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Describing periodic motion

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 y equal the distance from the tip of the hour hand to the ceiling x 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 ( 0 , 30 )
3 PM 54 in ( 3 , 54 )
6 PM 78 in ( 6 , 78 )
9 PM 54 in ( 9 , 54 )
Midnight 30 in ( 12 , 30 )

To model an equation, we first need to find the amplitude.

| A | = | 78 30 2 |      = 24

The clock’s cycle repeats every 12 hours. Thus,

B = 2 π 12     = π 6

The vertical shift is

D = 78 + 30 2     = 54

There is no horizontal shift, so C = 0. Since the function begins with the minimum value of y when x = 0 (as opposed to the maximum value), we will use the cosine function with the negative value for A . In the form y = A cos ( B x ± C ) + D , the equation is

y = −24 cos ( π 6 x ) + 54

See [link] .

Graph of the function y=-24cos(pi/6 x)+54 using the five key points: (0,30), (3,54), (6,78), (9,54), (12,30).
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Determining a model for tides

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

| A | = | ( 15 7 ) 2 |      = 4

The cycle repeats every 12 hours; therefore, B is

2 π 12 = π 6

There is a vertical translation of ( 15 + 8 ) 2 = 11.5. Since the value of the function is at a maximum at t = 0 , we will use the cosine function, with the positive value for A .

y = 4 cos ( π 6 ) t + 11

See [link] .

Graph of the function y=4cos(pi/6 t) + 11 from 0 to 12. The midline is y=11, three key points are (0,15), (6,7), and (12, 15).
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The daily temperature in the month of March in a certain city varies from a low of 24 °F to a high of 40 °F . Find a sinusoidal function to model daily temperature and sketch the graph. Approximate the time when the temperature reaches the freezing point 32 °F . Let t = 0 correspond to noon.

y = 8 sin ( π 12 t ) + 32
The temperature reaches freezing at noon and at midnight.

Graph of the function y=8sin(pi/12 t) + 32 for temperature. The midline is at 32. The times when the temperature is at 32 are midnight and noon.
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Interpreting the periodic behavior equation

The average person’s blood pressure is modeled by the function f ( t ) = 20 sin ( 160 π t ) + 100 , where f ( t ) represents the blood pressure at time t , measured in minutes. Interpret the function in terms of period and frequency. Sketch the graph and find the blood pressure reading.

The period is given by

2 π ω = 2 π 160 π       = 1 80

In a blood pressure function, frequency represents the number of heart beats per minute. Frequency is the reciprocal of period and is given by

ω 2 π = 160 π 2 π = 80

See the graph in [link] .

Graph of the function f(t) = 20sin(160 * pi * t) + 100 for blood pressure. The midline is at 100.
The blood pressure reading on the graph is 120 80   ( maximum minimum ) .
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Modeling harmonic motion functions

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
Sharon Reply
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
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.
Suppose P= {-3,1,3} Q={-3,-2-1} and R= {-2,2,3}.what is the intersection
Elaine Reply
can I get some pretty basic questions
Ama Reply
In what way does set notation relate to function notation
is precalculus needed to take caculus
Amara Reply
It depends on what you already know. Just test yourself with some precalculus questions. If you find them easy, you're good to go.
the solution doesn't seem right for this problem
Mars Reply
what is the domain of f(x)=x-4/x^2-2x-15 then
Conney Reply
x is different from -5&3
All real x except 5 and - 3
how to prroved cos⁴x-sin⁴x= cos²x-sin²x are equal
jeric Reply
Don't think that you can.
By using some imaginary no.
how do you provided cos⁴x-sin⁴x = cos²x-sin²x are equal
jeric Reply
What are the question marks for?
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
Abena Reply
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
find the equation of the line if m=3, and b=-2
Ashley Reply
graph the following linear equation using intercepts method. 2x+y=4
ok, one moment
how do I post your graph for you?
it won't let me send an image?
also for the first one... y=mx+b so.... y=3x-2
y=mx+b you were already given the 'm' and 'b'. so.. y=3x-2
Please were did you get y=mx+b from
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.
thanks Tommy
0=3x-2 2=3x x=3/2 then . y=3/2X-2 I think
co ordinates for x x=0,(-2,0) x=1,(1,1) x=2,(2,4)
"7"has an open circle and "10"has a filled in circle who can I have a set builder notation
Fiston Reply
Where do the rays point?
x=-b+_Гb2-(4ac) ______________ 2a
Ahlicia Reply
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
Carlos Reply
so good
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
strategies to form the general term
consider r(a+b) = ra + rb. The a and b are the trig identity.
How can you tell what type of parent function a graph is ?
Mary Reply
generally by how the graph looks and understanding what the base parent functions look like and perform on a graph
if you have a graphed line, you can have an idea by how the directions of the line turns, i.e. negative, positive, zero
y=x will obviously be a straight line with a zero slope
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
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.
yes, correction on my end, I meant slope of 1 instead of slope of 0
what is f(x)=
Karim Reply
I don't understand
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."
Sorry, I don't know where the "Â"s came from. They shouldn't be there. Just ignore them. :-)
It is the  that should not be there. It doesn't seem to show if encloses in quotation marks. "Â" or 'Â' ... Â
Now it shows, go figure?
Practice Key Terms 2

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