Periodic Functions Mathematics & Economics Math Problem (Math Problem Sample)

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Using trigonometric functions to model climate
Background
The sine and cosine functions can be used to model fluctuations in temperature data throughout the year.
An equation that can be used to model these data is of the form: y = A cos B(x - C) + D,
where A,B,C,D, are constants, y is the temperature in °C and x is the month (1–12).
Directions
Your task is to create a model of the data to predict the times during the year that a location would be pleasant to visit. This may be when the average monthly temperature is over 14°C.
MonthJanFebMarAprMayJunJulAugSepOctNovDec
°C17.817.916.614.412.010.29.59.911.312.914.516.4
Above is a table of average monthly temperatures at Wellington Airport from 1971-2000
Graph this data.
Write a trigonometric equation using the cosine function that best models this situation.
Rewrite the equation using the sine function.
For which places would the sine function be a more obvious model for the temperature data?
The long-term average temperatures for Wellington were given above.
Below is a table of mean monthly temperatures for Wellington Airport for the year 2000.
MonthJanFebMarAprMayJunJulAugSepOctNovDec
°C17.117.615.715.213.311.010.710.312.013.713.117.8
Graph this data and write a trigonometric equation to model this data.
Explain why this model does not fit as well as in the previous example.
What features of the graph determine the values for A, B, C and D?
Investigate temperature data for other places.
Solve the trigonometric equation derived above to find when the mean temperature will be over 14°C in Wellington.
How can the answer be checked on the graph?
Is it sensible to convert the solution to months and days?
Investigate a model for rainfall data or relative humidity.
How do these models relate to the equation for temperature?
Teacher Background
Objectives: