Question

On the bay of Lake Huron the tides vary between 2 feet and 8 feet. The tide is at its lowest point when time (t) is 0 and completes a full cycle in 16 hours. What is the amplitude, period, and midline of a function that would model this periodic phenomenon?

Answer 1

The most general form of a sne equaion is shown below

$\begin{gathered} y=Asin(B(x-C))+D \\ A\rightarrow\text{ amplitude} \\ B\rightarrow period=\frac{2\pi}{B} \\ C\rightarrow\text{ horizontal shift} \\ D\rightarrow\text{ vertical shift} \end{gathered}$

In our case, the midine of the function is at 8(8-2)/2=3; then, 2+3=5. The midline is at y=5.

Now, since at t=0 the function is at its lowest point,

$\Rightarrow y=3sin(Bt-\frac{\pi}{2})+5$

Finally, regarding the period of the function, since every 16 hours, the function completes a full cycle,

$B=\frac{2\pi}{16}$

Therefore,

$y=3sin(\frac{2\pi}{16}t-\frac{\pi}{2})+5$

The amplitude is 3, the period is 2pi/16 and the midline is y=5.