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# Math Portfolio Type II - Applications of Sinusoidal Functions

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Introduction

Portfolio Type II

Applications of Sinusoidal Functions

In this project, you will develop and apply sinusoidal functions that model the time of sunrise, the time of sunset, and the number of hours of daylight for Toronto, Ontario.

To study the relationship between latitude and the number of hours of daylight, data from Miami Florida, which is located at different latitude from Toronto, will also be used.

The location of Toronto and of Miami and data on their sunrise times and sunset times are given in the tables below.

Sunrise and Sunset times for 2007

 TorontoLocation: 44°N 79°W Date Day Rise (hh mm) Set (hh mm) January 1 1 0751 1648 February 1 32 0734 1726 March 1 60 0653 1804 April 1 91 0558 1843 May 1 121 0508 1918 June 1 152 0436 1951 July 1 182 0437 2003 August 1 213 0504 1941 September 1 244 0539 1852 October 1 274 0613 1757 November 1 305 0652 1707 December 1 335 0730 1639 MiamiLocation: 26°N 80°W Date Day Rise (hh mm) Set (hh mm) January 1 1 0707 1740 February 1 32 0704 1803 March 1 60 0644 1821 April 1 91 0611 1837 May 1 121 0543 1851 June 1 152 0528 1907 July 1 182 0532 1916 August 1 213 0546 1907 September 1 244 0600 1840 October 1 274 0612 1807 November 1 305 0628 1738 December 1 335 0650 1728

Note: Times are written in 24-hour form, eastern standard time

—Astronomical Applications Dept., U.S. Naval Observatory, Washington, DC

Part A

1.         Using a GDC or graphing software and the data for Toronto, plot the graph of day number against sunrise. Note that the times in the table are in (hh mm), so a time of 0751 (7:51 A.M.) must be converted to a decimal (7.85 h).

 TorontoLocation: 44°N 79°W Date Day Time of Sunrise January 1 1 7.85h February 1 32 7.57h March 1 60 6.88h April 1 91 5.97h May 1 121 5.13h June 1 152 4.60h July 1 182 4.62h August 1 213 5.07h September 1 244 5.65h October 1 274 6.22h November 1 305 6.87h December 1 335 7.50h All the values of the sunrise times for Toronto are converted to a decimal. The values of the sunrise times (y-value on the graph) and the number of days (x-value on the graph) are listed into Microsoft Excel.

Middle

a = 0.846

b = 0.015

c = 127.798

d = 6.362

The equation that represents the time of sunrise as a function of day number, n, for Miami through sinusoidal regression is g(n)= 0.846 sin[0.015(n –127.798)] + 6.362. This is found through using the TI-84 Plus graphing calculator. All the coordinates are listed in a table and the equation is found using sinusoidal regression, which is done with the graphing calculator.

6.         State the range, domain, and period of the function representing the time of sunrise for Miami that you found in question 5.

The domain of the function would be {x Є R | 1 ≤ x ≤ 365} and the range of the function would be {y Є R | 5.516 ≤ y ≤ 7.208}. The period of the function representing the time of sunrise for Miami is , which results in the answer 0.017. The domain represents the number of days, starting from day 1 to day 365, since day 0 does not exist. As a result, the domain is equal or in between 1 and 365. The range of the function represents the time of sunrise. Therefore, the minimum and maximum values are found out, equal or in between 5.516 and 7.208. The period of a sinusoidal function is , so the period equals to 365 (there is 365 days in a year). Therefore, the answer becomes , which is 0.017.

7.         The graph of function f, which represents the time of sunrise in Toronto, can be transformed to the graph of function g, which represents the time of sunrise in Miami. One of the transformations that occurs is a vertical stretch. Calculate the vertical stretch factor that would be required in the transformation of the graph of function f into the graph of function

Conclusion

n is the value of the phase shift.

9.        For Miami, consider June 21 (day 172), with 13.72 hours of daylight, as the day on which there is the maximum number of hours of daylight and December 22 (day 356), with 10.51 hours of daylight, as the day on which there is the minimum number of hours of daylight. Using the data from these days (not the regression data), algebraically determine a cosine equation in the form of h(n) = a cos[b(n – c)] + d for the number of hours of daylight. Explain how you determined each of the parameters, to the nearest thousandth, of h(n) = a cos[b(n c)] + d. Comment on how well your function fits the data.

In order to figure out the cosine equation, we must figure out all the parameters in the equation.

a = = = 1.605

b = = 0.017

d = = = 12.115

In order to find out the entire cosine equation, we must substitute in one of the coordinates into the equation in order to find the value of c.

h(n) = a cos[b(n – c)] + d

13.72 = 1.605 cos[0.017(172 – c)] + 12.115

1.605 = 1.605 cos[0.017(172 – c)]

1 = cos[0.017(172 – c)]

0 = 0.017(172 – c)

0 = 172 – c

-172 = - c

c = 172

Therefore, the cosine equation in the form of h(n) = a cos[b(n – c)] + d for the number of hours of daylight that is algebraically determined is h(n) = 1.605cos[0.017(n –172)] + 12.115.

The function fits extremely well into the data, since both functions look extremely identical to one another. All the coordinates fit in together and the graph looks like one grid with two unison sinusoidal functions. This shows that the cosine equation is extremely accurate, since it fits exactly on the graph shown on the TI-84 Plus graphing calculator. This student written piece of work is one of many that can be found in our AS and A Level Core & Pure Mathematics section.

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