Sun Elevation: An Equation

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The sun by definition is the star that is "the central body of the solar system, around which the planets revolve and from which they receive light and heat." This self-luminous star is not only the largest object in the solar system containing 99.8 percent of the total mass but is also a fundamental necessity to every human and living thing's life.

The position of the in relationship to the sun, Earth determines the time of the year or season. The problem states "assume that at this time of year the sinusoidal axis is at E = -5 degrees." From this statement, we could conclude that the time of year this problem took place when daylight is less than half of the day is the winter, since the sun shines longer in the summer then it does in the winter. So we assumed that + 5 degrees would equal the summer schedule, and - 5 degrees would equal the winter. The actual angle will require additional information. We concluded that the x, or t in this case, intercepts were points of the sunrise and sunset. When the line rises over the t-axis, then the sun was up or in other words daytime, and when the line goes below the t-axis, then the sun was not out or in other words nighttime.

The formula applied to the graph of this problem was Acos(Bx - C) + D, where A is the amplitude; B is the period; C is the horizontal shift; and D is the vertical shift. We found that the amplitude, as stated in the problem, is 60 degrees, and that the period is 24 hours. Since Period = 2p/B, B = 2p/24. The problem stated that the axis of the sinusoid is shifted to - 5 degrees, so D or the vertical shift equals - 5 degrees. To find the horizontal shift, we used the formula Starting Point = C/B. As given in the problem statement, the maximum angle of elevation occurs at 12:45 p.m. We took the time 12:45 p.m. where the maximum angle of elevation occurred and changed it to 12.75 because 45/60 is equal to .75/100. Therefore C= 12.75/ (2p24) or C = 17p/16. In conclusion, the...
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