# 7.6 Modeling with trigonometric equations  (Page 9/14)

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A muskrat population oscillates 33 above and below average during the year, reaching the lowest value in January. The average population starts at 900 muskrats and increases by 7% each month. Find a function that models the population, $\text{\hspace{0.17em}}P,$ in terms of months since January, $\text{\hspace{0.17em}}t.$

A fish population oscillates 40 above and below average during the year, reaching the lowest value in January. The average population starts at 800 fish and increases by 4% each month. Find a function that models the population, $\text{\hspace{0.17em}}P,$ in terms of months since January, $\text{\hspace{0.17em}}t.$

$P\left(t\right)=-40\mathrm{cos}\left(\frac{\pi }{6}t\right)+800{\left(1.04\right)}^{t}$

A spring attached to the ceiling is pulled 10 cm down from equilibrium and released. The amplitude decreases by 15% each second. The spring oscillates 18 times each second. Find a function that models the distance, $\text{\hspace{0.17em}}D,$ the end of the spring is from equilibrium in terms of seconds, $\text{\hspace{0.17em}}t,$ since the spring was released.

A spring attached to the ceiling is pulled 7 cm down from equilibrium and released. The amplitude decreases by 11% each second. The spring oscillates 20 times each second. Find a function that models the distance, $\text{\hspace{0.17em}}D,$ the end of the spring is from equilibrium in terms of seconds, $\text{\hspace{0.17em}}t,$ since the spring was released.

$D\left(t\right)=7{\left(0.89\right)}^{t}\mathrm{cos}\left(40\pi t\right)$

A spring attached to the ceiling is pulled 17 cm down from equilibrium and released. After 3 seconds, the amplitude has decreased to 13 cm. The spring oscillates 14 times each second. Find a function that models the distance, $\text{\hspace{0.17em}}D,$ the end of the spring is from equilibrium in terms of seconds, $\text{\hspace{0.17em}}t,$ since the spring was released.

A spring attached to the ceiling is pulled 19 cm down from equilibrium and released. After 4 seconds, the amplitude has decreased to 14 cm. The spring oscillates 13 times each second. Find a function that models the distance, $\text{\hspace{0.17em}}D,$ the end of the spring is from equilibrium in terms of seconds, $\text{\hspace{0.17em}}t,$ since the spring was released.

$D\left(t\right)=19{\left(0.9265\right)}^{t}\mathrm{cos}\left(26\pi t\right)$

For the following exercises, create a function modeling the described behavior. Then, calculate the desired result using a calculator.

A certain lake currently has an average trout population of 20,000. The population naturally oscillates above and below average by 2,000 every year. This year, the lake was opened to fishermen. If fishermen catch 3,000 fish every year, how long will it take for the lake to have no more trout?

Whitefish populations are currently at 500 in a lake. The population naturally oscillates above and below by 25 each year. If humans overfish, taking 4% of the population every year, in how many years will the lake first have fewer than 200 whitefish?

$20.1\text{\hspace{0.17em}}$ years

A spring attached to a ceiling is pulled down 11 cm from equilibrium and released. After 2 seconds, the amplitude has decreased to 6 cm. The spring oscillates 8 times each second. Find when the spring first comes between $\text{\hspace{0.17em}}-0.1\text{\hspace{0.17em}}$ and effectively at rest.

A spring attached to a ceiling is pulled down 21 cm from equilibrium and released. After 6 seconds, the amplitude has decreased to 4 cm. The spring oscillates 20 times each second. Find when the spring first comes between $\text{\hspace{0.17em}}-0.1\text{\hspace{0.17em}}$ and effectively at rest.

17.8 seconds

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
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
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
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
what is functions?
A mathematical relation such that every input has only one out.
Spiro
yes..it is a relationo of orders pairs of sets one or more input that leads to a exactly one output.
Mubita
Is a rule that assigns to each element X in a set A exactly one element, called F(x), in a set B.
RichieRich
If the plane intersects the cone (either above or below) horizontally, what figure will be created? By By By Mahee Boo By By Richley Crapo By Brooke Delaney By By OpenStax By OpenStax By Robert Murphy By OpenStax By OpenStax