In this project, I will be finding periodic data to create ...



In this project, I will be finding periodic data to create a sine or cosine model. I will then use that model to extrapolate information. The set of data that I found was the average monthly temperatures for Santiago, Chile.

[pic]

|Month |Average Monthly |

| |High Temperatures for |

| |Santiago, Chile |

|January |85 |

|February |85 |

|March |81 |

|April |72 |

|May |65 |

|June |58 |

|July |58 |

|August |61 |

|September |65 |

|October |72 |

|November |77 |

|December |83 |

Below is the data graphed in a scatter plot:

[pic]

This data is obviously in the shape of a cosine curve. This is due to the fact that the Southern Hemisphere has their coldest months in the middle of the year May – September. Using the steps to find a cosine model, I will find a model that best fits this data.

Step 1: Find the amplitude:

To find the amplitude one must take the maximum temperature – minimum temperature and divide it by 2. This is because the amplitude is the distance from the primary axis to the maximum or minimum point of the data.

Maximum Temperature: 85 Minimum Temperature: 58

[pic], so the value of a in the model: y = acos(bx + c) + d is 13.5.

Step 2: Find the vertical translation:

To find the vertical translation one must take the maximum temperature + the minimum temperature and divide it by 2. This is because the vertical shift moves the primary axis up however many units from the x – axis.

[pic], so the value of d in the model is 71.5.

So far, I have y = 13.5 cos(bx +c) + 71.5, now I must find the values of b and c.

Step 3: Find the period, hence finding the value of b:

To fine the period, I just look at how long it takes to complete one cycle. For this data, one cycle is completed in 1 year or twelve months. Therefore, the period is 12. So to find the value of b I must solve the equation:

[pic]. [pic]; therefore 12b = 360, and b = 30

So, my model is y = 13.5 cos (30x + c) + 71.5. Now to find my horizontal translation.

To do this, I must graph both the data and the model on the same window. Then I must find out how far to move the graph left or right by choosing the minimum values of both the data and the model.

[pic] [pic] Minimum (7, 58) Minimum (6, 58)

Therefore, the graph must be shifted 1 unit to the right. The horizontal shift is determined by the value of [pic]. Therefore, for my model I need [pic]= 1. I already know that b = 30, so [pic]and then c must = 30. Because I need the graph to shift right, the value of c must be negative.

So my final model is y = 13.5 cos (30x - 30) + 71.5, and the graph should look like:

[pic]

As you can see, this is a very good model to fit the data. Most points fall on the model, and it would be a great model to use to predict future temperature.

So, according to the model, I can figure the average temperature in the month of February to be:

Y = 13.5 (cos 30 (2) – 30) + 71.5. The average temperature according to my model is 83.2, and according to the data the average temperature for the data is 85. In fact, my model for July exactly matches the data for July. If I plug in the value of 7, I will get 58 with my model which is exactly what the data says! How cool!!

During this project, I learned how to use Microsoft Equation 3.0, and I learned that fitting a cosine model to periodic data is not as hard as it looks. I had fun taking pictures of my graphs, and using my calculator to guide me through the steps.

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