/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 56 A heavy wind is kicking up ocean... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 56

A heavy wind is kicking up ocean swells approximately \(10 \mathrm{ft}\) high (from crest to trough), with wavelengths of \(250 \mathrm{ft}\). (a) Find an equation that models these swells. (b) Graph the equation. (c) Determine the height of a wave measured \(200 \mathrm{ft}\) from the trough of the previous wave.

Short Answer

Expert verified
(a) Equation: \(y(x) = 5 \sin\left(\frac{2\pi}{250}x\right)\); (b) Graph: periodic wave; (c) Height at 200 ft: \(\approx 2.939\) ft.

Step by step solution

01

Understand the Wave Equation

Ocean swells can be modeled by the sine function: \(y(x) = A \sin\left(\frac{2\pi}{\lambda}x + \phi\right)\). Here, \(A\) is the amplitude, \(\lambda\) is the wavelength, and \(\phi\) is the phase shift.
02

Determine the Amplitude

The amplitude \(A\) is half the height from the crest to the trough. For a height of \(10 \text{ ft}\), the amplitude is \(A = \frac{10}{2} = 5 \text{ ft}\).
03

Determine the Wavelength

The wavelength \(\lambda\) is given as \(250 \text{ ft}\).
04

Write the Equation

Using the values found, the equation for the swells is \(y(x) = 5 \sin\left(\frac{2\pi}{250}x\right)\).
05

Graph the Equation

To graph the equation \(y(x) = 5 \sin\left(\frac{2\pi}{250}x\right)\), plot the sine wave with amplitude \(5\) and wavelength \(250\). Note the periodic pattern.
06

Calculate the Height at 200 ft

Using the equation, calculate the wave height at \(x = 200\): \(y(200) = 5 \sin\left(\frac{2\pi}{250} \times 200\right)\). Simplify to \(y(200) = 5 \sin\left(\frac{4\pi}{5}\right)\).
07

Compute the Sine Value

Using a calculator, find \(\sin\left(\frac{4\pi}{5}\right) \approx 0.5878\). Hence, \(y(200) = 5 \times 0.5878 \approx 2.939\).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Amplitude
When discussing waves, especially sinusoidal ones like ocean swells, the term 'amplitude' is essential. Amplitude refers to the maximum distance a point on the wave鈥攕uch as the crest or the trough鈥攎oves from its central point. In mathematical terms, it is the peak value of the function. More simply, it's half the height from the top of the crest to the bottom of the trough.
In the given exercise, the ocean waves have a total height of 10 feet from crest to trough. Therefore, to find the amplitude, you simply divide the total height by two. This gives an amplitude, or maximum deviation from the wave's midpoint, of 5 feet.
Understanding amplitude is crucial as it helps you measure how "tall" the waves are. This is an essential factor as it directly influences how dynamic or powerful the wave appears. In practical terms, the greater the amplitude, the more energy the wave carries.
Wavelength
Wavelength is a key concept in wave theory, representing the distance between two peaks (crests) or troughs in a wave. It's essentially the length of one complete wave cycle. In mathematical modeling of wave functions, the wavelength \(\lambda\) is vital.
In the context of the given problem, the wavelength is specified as 250 feet. This length is a measure of how far apart the waves are spread out from one another along the surface of the water.
Why is wavelength important? The wavelength determines how frequently the waves occur over a given distance. Longer wavelengths mean waves are spaced farther apart, while shorter wavelengths indicate that waves are closer together, impacting the wave's potential effect on boats or coastal structures.
Wave Height
Wave height refers to the total vertical distance from the crest of the wave to the trough. In essence, it measures how tall the wave appears from the lowest point in the trough to the highest point at the crest, making it a full measure of the wave's vertical size.
In the exercise example, the wave height is given as 10 feet. Knowing the wave height is important as it directly relates to the force or energy that the wave can exert. In practical terms, with coastal engineering or maritime navigation, wave height is a critical data point for safety and operational planning.
While wave height and amplitude are related, they are not the same. Amplitude is half the wave height and represents the maximum displacement of the wave, whereas wave height includes both the upward and downward swings of the wave from peak to trough.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

In Oslo, Norway, the number of hours of daylight reaches a low of \(6 \mathrm{hr}\) in January, and a high of nearly \(18.8 \mathrm{hr}\) in July. (a) Find a sinusoidal equation model for the number of daylight hours each month; (b) sketch the graph; and (c) approximate the number of days each year there are more than \(15 \mathrm{hr}\) of daylight. Use 1 month \(\approx 30.5\) days. Assume \(t=0\) corresponds to January 1 .

In what quadrant does the angle \(t=3.7\) terminate? What is the reference arc?

Temperature models: The average temperature on Valentine's Day in Sydney, Australia, can be modeled with the equation \(T(t)=4 \cos \left(\frac{\pi}{12} t+\frac{9 \pi}{12}\right)+22\), where \(T\) is the temperature in Celsius and \(t\) is the time of day ( \(t=0\) corresponds to midnight). Use the model to (a) find the period of the model; (b) find the average minimum and maximum temperature; and (c) when these "extreme" temperatures occur.

Clearly state the period of each function, then match it with the corresponding graph. $$ y=2 \csc \left(\frac{1}{2} t\right) $$

Wave height: A data buoy placed off the coast of Santa Cruz, California, measures wave height and transmits the information to a monitoring station. For the minute 12:28 PDT (low tide), the wave height can be modeled with the equation \(y=2.6 \sin \left(\frac{2 \pi}{6} t\right)-0.6\), where \(t\) is measured in seconds and \(y\) is in feet \((y=0\) corresponds to the height of calm sea between high and low tide). Use the model to find (a) the time between each wave and (b) wave height from peak to trough.
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Geometry

Read Explanation

Probability and Statistics

Read Explanation

Decision Maths

Read Explanation

Calculus

Read Explanation

Logic and Functions

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.