The height of a piston in a cylinder can be modeled by a sine or cosine function. A piston is at its lowest point in a cylinder, 8cm from the bottom, at t=3.2 seconds.
The piston is at its highest position, 39cm from the bottom, at t=3.6 seconds. Find an equation for the height of the piston, in cm, at any given time t.
We will use a cosine model. The general model is `y=Acos(B(x-h))+k` where A is the amplitude, B is associated with the period (it is the horizontal compression/stretch), h is the horizontal translation, and k the vertical translation.
(a) The amplitude is `A=(39-8)/2=15.5`
(b) The vertical translation is `k=8+15.5=39-15.5=23.5`
(c) The horizontal translation is h=.4 It takes .4sec to go through 1/2 of a period -- since it starts at the lowest point we can shift the graph left or right .4 units, or we could multiply by -15.5 to reflect the graph.
(d) To find B we use `B=(2pi)/p` where p is the period. Since the full period is .8 seconds we have `B=(2pi)/(4/5)=(5pi)/2`
The equation: `y=15.5cos((5pi)/2(x-.4))+23.5`
As stated, there are many other possible equations for this graph.