Jami is trying to model the movement of a tire. In order to properly model the movement, she is going to focus on the movement of the tire's air nozzle. The nozzle is on the right side of the tire and she is going to treat that position as the equilibrium position. She then slowly rotates the tire so that the nozzle goes up at the beginning of the rotation, and follows its change in height.
The distance of the nozzle from the center of rotation is nine inches. She finds that it takes eight seconds for the tire to make a single revolution.
Which of the following functions best models the position of the nozzle?
A. s(t)=9sin(4pi t)
B.s(t)=9sin(pi/4 t)
C.s(t)=9cos(4pi t)
D. s(t)=9cos(pi/4 t)

Some events are more adequate to be modeled with trigonometric equations, especially sinusoids which run from a minimum to a maximum value periodically. The sine and cosine functions can work in such cases, and they usually are chosen exclusively by they behavior in time. The sine function starts in the equilibrium point at t=0, the cosine starts at the maximum point at t=0. Of course, we're assuming there is no phase shift.

Carefully reading the conditions of the problem, we learn that the height is being measured from its maximum height. It makes the cosine as the required function. We only need to find its angular frequency. The cosine function can be expressed as.

[tex]s(t)=Acos(wt)[/tex]

Where A is the amplitude and w is the angular frequency. The amplitude is obviously 9 since it's the distance from the center to the nozzle. The angular frequency can be computed by

[tex]\displaystyle w=\frac{2\pi}{T}[/tex]

where T is the period or the time taken to complete a whole revolution. That time is given as 8 seconds, so

[tex]\displaystyle w=\frac{2\pi}{8}=\frac{\pi}{4}[/tex]

Thus the function that models the phenomena is

[tex]\displaystyle \boxed{s(t)=9cos\left(\frac{\pi}{4}t\right)}[/tex]

Option: D.