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.
Questions & Answers
can you not take the square root of a negative number
No because a negative times a negative is a positive. No matter what you do you can never multiply the same number by itself and end with a negative
lurverkitten
Actually you can. you get what's called an Imaginary number denoted by i which is represented on the complex plane. The reply above would be correct if we were still confined to the "real" number line.
Liam
Suppose P= {-3,1,3} Q={-3,-2-1} and R= {-2,2,3}.what is the intersection
Someone should please solve it for me
Add 2over ×+3 +y-4 over 5
simplify (×+a)with square root of two -×root 2 all over a
multiply 1over ×-y{(×-y)(×+y)} over ×y
For the first question, I got (3y-2)/15
Second one, I got Root 2
Third one, I got 1/(y to the fourth power)
I dont if it's right cause I can barely understand the question.
Is under distribute property, inverse function, algebra and addition and multiplication function; so is a combined question
graph the following linear equation using intercepts method.
2x+y=4
Ashley
how
Wargod
what?
John
ok, one moment
UriEl
how do I post your graph for you?
UriEl
it won't let me send an image?
UriEl
also for the first one... y=mx+b so.... y=3x-2
UriEl
y=mx+b
you were already given the 'm' and 'b'.
so..
y=3x-2
Tommy
Please were did you get y=mx+b from
Abena
y=mx+b is the formula of a straight line.
where m = the slope & b = where the line crosses the y-axis. In this case, being that the "m" and "b", are given, all you have to do is plug them into the formula to complete the equation.
Tommy
thanks Tommy
Nimo
0=3x-2
2=3x
x=3/2
then .
y=3/2X-2
I think
Given
co ordinates for x
x=0,(-2,0)
x=1,(1,1)
x=2,(2,4)
neil
"7"has an open circle and "10"has a filled in circle who can I have a set builder notation
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
strategies to form the general term
carlmark
consider r(a+b) = ra + rb. The a and b are the trig identity.
Mike
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