The following problems rely on the data contained in this spreadsheet. It has four tabs, and each problem will refer you to the appropriate tab. Note that this is real world data, not data generated for an “easy” math problem. Your model is NOT likely to hit all the data points perfectly.
Sunrise
1 – Copy the full year of Sunrise data from columns B and C on the “Sun + Moon” tab of the spreadsheet linked to above into your preferred graphing software.
2 – Analyze the data and develop a simple model for sunrise times in Portland, Maine as a function of the Day #, using one Sine or Cosine function and only one instance of your variable.
3 – Graph the sunrise times and your model’s results on the same graph
4 – What does your model predict well? Why?
5 – What does your model not predict well? What do you know (or learn from research) that would explain why sunrise times in some seasons in Maine differ from your simple model?
Birthday Tides
1a – Open the “All Tides” tab in the spreadsheet that is linked to above, find your birthday in the left column, and make a note of the “Elapsed Days” and “Water Height” of the first four tides on your birthday on your working document. The first tide could be high or low, depending on the time of year.
1b – Develop a simple model for the four tide Heights you noted as a function of Elapsed Days, using one Sine or Cosine function and only one instance of your variable.
1c – Use a calculator or software to overlay a graph of your model with a graph of the four points
1d – Which points does your model predict well? Why?
1e – Which points does your model not predict well? Why?
1f – Can you think of a way to improve your model if the restrictions were dropped, i.e. using more than one Sine or Cosine function and/or more than one instance of your variable?
2a – Starting with your model from question 1, but without restricting yourself to one Sine or Cosine function and only one instance of your variable, improve your model’s predictions using the ideas you came up with in the last step of question 1. In particular, can you improve your model by using functions of your variable instead of constants for the amplitude and/or vertical translation?
2b – Use a calculator or graphing software to overlay a graph of both models with a graph of your four data points
2c – Which model fits your four data points data better?
2d – Which model do you think will predict earlier or later data points better? Why?
3a – Expand your data set by including either the previous or the next 8 tide Elapsed Days and Heights (before or after those used above) from the “All Tides” spreadsheet.
3b – Add the 8 new points to your graph of both models
3c – Which model fits all 12 data points best? Why?
3d – Develop a model that fits all 12 data points better than either of the above models by making the amplitude a periodic function of your variable.
3f – Use your calculator or graphing software to compare the accuracy of your models’ predictions. Which model looks as though it fits the data better?
MathMaine 