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Chapter 5 Project
Modeling the Illumination of the Moon
From the earth, the Moon appears to be a circular disk in the sky that is illuminated to varying degrees by direct sunlight. During each lunar orbit the Moon varies from a status of being a New Moon with no visible illumination to that of a Full Moon which is fully illuminated by direct sunlight. The United States Naval Observatory has developed a mathematical model to find the fraction of the Moon's visible disk that is illuminated by the Sun. The data in the table below (obtained from the U.S. Naval Observatory web site, http://www.usno.navy.mil/, Astronomical Applications Department) shows the fraction of the Moon illuminated at midnight for each day in January 2000.
| Fraction of the Moon illuminated, January 2000 | |||||||
|---|---|---|---|---|---|---|---|
| Day# | Fraction Eliminated |
Day# | Fraction Eliminated |
Day# | Fraction Eliminated |
Day# | Fraction Eliminated |
| 1 | 0.25 | 9 | 0.05 | 17 | 0.78 | 25 | 0.80 |
| 2 | 0.18 | 10 | .11 | 18 | 0.87 | 26 | 0.71 |
| 3 | 0.11 | 11 | 0.18 | 19 | 0.94 | 27 | 0.61 |
| 4 | 0.06 | 12 | 0.26 | 20 | 0.99 | 28 | 0.51 |
| 5 | 0.02 | 13 | 0.36 | 21 | 1.00 | 29 | 0.42 |
| 6 | 0.00 | 14 | 0.46 | 22 | 0.99 | 30 | 0.32 |
| 7 | 0.00 | 15 | 0.57 | 23 | 0.94 | 31 | 0.24 |
| 8 | 0.02 | 16 | 0.68 | 24 | 0.88 |
||
EXPLORATIONS
- Enter the data in the table above
into your graphing calculator or computer. Create a scatter plot of
the data.
- Find values for a, b, h, and k so
the equation
models the data in the data plot.
- Verify graphically the cofunction
identity
by substituting 
in the model above and using sin instead of cos. 
Observe how well this new model fits the data.
- Verify graphically the odd-even identity
for the model in #2 by substituting
and observing how well the graph fits the data.
- Find values for a, b, h, and k so
the equation
fits the data in the data plot.
- Verify graphically the cofunction
identity
by substituting
in the model above and using cos rather than sin 
.
Observe the fit of this model to the data.
- Verify graphically the odd-even identity
for the model in #5 by substituting
and graphing 
.
How does this model compare to the original one?
FURTHER EXPLORATIONS
- Use sinusoidal regression to find
a model for the data in the table.
- Compare the model you found in #5
with the one you found in #8.
- Use the model you found in #8 to create
a table of the fraction of the Moon illuminated for each day in March
2000.
- Compare your predictions in #10 with
the data at the U.S. Naval Observatory Astronomical Applications web
site (http://aa.usno.navy.mil/AA/).
- Use the model you found in #8 to create
a table of the fraction of the Moon illuminated for each day in the
current month.
- Compare your predictions in #12 with
the data at the U.S. Naval Observatory web site.