7.6 Modeling with trigonometric equations (Page 3/14)

Precalculus

Page 3 / 14

Describing periodic motion

The hour hand of the large clock on the wall in Union Station measures 24 inches in length. At noon, the tip of the hour hand is 30 inches from the ceiling. At 3 PM, the tip is 54 inches from the ceiling, and at 6 PM, 78 inches. At 9 PM, it is again 54 inches from the ceiling, and at midnight, the tip of the hour hand returns to its original position 30 inches from the ceiling. Let $y$ equal the distance from the tip of the hour hand to the ceiling $x$ hours after noon. Find the equation that models the motion of the clock and sketch the graph.

Begin by making a table of values as shown in [link] .

$x$	$y$	Points to plot
Noon	30 in	$(0, 30)$
3 PM	54 in	$(3, 54)$
6 PM	78 in	$(6, 78)$
9 PM	54 in	$(9, 54)$
Midnight	30 in	$(12, 30)$

To model an equation, we first need to find the amplitude.

\begin{array}{l} \begin{array}{l} | A | = | \frac{78 - 30}{2} | \\ = 24 \end{array} \end{array}

The clock’s cycle repeats every 12 hours. Thus,

\begin{array}{l} \begin{array}{l} B = \frac{2 π}{12} \\ = \frac{π}{6} \end{array} \end{array}

The vertical shift is

\begin{array}{l} \begin{array}{l} D = \frac{78 + 30}{2} \\ = 54 \end{array} \end{array}

There is no horizontal shift, so $C = 0.$ Since the function begins with the minimum value of $y$ when $x = 0$ (as opposed to the maximum value), we will use the cosine function with the negative value for $A .$ In the form $y = A \cos (B x \pm C) + D,$ the equation is

y = −24 \cos (\frac{π}{6} x) + 54

See [link] .

Graph of the function y=-24cos(pi/6 x)+54 using the five key points: (0,30), (3,54), (6,78), (9,54), (12,30).

Got questions? Get instant answers now!

Determining a model for tides

The height of the tide in a small beach town is measured along a seawall. Water levels oscillate between 7 feet at low tide and 15 feet at high tide. On a particular day, low tide occurred at 6 AM and high tide occurred at noon. Approximately every 12 hours, the cycle repeats. Find an equation to model the water levels.

As the water level varies from 7 ft to 15 ft, we can calculate the amplitude as

\begin{array}{l} \begin{array}{l} | A | = | \frac{(15 - 7)}{2} | \\ = 4 \end{array} \end{array}

The cycle repeats every 12 hours; therefore, $B$ is

\frac{2 π}{12} = \frac{π}{6}

There is a vertical translation of $\frac{(15 + 8)}{2} = 11.5.$ Since the value of the function is at a maximum at $t = 0,$ we will use the cosine function, with the positive value for $A .$

y = 4 \cos (\frac{π}{6}) t + 11

See [link] .

Graph of the function y=4cos(pi/6 t) + 11 from 0 to 12. The midline is y=11, three key points are (0,15), (6,7), and (12, 15).

Got questions? Get instant answers now!

Try It

The daily temperature in the month of March in a certain city varies from a low of $24 °F$ to a high of $40 °F .$ Find a sinusoidal function to model daily temperature and sketch the graph. Approximate the time when the temperature reaches the freezing point $32 °F .$ Let $t = 0$ correspond to noon.

$y = 8 \sin (\frac{π}{12} t) + 32$
The temperature reaches freezing at noon and at midnight.

Graph of the function y=8sin(pi/12 t) + 32 for temperature. The midline is at 32. The times when the temperature is at 32 are midnight and noon.

Got questions? Get instant answers now!

Interpreting the periodic behavior equation

The average person’s blood pressure is modeled by the function $f (t) = 20 \sin (160 π t) + 100,$ where $f (t)$ represents the blood pressure at time $t,$ measured in minutes. Interpret the function in terms of period and frequency. Sketch the graph and find the blood pressure reading.

The period is given by

\begin{array}{l} \begin{array}{l} \frac{2 π}{ω} = \frac{2 π}{160 π} \\ = \frac{1}{80} \end{array} \end{array}

In a blood pressure function, frequency represents the number of heart beats per minute. Frequency is the reciprocal of period and is given by

\begin{matrix} \frac{ω}{2 π} = \frac{160 π}{2 π} \\ = 80 \end{matrix}

See the graph in [link] .

Graph of the function f(t) = 20sin(160 * pi * t) + 100 for blood pressure. The midline is at 100. — The blood pressure reading on the graph is $\frac{120}{80} (\frac{maximum}{minimum}) .$

Got questions? Get instant answers now!

Modeling harmonic motion functions

Harmonic motion is a form of periodic motion, but there are factors to consider that differentiate the two types. While general periodic motion applications cycle through their periods with no outside interference, harmonic motion requires a restoring force. Examples of harmonic motion include springs, gravitational force, and magnetic force.

Questions & Answers

how do you get the 2/50

Abba Reply

number of sport play by 50 student construct discrete data

Aminu Reply

width of the frangebany leaves on how to write a introduction

Theresa Reply

Solve the mean of variance

Veronica Reply

Step 1: Find the mean. To find the mean, add up all the scores, then divide them by the number of scores. ... Step 2: Find each score's deviation from the mean. ... Step 3: Square each deviation from the mean. ... Step 4: Find the sum of squares. ... Step 5: Divide the sum of squares by n – 1 or N.

kenneth

what is error

Yakuba Reply

Is mistake done to something

Vutshila

anas

What is the life teble

anas

Jibrin

statistics is the analyzing of data

Tajudeen Reply

what is statics?

Zelalem Reply

how do you calculate mean

Gloria Reply

diveving the sum if all values

Shaynaynay

let A1,A2 and A3 events be independent,show that (A1)^c, (A2)^c and (A3)^c are independent?

Fisaye Reply

what is statistics

Akhisani Reply

data collected all over the world

Shaynaynay

construct a less than and more than table

Imad Reply

The sample of 16 students is taken. The average age in the sample was 22 years with astandard deviation of 6 years. Construct a 95% confidence interval for the age of the population.

Aschalew Reply

Bhartdarshan' is an internet-based travel agency wherein customer can see videos of the cities they plant to visit. The number of hits daily is a normally distributed random variable with a mean of 10,000 and a standard deviation of 2,400 a. what is the probability of getting more than 12,000 hits? b. what is the probability of getting fewer than 9,000 hits?

Akshay Reply

Bhartdarshan'is an internet-based travel agency wherein customer can see videos of the cities they plan to visit. The number of hits daily is a normally distributed random variable with a mean of 10,000 and a standard deviation of 2,400. a. What is the probability of getting more than 12,000 hits

Akshay

Bright

Sorry i want to learn more about this question

Bright

Someone help

Bright

a= 0.20233 b=0.3384

Sufiyan

Shaynaynay

How do I interpret level of significance?

Mohd Reply

It depends on your business problem or in Machine Learning you could use ROC- AUC cruve to decide the threshold value

Shivam

how skewness and kurtosis are used in statistics

Owen Reply

yes what is it

Taneeya

Got questions? Join the online conversation and get instant answers!

Jobilize.com Reply

<< Chapter < Page Page > Chapter >>

Practice Key Terms 2

Read also:

Get Jobilize Job Search Mobile App in your pocket Now!

100% Free Mobile Applications
Receive real-time job alerts and never miss the right job again

Source: OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6

Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Precalculus' conversation and receive update notifications?

Ask

	5 Simple harmonic motion By OpenStax Read Online Course
	3 Modeling periodic behavior By OpenStax Read Online Course
	1 Neuroanatomy 01 The Cranial Nerves By Stephen Voron Start Quiz
	15 Dr. Amberg Pharm quiz By Brooke Delaney Start Exam
©flickr: Abraham	2 Biology 2 By Sarah Warren Start Test
	5 Arts Society: Theater 5 By Jonathan Long Start Quiz
	NCE Ch 08 Appraisal By Anh Dao Start Quiz
	Concepts of biology By OpenStax Read Online Course
	NCE Ch 03 Human Growth and Develoment By Anh Dao Start Test
	2 Gastrointestinal Pathophysiology Self-Assessment By Laurence Bailen Start Assessment
	1 Pharmacology Nervous System MCQ By Rohini Ajay Start Quiz
	Microeconomics Practice MCQ By Frank Levy Start Test