# 7.6 Modeling with trigonometric equations  (Page 5/14)

 Page 5 / 14

## Finding a cosine function that models damped harmonic motion

Find and graph a function of the form $\text{\hspace{0.17em}}y=a{e}^{-ct}\mathrm{cos}\left(\omega t\right)\text{\hspace{0.17em}}$ that models the information given.

1. $a=20,c=0.05,p=4$
2. $a=2,c=1.5,f=3$

Substitute the given values into the model. Recall that period is $\text{\hspace{0.17em}}\frac{2\pi }{\omega }\text{\hspace{0.17em}}$ and frequency is $\text{\hspace{0.17em}}\frac{\omega }{2\pi }.$

1. $y=20{e}^{-0.05t}\mathrm{cos}\left(\frac{\pi }{2}t\right).\text{\hspace{0.17em}}$ See [link] .
2. $y=2{e}^{-1.5t}\mathrm{cos}\left(6\pi t\right).\text{\hspace{0.17em}}$ See [link] .

The following equation represents a damped harmonic motion model: Find the initial displacement, the damping constant, and the frequency.

initial displacement =6, damping constant = -6, frequency = $\frac{2}{\pi }$

## Finding a sine function that models damped harmonic motion

Find and graph a function of the form $\text{\hspace{0.17em}}y=a{e}^{-ct}\mathrm{sin}\left(\omega t\right)\text{\hspace{0.17em}}$ that models the information given.

1. $a=7,c=10,p=\frac{\pi }{6}$
2. $a=0.3,c=0.2,f=20$

Calculate the value of $\text{\hspace{0.17em}}\omega \text{\hspace{0.17em}}$ and substitute the known values into the model.

1. As period is $\text{\hspace{0.17em}}\frac{2\pi }{\omega },$ we have

The damping factor is given as 10 and the amplitude is 7. Thus, the model is $\text{\hspace{0.17em}}y=7{e}^{-10t}\mathrm{sin}\left(12t\right).\text{\hspace{0.17em}}$ See [link] .

2. As frequency is $\text{\hspace{0.17em}}\frac{\omega }{2\pi },$ we have

The damping factor is given as $\text{\hspace{0.17em}}0.2\text{\hspace{0.17em}}$ and the amplitude is $\text{\hspace{0.17em}}0.3.\text{\hspace{0.17em}}$ The model is $\text{\hspace{0.17em}}y=0.3{e}^{-0.2t}\mathrm{sin}\left(40\pi t\right).\text{\hspace{0.17em}}$ See [link] .

Write the equation for damped harmonic motion given $\text{\hspace{0.17em}}a=10,c=0.5,$ and $\text{\hspace{0.17em}}p=2.$

$y=10{e}^{-0.5t}\mathrm{cos}\left(\pi t\right)$

## Modeling the oscillation of a spring

A spring measuring 10 inches in natural length is compressed by 5 inches and released. It oscillates once every 3 seconds, and its amplitude decreases by 30% every second. Find an equation that models the position of the spring $\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ seconds after being released.

The amplitude begins at 5 in. and deceases 30% each second. Because the spring is initially compressed, we will write A as a negative value. We can write the amplitude portion of the function as

$A\left(t\right)=5{\left(1-0.30\right)}^{t}$

We put $\text{\hspace{0.17em}}{\left(1-0.30\right)}^{t}\text{\hspace{0.17em}}$ in the form $\text{\hspace{0.17em}}{e}^{ct}\text{\hspace{0.17em}}$ as follows:

Now let’s address the period. The spring cycles through its positions every 3 seconds, this is the period, and we can use the formula to find omega.

$\begin{array}{l}\hfill \\ 3=\frac{2\pi }{\omega }\hfill \\ \omega =\frac{2\pi }{3}\hfill \end{array}$

The natural length of 10 inches is the midline. We will use the cosine function, since the spring starts out at its maximum displacement. This portion of the equation is represented as

$y=\mathrm{cos}\left(\frac{2\pi }{3}t\right)+10$

Finally, we put both functions together. Our the model for the position of the spring at $\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ seconds is given as

$y=-5{e}^{-0.357t}\mathrm{cos}\left(\frac{2\pi }{3}t\right)+10$

See the graph in [link] .

A mass suspended from a spring is raised a distance of 5 cm above its resting position. The mass is released at time $\text{\hspace{0.17em}}t=0\text{\hspace{0.17em}}$ and allowed to oscillate. After $\text{\hspace{0.17em}}\frac{1}{3}\text{\hspace{0.17em}}$ second, it is observed that the mass returns to its highest position. Find a function to model this motion relative to its initial resting position.

$y=5\mathrm{cos}\left(6\pi t\right)$

## Finding the value of the damping constant c According to the given criteria

A guitar string is plucked and vibrates in damped harmonic motion. The string is pulled and displaced 2 cm from its resting position. After 3 seconds, the displacement of the string measures 1 cm. Find the damping constant.

The displacement factor represents the amplitude and is determined by the coefficient $\text{\hspace{0.17em}}a{e}^{-ct}\text{\hspace{0.17em}}$ in the model for damped harmonic motion. The damping constant is included in the term $\text{\hspace{0.17em}}{e}^{-ct}.\text{\hspace{0.17em}}$ It is known that after 3 seconds, the local maximum measures one-half of its original value. Therefore, we have the equation

$a{e}^{-c\left(t+3\right)}=\frac{1}{2}\text{\hspace{0.17em}}a{e}^{-ct}$

Use algebra and the laws of exponents to solve for $\text{\hspace{0.17em}}c.$

Then use the laws of logarithms.

The damping constant is $\text{\hspace{0.17em}}\frac{\mathrm{ln}\text{\hspace{0.17em}}2}{3}.$

#### Questions & Answers

how fast can i understand functions without much difficulty
what is set?
a colony of bacteria is growing exponentially doubling in size every 100 minutes. how much minutes will it take for the colony of bacteria to triple in size
I got 300 minutes. is it right?
Patience
no. should be about 150 minutes.
Jason
It should be 158.5 minutes.
Mr
ok, thanks
Patience
100•3=300 300=50•2^x 6=2^x x=log_2(6) =2.5849625 so, 300=50•2^2.5849625 and, so, the # of bacteria will double every (100•2.5849625) = 258.49625 minutes
Thomas
what is the importance knowing the graph of circular functions?
can get some help basic precalculus
What do you need help with?
Andrew
how to convert general to standard form with not perfect trinomial
can get some help inverse function
ismail
Rectangle coordinate
how to find for x
it depends on the equation
Robert
yeah, it does. why do we attempt to gain all of them one side or the other?
Melissa
whats a domain
The domain of a function is the set of all input on which the function is defined. For example all real numbers are the Domain of any Polynomial function.
Spiro
Spiro; thanks for putting it out there like that, 😁
Melissa
foci (–7,–17) and (–7,17), the absolute value of the differenceof the distances of any point from the foci is 24.
difference between calculus and pre calculus?
give me an example of a problem so that I can practice answering
x³+y³+z³=42
Robert
dont forget the cube in each variable ;)
Robert
of she solves that, well ... then she has a lot of computational force under her command ....
Walter
what is a function?
I want to learn about the law of exponent
explain this