• Join over 1.2 million students every month
  • Accelerate your learning by 29%
  • Unlimited access from just £6.99 per month

IB Math Portfolio: Light of My City

Extracts from this document...

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

...read more.

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.

...read more.

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.

This student written piece of work is one of many that can be found in our International Baccalaureate Maths section.

Found what you're looking for?

  • Start learning 29% faster today
  • 150,000+ documents available
  • Just £6.99 a month

Not the one? Search for your essay title...
  • Join over 1.2 million students every month
  • Accelerate your learning by 29%
  • Unlimited access from just £6.99 per month

See related essaysSee related essays

Related International Baccalaureate Maths essays

  1. This portfolio is an investigation into how the median Body Mass Index of a ...

    When the functions are allowed to extrapolate beyond the data points provided, one can see the great difference between the two: As expected, the sinusoidal function curves down, increasing at the rate it is decreasing from its max point at Age = 20.

  2. Extended Essay- Math

    0�""��%2��'L��t�_G;�]�o���"_�.��g��4 [`v�2�G�"��Y��vY�_�9�t*< "p=�*~@'6US"��C��-]�(tm)����v+~`FZuв�W$�)��6CR�ibG"B ��p0;S��]=d�ѭ��!E�-(���5=�$���QG�3����1/2~� i��g�{� C#�p� ���suA�T-1/4...�Ta 8w�_�v L�6\�)�9 =.����ÊW�>gn1/2" ^�B��...�[g�- ���D��jάG����[W ...�+��� {k��ÜYQÖª|a^��(*�{:�'' �� �"���(,W>����s��ѯ���Q97*�M�/*�V(tm)�1�Æ��Qm"g'� ��n[owKn��"=�_.���"d�@ !.Û°...�"��FVJ �ѱM1/2�%"� �V r�])-���N�*Ð�2���)�y ��T�t��"�w�"sW�Q"�3}���k7���'.�ef��=�8�M�p �LAq0��Ë��h��L�"$�N�g��9��s5'�&�|.6v��4"� -a&#��?�U?��1/2pU�1�"t�Jӭ�(r)v�� N��oU��p"�p...'{�" &]êºï¿½ï¿½C ��� tZ�v 0u+AR�!F�1/41/2��Õ�fOT=H�!e(tm)��5�#�-�<��S]�X+�̪�je%A\"�O �2...Lx X��k-×�;� 5��Z���...�^{DH��Y�>9�tY��\��?�?��g�.�y;����"G�l�,�"g...��� � B��L��É��䦰/I,�'�����v����}�Df��!N]�� �K"N!���q�r8�cw��(c)�1/2'c|�.>y@��$}�c�W&���M��b;�m!ï¿½È ï¿½):A)~!" "Å�b��?:]�1;t���lp˶�0Wo�C���:�{v"��q����G<y�o�@��d4/[�(tm)i����.����Ì�Z'"Lt�qn�(tm)fR�bT����m��^�W/�������h...�}䬬�oÄL��"�HNi%=�3/4& @\]t�� �C3/4f��ƺ��� .�uTx�fn�"Vw&���?O��m�s@2V��t"ئ6&�U�u'B�,Y%�����u�j8 d[m�v��/�oa0�"$�/j��(tm)��(r)� �"#�3/4-O:��Ê�W��a-�I��(tm)��n�p�n,�0"�Oa���d3/4�?��/;UÄ W���S���z�?�����[dΕeæ´l��u� \C'(r)\u�'�y�K� �-�R�d��-~,�@�#,��zE(c)}ψS�É�g�)�/~i��z�:����Z...$'��S9�;3�(c)

  1. Ib math HL portfolio parabola investigation

    would intersect the two lines at four distinct points and I have found these four points, labeling the intersections with y=Px+p as X2 and X3 and the intersections with y=Nx+n as X1 and X4. Using these four points I will again find the value of D = | (X2 - X1)

  2. Math Portfolio: trigonometry investigation (circle trig)

    Therefore, the amplitude of y=?sinx and y=?cosx will be the largest value of y and will be given by amplitude= and sets the range of the graph by giving the values of both maxima and the minima. (The curvy scribble looking line.)

  1. Salida del sol en NY

    o al oeste mientras no cambiaras la latitud, no habr�a cambios en la salida del sol Al analizar la siguiente grafica se buscara una serie de resultados a trav�s del an�lisis grafico de la misma: Gr�fica 3. Grafica tendencia de puesta del sol.

  2. Math Portfolio Type II

    (un, rn) = (60000, 1) according to the description. We can get the equation of the linear growth factor by entering these sets of ordered pairs in the STAT mode of the GDC Casio CFX-9850GC PLUS. The STAT mode looks as follows: - Thus, we obtain a linear graph which

  1. Math Portfolio Type II Gold Medal heights

    Now using the just attained new equation the functions graph will added to the graph already present Figure 3.0's draw a comparison between the two and prove that an averaged function would provide the best fit.

  2. Shadow Functions Maths IB HL Portfolio

    Part B: Cubic Polynomials Now, considering the cubic function The shadow function in this case is another cubic function which shares two points with , has opposite concavity and its zeros are -2 and 32. The shadow generating function is a line passing through the two points of intersection.

  • Over 160,000 pieces
    of student written work
  • Annotated by
    experienced teachers
  • Ideas and feedback to
    improve your own work