simusoidal functions

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rachelmaddie

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Hi. Can someone please check my work for this problem?
AA5CE535-7A54-45EF-96B4-48B258BD0806.jpeg
a
Amplitude = maximum - minimum/2 = (77-40)/2 = 18.5

b
Vertical shift = maximum +minimum/2. (77 + 40)/2 = 58.5

c
The period of the function is (data range/# of cycles), which in this case 24 hrs/1 = 24 hrs

d
We know that t =0 at 5 pm, so we want to find a formula for the temperature as a function of time f(t) where f(0) = the temperature at 5 pm (77 degrees)

So now we need to calculate the actual value of our phase shift, which is 25%*period = 25%*24 = 6
So our period = 24 = 2*pi/B
We start with:
24 = 2*pi/B

Multiply both sides by B:
24*B = (2*pi/B)*B

Simplify - the B's on the right side cancel out:
24*B = 2*pi

Divide both sides by 24:
(24*B)/24 = (2*pi)/24

Simplify again - the 24s on the left side cancel out:
B = pi/12

To write a sinusoidal function that models this temp. variation use the formula:
y = A sin(B(x + C)) + D
where Amplitude is A
Period is 2pi/B
Phase shift is C (positive to the left)
Vertical shift is D

Substituting this into formula we get:
y = 18.5*sin((pi/12)*(t + 6)) + 58.5

e
Find the value of t at 10 AM, replace t in the equation we found with that value, calculate the value of the equation with that t, and then compare the result to the recorded temperature at 10 AM
Since t = 0 at 5 pm, then t = -1 at 4 pm, -2 at 3 pm, t = -7 at 10 am.
Replace t with -7 in the equation we found for part d and calculate the value of the equation.
y = 18.5*sin((pi/12)*(-7 + 6)) + 58.5
y = 53.7118476656

The model’s temperature at 10AM is 52 degrees.
In comparison, the numbers are close but not identical, which is to be expected because one is a real measurement and the other is from a model.
 
rachelmaddie said:
a
Amplitude = maximum - minimum/2 = (77-40)/2 = 18.5

b
Vertical shift = maximum +minimum/2. (77 + 40)/2 = 58.5

...

In comparison, the numbers are close but not identical, which is to be expected because one is a real measurement and the other is from a model.
Click to expand...
This is the question I asked when you stated so precisely a) and b). It's a model. One would not expect it to hit values exactly, and yet you used exact values to produce Amplitude and Vertical Shift. It's probably okay for this assignment, but it is not a good way to build a model in general.

For example, I took the average of all the values and produced Vertical Shift = 57.4º. That may be better or worse than your average of Min and Max.

I played around with it for a while and found a good approximation to be:
Vertical Shift = 57.4
Amplitude = 19.2
Phase Shift = 2.7 (About 10.3 hours)

1597108931449.png

In other words, yours is pretty good, excepting the phase shift. That could be improved. 25% of the period is a very arbitrary assumption. Where does it actually cross the Vertical Shift line? That would be a better estimate. You can see that it crosses your 58.5º between 10-11 AM and between 9-10 PM. This lines up nicely with my 10.3 hours.

Anyway, let's learn two things:
1) There is A LOT to learn about this sort of thing. No need to try to cram it all in your head, today. I was just having some fun. :)
2) What you did was GREAT. Let's work on the Phase Shift a little.
 
tkhunny said:
This is the question I asked when you stated so precisely a) and b). It's a model. One would not expect it to hit values exactly, and yet you used exact values to produce Amplitude and Vertical Shift. It's probably okay for this assignment, but it is not a good way to build a model in general.

For example, I took the average of all the values and produced Vertical Shift = 57.4º. That may be better or worse than your average of Min and Max.

I played around with it for a while and found a good approximation to be:
Vertical Shift = 57.4
Amplitude = 19.2
Phase Shift = 2.7 (About 10.3 hours)

View attachment 20944

In other words, yours is pretty good, excepting the phase shift. That could be improved. 25% of the period is a very arbitrary assumption. Where does it actually cross the Vertical Shift line? That would be a better estimate. You can see that it crosses your 58.5º between 10-11 AM and between 9-10 PM. This lines up nicely with my 10.3 hours.

Anyway, let's learn two things:
1) There is A LOT to learn about this sort of thing. No need to try to cram it all in your head, today. I was just having some fun. :)
2) What you did was GREAT. Let's work on the Phase Shift a little.
Click to expand...
Would I still receive full credit for my work even if it’s not as precise from your estimations?
 
rachelmaddie said:
Would I still receive full credit for my work even if it’s not as precise from your estimations?
Click to expand...
You would have to ask the rubric for that. It's a fund idea. Have a class contest and see who can get the closest (based on some defined measure of "close").

For me, as long as you exposed your assumptions and gave it a good effort, consistent with class discussion and/or textbook presentation, it shouldn't matter how "close" it is. You're learning, remember? When it's your job, this will all be different. :)
 
Amplitude = maximum - minimum/2 = (77-40)/2 = 18.5

Vertical shift = maximum +minimum/2 = (77 + 40)/2 = 58.5

There are 6 equal signs above. 4 of them are not valid! Can you find them?
 
Jomo said:
Amplitude = maximum - minimum/2 = (77-40)/2 = 18.5

Vertical shift = maximum +minimum/2 = (77 + 40)/2 = 58.5

There are 6 equal signs above. 4 of them are not valid! Can you find them?
Click to expand...
Yes.
Amplitude:
maximum - minimum/2
(77-40)/2 = 18.5

Vertical shift
maximum +minimum/2
(77 + 40)/2 = 58.5
 
rachelmaddie said:
Yes.
Amplitude:
maximum - minimum/2
(77-40)/2 = 18.5

Vertical shift
maximum +minimum/2
(77 + 40)/2 = 58.5
Click to expand...
No sorry, but maximum - minimum/2 = (77-40)/2 is simply not true. From what I read you have been told about this type of error many times!
Also, maximum +minimum/2 = (77 + 40)/2 is NOT correct!
 
Jomo said:
No sorry, but maximum - minimum/2 = (77-40)/2 is simply not true. From what I read you have been told about this type of error many times!
Also, maximum +minimum/2 = (77 + 40)/2 is NOT correct!
Click to expand...
What did I do wrong?
 
tkhunny said:
You fixed it around "77-40", making sure it wasn't 77 - 40/2 = 77 - 20 = 57
Why not do the same around "maximum - minimum"?
Click to expand...
I still do not understand. You mean like this?

(maximum - minimum)/2 = (77-40)/2
Also, (maximum +minimum)/2 = (77 + 40)/2
 
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