,
frequency, or wavelength.