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
$\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ equal the distance from the tip of the hour hand to the ceiling
$\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ 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
$\left(0,30\right)$
3 PM
54 in
$\left(3,54\right)$
6 PM
78 in
$\left(6,78\right)$
9 PM
54 in
$\left(9,54\right)$
Midnight
30 in
$\left(12,30\right)$
To model an equation, we first need to find the amplitude.
There is no horizontal shift, so
$\text{\hspace{0.17em}}C=0.\text{\hspace{0.17em}}$ Since the function begins with the minimum value of
$\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ when
$\text{\hspace{0.17em}}x=0\text{\hspace{0.17em}}$ (as opposed to the maximum value), we will use the cosine function with the negative value for
$\text{\hspace{0.17em}}A.\text{\hspace{0.17em}}$ In the form
$\text{\hspace{0.17em}}y=A\text{\hspace{0.17em}}\mathrm{cos}(Bx\pm C)+D,\text{\hspace{0.17em}}$ the equation is
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
The cycle repeats every 12 hours; therefore,
$\text{\hspace{0.17em}}B\text{\hspace{0.17em}}$ is
$\frac{2\pi}{12}=\frac{\pi}{6}$
There is a vertical translation of
$\text{\hspace{0.17em}}\frac{(15+8)}{2}=\mathrm{11.5.}\text{\hspace{0.17em}}$ Since the value of the function is at a maximum at
$\text{\hspace{0.17em}}t=0,$ we will use the cosine function, with the positive value for
$\text{\hspace{0.17em}}A.$
The daily temperature in the month of March in a certain city varies from a low of
$\text{\hspace{0.17em}}24\text{\xb0F}\text{\hspace{0.17em}}$ to a high of
$\text{\hspace{0.17em}}40\text{\xb0F}\text{.}\text{\hspace{0.17em}}$ Find a sinusoidal function to model daily temperature and sketch the graph. Approximate the time when the temperature reaches the freezing point
$\text{\hspace{0.17em}}32\text{\xb0F}\text{.}\text{\hspace{0.17em}}$ Let
$\text{\hspace{0.17em}}t=0\text{\hspace{0.17em}}$ correspond to noon.
$y=8\mathrm{sin}\left(\frac{\pi}{12}t\right)+32$ The temperature reaches freezing at noon and at midnight.
The average person’s blood pressure is modeled by the function
$\text{\hspace{0.17em}}f\left(t\right)=20\text{\hspace{0.17em}}\mathrm{sin}\left(160\pi t\right)+100,\text{\hspace{0.17em}}$ where
$\text{\hspace{0.17em}}f\left(t\right)\text{\hspace{0.17em}}$ represents the blood pressure at time
$\text{\hspace{0.17em}}t,$ measured in minutes. Interpret the function in terms of period and frequency. Sketch the graph and find the blood pressure reading.
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.
I've run into this:
x = r*cos(angle1 + angle2)
Which expands to:
x = r(cos(angle1)*cos(angle2) - sin(angle1)*sin(angle2))
The r value confuses me here, because distributing it makes:
(r*cos(angle2))(cos(angle1) - (r*sin(angle2))(sin(angle1))
How does this make sense? Why does the r distribute once
this is an identity when 2 adding two angles within a cosine. it's called the cosine sum formula. there is also a different formula when cosine has an angle minus another angle it's called the sum and difference formulas and they are under any list of trig identities
Brad
How can you tell what type of parent function a graph is ?
generally by how the graph looks and understanding what the base parent functions look like and perform on a graph
William
if you have a graphed line, you can have an idea by how the directions of the line turns, i.e. negative, positive, zero
William
y=x will obviously be a straight line with a zero slope
William
y=x^2 will have a parabolic line opening to positive infinity on both sides of the y axis
vice versa with y=-x^2 you'll have both ends of the parabolic line pointing downward heading to negative infinity on both sides of the y axis
William
y=x will be a straight line, but it will have a slope of one. Remember, if y=1 then x=1, so for every unit you rise you move over positively one unit. To get a straight line with a slope of 0, set y=1 or any integer.
Aaron
yes, correction on my end, I meant slope of 1 instead of slope of 0
Typically a function 'f' will take 'x' as input, and produce 'y' as output. As
'f(x)=y'.
According to Google,
"The range of a function is the complete set of all possible resulting values of the dependent variable (y, usually), after we have substituted the domain."
Thomas
Sorry, I don't know where the "Â"s came from. They shouldn't be there. Just ignore them. :-)
Thomas
GREAT ANSWER THOUGH!!!
Darius
Thanks.
Thomas
Â
Thomas
It is the Â that should not be there. It doesn't seem to show if encloses in quotation marks.
"Â" or 'Â' ... Â
I've been struggling so much through all of this. my final is in four weeks 😭
Tiffany
this book is an excellent resource! have you guys ever looked at the online tutoring? there's one that is called "That Tutor Guy" and he goes over a lot of the concepts
Darius
thank you I have heard of him. I should check him out.
Tiffany
is there any question in particular?
Joe
I have always struggled with math. I get lost really easy, if you have any advice for that, it would help tremendously.
Tiffany
Sure, are you in high school or college?
Darius
Hi, apologies for the delayed response. I'm in college.
The center is at (3,4) a focus is at (3,-1) and the lenght of the major axis is 26 what will be the answer?
Rima
I done know
Joe
What kind of answer is that😑?
Rima
I had just woken up when i got this message
Joe
Can you please help me. Tomorrow is the deadline of my assignment then I don't know how to solve that
Rima
i have a question.
Abdul
how do you find the real and complex roots of a polynomial?
Abdul
@abdul with delta maybe which is b(square)-4ac=result then the 1st root -b-radical delta over 2a and the 2nd root -b+radical delta over 2a. I am not sure if this was your question but check it up
Nare
This is the actual question: Find all roots(real and complex) of the polynomial f(x)=6x^3 + x^2 - 4x + 1
Abdul
@Nare please let me know if you can solve it.
Abdul
I have a question
juweeriya
hello guys I'm new here? will you happy with me
mustapha
The average annual population increase of a pack of wolves is 25.
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