Guidance on modeling a function

[Simplemind]

Good afternoon!

I need some help understanding what I am doing on a project. I am placing below what is being asked and what I have done to the point where I am stuck. Some gentle guidance to get me back on track would be greatly appreciated because learning this is the most important thing to me, not just getting answers. That way if there is an extra credit problem later I can do it on my own

Okay the project is about Therms and Temperature in a 24 month period. Basically the information for month 15 is missing and at the end of this project we should be able to model it well enough to figure out what the therm and temperature was for the missing month. I have to place the data in here so that my answer can be checked. I will try to keep it simple.

2003
Months 1 2 3 4 5 6 7 8 9 10 11 12
Therms 81 75 57 37 17 13 11 10 11 17 22 48
Temp 18 15 28 41 54 61 62 65 62 55 52 36

2004
Months 13 14 15 16 17 18 19 20 21 22 23 24
Therms 72 68 ? 40 30 14 10 10 11 12 27 54
Temp 32 23 ? 40 53 60 63 65 62 59 46 29

I plotted them like it asked and I got a sinsusoidal graph proportionate to what I should expect from gas usage vs temperature. I feel very confident I did this correct. Here is where it got confusing to me and I would love someone to check over my work and tell me what I need to look back at, as well, help me figure out what one of the questions is asking me to do in my preliminary formula.

For the function F(t) = Asin[B(t-C)]+D GAS-USAGE ONLY(therms) I had to find A the amplitude B the period C the phase shift D is the vertical shift

D seemed like the first thing to try to find. It is the average of the average max and average min The average Max was 76.5 ( I took the two highest temperatures and averaged them) The Min was 10 (similarly I took the 2 min and averaged) now the average of 76.5 and 10 is 43.25 So for D I have D= 43.25

The amplitude is the absolute value of the min height minus D and/or the absolute value of the max height minus D...It should be the same answer so I did
I 10-43.25I = I-33.25I = 33.25
I 76.5 - 43.25 I = 33.25
A = 33.25

For the period B it is 2pi / the number of months in the cycle which when I graph it one cycle looks like 12 months. This would calculate as 2pi/12 and reduced would = pi/6 So B = pi/6

Then it said write a preliminary formula for F letting C = 0 for the moment
That seems easy enough I get F(t) = 33.25sin[pi/6(t-0)]+43.25

Here is where I get stuck because now I have to figure out C and C/B will equal my phase shift

The next question says that on my chart above the max gas usage first occurs when t = 1. When you look at it month 1 is 81 degrees and that is the highest temperature recorded for the charts. Makes sense. Then here is where I get stuck. it asks for what value of t does my preliminary version of F first reach its maximum value? Then I am suppose to use that information to find a good value for C. I honestly have looked at this for 2 weeks now and it just isnt clicking what I am suppose to do here and I bet it is something simple. Like I said before I am a big believer in learning and I want to understand how this works so I am not looking for someone to give me any answers...just some guidance. This class has been the first one that has made me feel like I might not be smart enough to go on with school so I've been down in the dumps over not getting this on my own. I appreciate any and all responses even if it is just positive feedback telling me I can do this.

Thanks
SM

 

[wjm11]

Okay the project is about Therms and Temperature in a 24 month period. Basically the information for month 15 is missing and at the end of this project we should be able to model it well enough to figure out what the therm and temperature was for the missing month. I have to place the data in here so that my answer can be checked. I will try to keep it simple.

2003
Months 1 2 3 4 5 6 7 8 9 10 11 12
Therms 81 75 57 37 17 13 11 10 11 17 22 48
Temp 18 15 28 41 54 61 62 65 62 55 52 36

2004
Months 13 14 15 16 17 18 19 20 21 22 23 24
Therms 72 68 ? 40 30 14 10 10 11 12 27 54
Temp 32 23 ? 40 53 60 63 65 62 59 46 29

I plotted them like it asked and I got a sinsusoidal graph proportionate to what I should expect from gas usage vs temperature. I feel very confident I did this correct. Here is where it got confusing to me and I would love someone to check over my work and tell me what I need to look back at, as well, help me figure out what one of the questions is asking me to do in my preliminary formula.

For the function F(t) = Asin[B(t-C)]+D GAS-USAGE ONLY(therms) I had to find A the amplitude B the period C the phase shift D is the vertical shift

D seemed like the first thing to try to find. It is the average of the average max and average min The average Max was 76.5 ( I took the two highest temperatures and averaged them) The Min was 10 (similarly I took the 2 min and averaged) now the average of 76.5 and 10 is 43.25 So for D I have D= 43.25

The amplitude is the absolute value of the min height minus D and/or the absolute value of the max height minus D...It should be the same answer so I did
I 10-43.25I = I-33.25I = 33.25
I 76.5 - 43.25 I = 33.25
A = 33.25

For the period B it is 2pi / the number of months in the cycle which when I graph it one cycle looks like 12 months. This would calculate as 2pi/12 and reduced would = pi/6 So B = pi/6

Then it said write a preliminary formula for F letting C = 0 for the moment
That seems easy enough I get F(t) = 33.25sin[pi/6(t-0)]+43.25

Here is where I get stuck because now I have to figure out C and C/B will equal my phase shift

The next question says that on my chart above the max gas usage first occurs when t = 1. When you look at it month 1 is 81 degrees and that is the highest temperature recorded for the charts. Makes sense. Then here is where I get stuck. it asks for what value of t does my preliminary version of F first reach its maximum value? Then I am suppose to use that information to find a good value for C. I honestly have looked at this for 2 weeks now and it just isnt clicking what I am suppose to do here and I bet it is something simple.

Hello, SM,

You’re off to a great start. Let’s consider where you are right now:

F(t) = 33.25sin[pi/6(t-C)]+43.25

[Personally, I prefer a slightly different form of F(t) = Asin[B(t-C)]+D. I prefer to write it as F(t) = Asin[Bt-C]+D. That just makes C the phase shift instead of the “C/B” term that you are using. No matter.]

First let’s talk about sinusoids in general. Keep in mind that a sinusoid can be written as either a sine function or a cosine function. YOU get to choose. The data you have can be expressed either way.

You mentioned month 1, so let’s look there; the therm value is at a peak at month 1. Which function, sine or cosine, starts at a peak? That would be the cosine function, cos(0) = 1. (The sine function, on the other hand, starts at the midrange of the data, sin(0) = 0.) Since you are looking at the month 1 peak, you might choose the cosine function for your equation:

F(t) = 33.25cos[pi/6(t-C)]+43.25

You can see that this peak starting point is at month 1 (and not at month 0), so this cosine function has been “moved” (phase-shifted/translated) 1 unit to the right. Therefore, your C/B term equals 1. Rearranging, we find that C = 1/B = 6/pi.

F(t) = 33.25cos[pi/6(t-(6/pi))]+43.25

[I will mention at this point why I prefer “my” form of the equation; if you distribute the “pi/6” term, the equation becomes simply

33.25cos[(pi/6)t - 1]+43.25

In this form, the phase shift of 1 can be easily read right in the equation.]

If you wish to write this equation using the sine function, you need to find a month that represent midpoint therm levels – perhaps month 4. You would then use 4 as your phase shift.

Make sense?