# 7.6 Modeling with trigonometric equations  (Page 8/14)

 Page 8 / 14

The sea ice area around the South Pole fluctuates between about 18 million square kilometers in September to 3 million square kilometers in March. Assuming a sinusoidal fluctuation, when are there more than 15 million square kilometers of sea ice? Give your answer as a range of dates, to the nearest day.

From July 8 to October 23

During a 90-day monsoon season, daily rainfall can be modeled by sinusoidal functions. If the rainfall fluctuates between a low of 2 inches on day 10 and 12 inches on day 55, during what period is daily rainfall more than 10 inches?

During a 90-day monsoon season, daily rainfall can be modeled by sinusoidal functions. A low of 4 inches of rainfall was recorded on day 30, and overall the average daily rainfall was 8 inches. During what period was daily rainfall less than 5 inches?

From day 19 through day 40

In a certain region, monthly precipitation peaks at 8 inches on June 1 and falls to a low of 1 inch on December 1. Identify the periods when the region is under flood conditions (greater than 7 inches) and drought conditions (less than 2 inches). Give your answer in terms of the nearest day.

In a certain region, monthly precipitation peaks at 24 inches in September and falls to a low of 4 inches in March. Identify the periods when the region is under flood conditions (greater than 22 inches) and drought conditions (less than 5 inches). Give your answer in terms of the nearest day.

Floods: July 24 through October 7. Droughts: February 4 through March 27

For the following exercises, find the amplitude, period, and frequency of the given function.

The displacement $\text{\hspace{0.17em}}h\left(t\right)\text{\hspace{0.17em}}$ in centimeters of a mass suspended by a spring is modeled by the function $\text{\hspace{0.17em}}h\left(t\right)=8\mathrm{sin}\left(6\pi t\right),$ where $\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ is measured in seconds. Find the amplitude, period, and frequency of this displacement.

The displacement $\text{\hspace{0.17em}}h\left(t\right)\text{\hspace{0.17em}}$ in centimeters of a mass suspended by a spring is modeled by the function $\text{\hspace{0.17em}}h\left(t\right)=11\mathrm{sin}\left(12\pi t\right),$ where $\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ is measured in seconds. Find the amplitude, period, and frequency of this displacement.

Amplitude: 11, period: $\text{\hspace{0.17em}}\frac{1}{6},\text{\hspace{0.17em}}$ frequency: 6 Hz

The displacement $\text{\hspace{0.17em}}h\left(t\right)\text{\hspace{0.17em}}$ in centimeters of a mass suspended by a spring is modeled by the function $\text{\hspace{0.17em}}h\left(t\right)=4\mathrm{cos}\left(\frac{\pi }{2}t\right),$ where $\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ is measured in seconds. Find the amplitude, period, and frequency of this displacement.

For the following exercises, construct an equation that models the described behavior.

The displacement $\text{\hspace{0.17em}}h\left(t\right),$ in centimeters, of a mass suspended by a spring is modeled by the function $\text{\hspace{0.17em}}h\left(t\right)=-5\text{\hspace{0.17em}}\mathrm{cos}\left(60\pi t\right),$ where $\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ is measured in seconds. Find the amplitude, period, and frequency of this displacement.

Amplitude: 5, period: $\text{\hspace{0.17em}}\frac{1}{30},$ frequency: 30 Hz

For the following exercises, construct an equation that models the described behavior.

A deer population oscillates 19 above and below average during the year, reaching the lowest value in January. The average population starts at 800 deer and increases by 160 each year. Find a function that models the population, $\text{\hspace{0.17em}}P,$ in terms of months since January, $\text{\hspace{0.17em}}t.$

A rabbit population oscillates 15 above and below average during the year, reaching the lowest value in January. The average population starts at 650 rabbits and increases by 110 each year. Find a function that models the population, $\text{\hspace{0.17em}}P,$ in terms of months since January, $\text{\hspace{0.17em}}t.$

$P\left(t\right)=-15\mathrm{cos}\left(\frac{\pi }{6}t\right)+650+\frac{55}{6}t$

The average annual population increase of a pack of wolves is 25.
how do you find the period of a sine graph
Period =2π if there is a coefficient (b), just divide the coefficient by 2π to get the new period
Am
if not then how would I find it from a graph
Imani
by looking at the graph, find the distance between two consecutive maximum points (the highest points of the wave). so if the top of one wave is at point A (1,2) and the next top of the wave is at point B (6,2), then the period is 5, the difference of the x-coordinates.
Am
you could also do it with two consecutive minimum points or x-intercepts
Am
I will try that thank u
Imani
Case of Equilateral Hyperbola
ok
Zander
ok
Shella
f(x)=4x+2, find f(3)
Benetta
f(3)=4(3)+2 f(3)=14
lamoussa
14
Vedant
pre calc teacher: "Plug in Plug in...smell's good" f(x)=14
Devante
8x=40
Chris
Explain why log a x is not defined for a < 0
the sum of any two linear polynomial is what
Momo
how can are find the domain and range of a relations
A cell phone company offers two plans for minutes. Plan A: $15 per month and$2 for every 300 texts. Plan B: $25 per month and$0.50 for every 100 texts. How many texts would you need to send per month for plan B to save you money?
6000
Robert
more than 6000
Robert
can I see the picture
How would you find if a radical function is one to one?
how to understand calculus?
with doing calculus
SLIMANE
Thanks po.
Jenica
Hey I am new to precalculus, and wanted clarification please on what sine is as I am floored by the terms in this app? I don't mean to sound stupid but I have only completed up to college algebra.
I don't know if you are looking for a deeper answer or not, but the sine of an angle in a right triangle is the length of the opposite side to the angle in question divided by the length of the hypotenuse of said triangle.
Marco
can you give me sir tips to quickly understand precalculus. Im new too in that topic. Thanks
Jenica
if you remember sine, cosine, and tangent from geometry, all the relationships are the same but they use x y and r instead (x is adjacent, y is opposite, and r is hypotenuse).
Natalie
it is better to use unit circle than triangle .triangle is only used for acute angles but you can begin with. Download any application named"unit circle" you find in it all you need. unit circle is a circle centred at origine (0;0) with radius r= 1.
SLIMANE
What is domain
johnphilip
the standard equation of the ellipse that has vertices (0,-4)&(0,4) and foci (0, -15)&(0,15) it's standard equation is x^2 + y^2/16 =1 tell my why is it only x^2? why is there no a^2?
what is foci?
This term is plural for a focus, it is used for conic sections. For more detail or other math questions. I recommend researching on "Khan academy" or watching "The Organic Chemistry Tutor" YouTube channel.
Chris
how to determine the vertex,focus,directrix and axis of symmetry of the parabola by equations