Chapter 6.6, Problem 33AYU

(a)

To determine

To find: The time for the next tide.

Answer to Problem 33AYU

The next high tide is $11:55\text{PM}$ .

Explanation of Solution

Given:

The time between the two consecutive high tides is of 12 hours 25 minutes.

The time of high tide on July 25 is at 11:20 Am and the low tide is at 5:31 PM.

The height of the high tide is $5.84\text{feet}$ .

The height of the water at low tide is $-0.37\text{foot}$ .

Calculation:

Consider the period between the two high tides is of $12\text{hours}$ and $25$ minutes. As the first high tide occurs at 11:30 AM. So the next high tide occurs at,

  $11:30+12:25=23:55\text{hours}$

Thus, the next high tide is $11:55\text{PM}$ .

(b)

To determine

To find: The sinusoidal function of the form $y=A\sin \left(\omega x-\varphi \right)+B$ that models the data..

Answer to Problem 33AYU

The sinusoidal model is $y=3.1041\sin \left(\frac{24\pi }{149}\left(x-8.3959\right)\right)+2.735$ .

Explanation of Solution

Consider the form of the sinusoidal function is,

  $y=A\sin \left(\omega x-\varphi \right)+B$

Consider the amplitude of the sinusoidal function is,

  $\begin{array}{c}A=\frac{5.84-\left(-0.37\right)}{2}\\ =3.105\end{array}$

Consider the value of the vertical shift is,

  $\begin{array}{c}V=\frac{5.84+\left(-0.37\right)}{2}\\ =2.735\end{array}$

Consider the time period of the sinusoidal function is,

  $\begin{array}{l}T=\frac{2\pi }{\omega }\\ \frac{149}{12}=\frac{2\pi }{\omega }\\ \omega =\frac{24\pi }{149}\end{array}$

The time is divided into four sub intervals as $\left[0,3.1041\right],\left[3.1041,6.2082\right],\left[6.2082,9.3123\right],\left[9.3123,12.4166\right]$ , the time is increasing at the interval $\left(0,3.1041\right)$ and decreasing at the interval $\left[3.1041,6.2082\right]$ .

The local maximum is at $x=3.1041$ .

The local minimum is at $x=11.5$ .

The required horizontal shift is $x=11.5-3.1041$ that is $8.3959$ to the right.

So, the sinusoidal model is,

  $y=3.1041\sin \left(\frac{24\pi }{149}\left(x-8.3959\right)\right)+2.735$

(c)

To determine

To find: The height of the water at 3 PM on July 25.

Answer to Problem 33AYU

The height of the water is $2.12\text{feet}$ .

Explanation of Solution

Consider the sinusoidal function is,

  $y=3.1041\sin \left(\frac{24\pi }{149}\left(x-8.3959\right)\right)+2.735$

Consider at 3:00 AM the sine wave function is,

  $\begin{array}{c}y=3.1041\sin \left(\frac{24\pi }{149}\left(15-8.3959\right)\right)+2.735\\ =2.12\text{feet}\end{array}$