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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

Toronto

Location: 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

Miami

Location: 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).

Toronto

Location: 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


image00.png

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.

...read more.

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 image02.png, 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 image10.png, so the period equals to 365 (there is 365 days in a year). Therefore, the answer becomes image02.png, 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

...read more.

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 = image05.png=image06.png= 1.605

b = image02.png= 0.017

d = image07.png= image08.png= 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.

image09.png

...read more.

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