Question

In Pensacola in June, high tide was at noon. The water level at high tide was 12 feet and 2 feet at low tide. Assuming the next high tide is exactly 12 hours later and that the height of the water can be modeled by a cosine curve, find an equation for water level in June for Pensacola as a function of time (t).

Answer 1

`\displaystyle h(t)=5cos\left((\pi)/(6)t\right)+7`

Step-by-step explanation:

Function Modeling

Scientists are always looking to find better ways to express in models what is happening in reality. Mathematics models find numerical relationships by using a variety of functions, tables, graphs, and rules to accurately describe the real world's magnitudes.

The problem at hand requires us to model the height of the water as a function of time t by using a cosine function. The general expression for a cosine function is

`h(t)=Acos(wt+\phi)+b`

Where A is the amplitude, w is the angular frequency,
`\phi` is the phase shift and b is the midline or offset of the sinusoid with respect to the x-axis line.

We can easily find both A and b by knowing the max level is 12 feet and the min level is 2 feet. The center level gives us the midline:

`\displaystyle b=(12+2)/(2)=7\ feet`

And the difference between the max or min levels (in absolute value) gives us the amplitude

`A=12-7=5\ feet`

Thus, the equation is now

`h(t)=5cos(wt+\phi)+7`

We know that at t=0 (noon) the tide was high at 12 feet:

`12=5cos(0+\phi)+7`

Rearranging and simplifying

`cos\phi = 1`

`\phi=0`

Substituting into the equation

`h(t)=5cos(wt)+7`

We only need to find w and it will be done by knowing that the next high tide occurs 12 hours later from the first one. It means the cycle repeats after 12 hours, or the period is T=12. We know

`\displaystyle w=(2\pi)/(T)`

We compute

`\displaystyle w=(2\pi)/(12)=(\pi)/(6)`

The final expression for the function is

`\boxed{\displaystyle h(x)=5cos\left((\pi)/(6)t\right)+7}`