Question

Low tide in the ocean occurs at 8:00am. The depth of the water at this time is 4 meters. High tide occurs at 4:00pm and the depth of the water at this time is 38 meters. Assume that the 12:00am (prior to the first low tide) is t = 0. Sketch a sine/cosine wave that represents the height of the water as a function of time. Make a table of values for two full cycles. You do not need to record the height at every hour, but you should be recording the times for maximums and minimums and the axis of the wave. Write an equation for your graph.

Answer 1

I've made a really ugly (I'm sorry) drawing of the situation of the tide at the two "principal hours" of the day. What we want is a function

$f(x)$

that explains the behavior of the tide of the ocean. Fortunately, this phenomenon has been studied from many years ago, and we have the following possible skeletons of the desired function:

$f(t)=A\cdot\cos (B\cdot t)+C,\text{ or }f(t)=\text{ A}\cdot\sin (B\cdot t)+C$

Using the given values we need to find the models best fitting our case, and then calculate the constants. Now, for the model is a cyclic function, we expect it to be "centered" at the midpoint (M) between 4 and 38:

$M=\frac{38+4}{2}=21$

This midpoint represents C. The value of A can be interpreted as the "high or depth" of the cyclic function; namely,

$A=38-21=17,A=21-4=17$

Then, our possible models are

$f(t)=17\cdot\cos (B\cdot t)+21,\text{ or }f(t)=17\cdot\sin (B\cdot t)+21$

Now, I must say that both models work! (but there is a particular B for each of them). We need to choose one; I'm going to take "cos model" out. For the minimum value os cos(...) is -1, we need

$\cos (B\cdot t)=-1\text{ when }t=8$

(t represents the number of hours after 12:00 am as you can check in the statement of the problem), in order to obtain f(8)=4. At the same time, we need

$\cos (B\cdot t)=1\text{ when }t=16$

(t=16 is equivalent to be at 4:00pm), in order to obtain f(16)=38. This has a really profound meaning.

The yellow curve is half of a cycle, and its "length" is 8 (it goes from t=8 to t=16). Then, the length of a complete cycle is 16. This implies that the period (T) of the function f is

$T=\frac{2\pi}{16}=\frac{\pi}{8}$

This period is the meaning of B. Then, our complete model is

$f(t)=17\cdot\cos (\frac{\pi}{8}\cdot t)+21$

This model looks like

(the blue line (y=21) represents what the statement of the problem called "the axis of the wave")

The task now is to record in the table the value of the model at the important t's for two cycles. For example, let's take the two cycles from 0 to 32 (Remember: we found that a cycle has a length of 16).

(That's the table)