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IB Math Portfolio: Light of My City

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Introduction

IB Math HL

Portfolio Type II: The Light of My City

image20.jpg

For this project, I chose the glorious city of Pyongyang, the capital city of what is now the Democratic People’s Republic of Korea. Other than its historic importance and communist architectural grandeur, Pyongyang is noted for having a clear distinction during its four seasons, and a clear variation of sunlight depending on the season and time. In regards to the data, I used the year of 2008 as the basis for the calculations and data.

PART I: The Data

Find the shortest day of the year and the amount of sunlight on that day.

The shortest day of the year coincides with the December solstice which occurred on Sunday, December 21st in the year of 2008 at 9:04 PM. At this time, the length of the day (the amount of sunlight) totally at exactly 9h 25m 41s from 7:52 AM when the sun rose to 5:18 PM when the sun set.

Find the longest day of the year and the amount of sunlight on that day.

The longest day of the year always coincides with the June solstice which occurred on Saturday, June 21st in the year of 2008 at 8:59 AM. The length of the day totaled at precisely 14h 54m 33s, from 5:11 AM when the sun rose, to 8:06 PM when the sun set.

The difference as we can see between the two extremes of the year is 5 hours 28 minutes and 52 seconds, which is quite a difference.

year

Equinox
Mar

Solstice
June

Equinox
Sept

Solstice
Dec

day

time

day

time

day

time

day

time

2007

21

09:07

22

03:06

23

18:51

22

15:08

2008

20

14:48

21

08:59

23

12:44

21

09:04

2009

20

08:44

21

14:45

23

06:18

22

02:47

2010

21

02:32

21

20:28

23

12:09

22

08:38

Analysis

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Middle

, whereas the sine function is image23.png. The difference between the two is the horizontal shift shown by the constants image24.png for cosine and image25.png for sine. Because of the shape of these graphs, cosine requires a larger shift to enter its position while sine requires less.

To find the amount of sunlight on April 27th, I first needed to convert this date in terms of a decimal in order put it into a x input. Since December 21 was equal to 4 and March 20 equal to 1, I had to find out what April 27th was. If one year between December 21 2007 to December 21 2008 was divided exactly in 4, it would equal 366 days (2008 is a leap year) divided by 4.

image26.png←The 127comes from the number of days from December 21st to April 27th

If we add this to 1 (the time from December 21st to March 20th), we get 1.378. Now I plugged into the equation to get the actual amount of sunlight on that exact day of April 27th.

image27.pngimage28.pngimage15.png

Whew… And the result was 13.59 according to the GraphingPackage program for both sine and cosine functions . To verify if this was accurate, I went back to the Sunrise and Sunset program.

According to the site, Pyongyang had the following statistics of sunlight on April 27.

Apr 27, 2008

5:44 AM

7:26 PM

13h 41m 39s

However, I needed to compare 13.59 with the time in 13:41:39. So I converted the latter into decimal form.

image29.png

Although the two amounts of time weren’t identical, it was pretty clear that it was darn close.

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Conclusion

image40.png(3)                image41.png(4)

(4) Finally I let the calculator graph this equation, rendering a nice sine regression function concerning all the monthly values of sunlight amount.

Comparison:

This last graph I got from the calculator almost amazingly similar to the “theoretical” sine graph I had made using only the shortest and longest days of the year. However, for the one I made, for the sake of facility I used 4 as the input number. Thus the x was actually seasonal rather than monthly. However, that didn’t affect the results all that much because to calculate the values I divided all my results by 4 to make it compatible with my graph.

image42.pngimage09.pngimage00.png

What is interesting with the calculator’s graph is the median amount of sunlight, which is given by the “d” or the vertical shift value in Figure (3). The median sunlight I calculated with my own graph was very close, which was 12.169, compared to the calculators 12.134. That, I feel, gives credibility to my own results. Since the vertical shift was similar between the two graphs, no doubt the amplitude would be similar, and it was. The calculator gave it at 2.693, while mine was 2.741.

Ultimately, other than the input increment, which mine was at 4 and the graph’s was at 12, there was no  great difference, making this an overall worthwhile effort.

Bibliography:

The amazingly vital data: Time and Date.com

http://www.timeanddate.com/worldclock/astronomy.html?n=205&month=3&year=2008&obj=sun&afl=-11&day=1

Information on Solstices and Equinoxes:

http://en.wikipedia.org/wiki/Solstice

http://en.wikipedia.org/wiki/Equinox

TI-84 Plus Calculator for calculations

GraphingPackage program for graphs

Microsoft Excel for Charts

...read more.

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