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=(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-(\pi)/(2))+5`

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

`B=(2\pi)/(16)`

Therefore,

`y=3sin((2\pi)/(16)t-(\pi)/(2))+5`

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