Chapter 9.6, Problem 31E

a.

To determine

To find: A sinusoidal model for the tide depth $d$ (in feet) as a function of time $t$ (in hours).

Answer to Problem 31E

  $y=-6.5\cos \left(\frac{\pi }{6}t\right)+10$ where $t=0$ represents midnight.

Explanation of Solution

Given information:

The low tide occurs at midnight and is at $3.5$ feet. After $6$ hours high tide occurs which is at $16.5$ feet.

Calculation:

As per the given information −

The low tide occurs at midnight and is at $3.5$ feet. After $6$ hours high tide occurs which is at $16.5$ feet.

Step 1: Find the maximum and minimum value.

Here, maximum value is $16.5$ and minimum value is $3.5$ .

Step 2: Identify the vertical shift $k$ .

The value of $k$ is the mean of maximum and minimum value, i.e.

  $k=\frac{16.5+3.5}{2}=10$

Step 3: Find the amplitude and period.

Function goes from minimum to maximum in $6$ hours and then again maximum to minimum in next $6$ hours. So, period would be $12$ hours.

  $\begin{array}{l}\text{i}\text{.e}\text{. }12=\frac{2\pi }{b}\\ \Rightarrow b=\frac{\pi }{6}\end{array}$

Also at $t=0$ , function would be at its minimum value. So, cosine function reflected across $x-$ axis can be taken as model.

  $\begin{array}{l}\left|a\right|=\frac{\text{Maximum value}-\text{Minimum value}}{2}=\frac{16.5-3.5}{2}=6.5\\ \text{Since graph is a reflection, }a>0\\ \Rightarrow a=-6.5\end{array}$

Step $5$ : The function graph would be given as −

  $y=-6.5\cos \left(\frac{\pi }{6}t\right)+10$

  

Hence, desired model function is $y=-6.5\cos \left(\frac{\pi }{6}t\right)+10$ where $t=0$ represents midnight.

b.

To determine

To find: All the time when low and high tide occurs in $24$ hour period.

Answer to Problem 31E

Low tide occurs at midnight, $12$ noon and again at midnight.

Similarly, high tide occurs at $6$ AM, $6$ PM.

Explanation of Solution

Given information:

Same as part $a$ of this exercise.

Calculation:

From part $a$ of this exercise the model function would be given as −

  $y=-6.5\cos \left(\frac{\pi }{6}t\right)+10$ where $t=0$ represents midnight.

Below is the function graph for a $24$ hour period.

  

From above graph it can be observed that low tide occurs at midnight, $12$ noon and again at midnight.

Similarly, high tide occurs at $6$ AM, $6$ PM.

c.

To determine

To find: The relationship between graph from part $b$ and graph of tide depth $t$ hours after $3:00$ AM.

Explanation of Solution

Given information:

Same as part $a$ of this exercise.

Calculation:

From part $a$ of this exercise the model function would be given as −

  $y=-6.5\cos \left(\frac{\pi }{6}t\right)+10$ where $t=0$ represents midnight.

Below is the function graph for a $24$ hour period.

  

When $t=0$ is at $3:00$ AM. Tide depth is halfway through maximum. So, a sine function can be modeled.

  $y=6.5\sin \left(\frac{\pi }{6}t\right)+10$ where $t=0$ represents $3:00$ AM.