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DAYLIGHT HOURS IN DUNEDIN

IB Math SL Writing Task

Through this investigation I will be exploring the

mathematics of the sunrise times for dunedin and developing a

model for the prediction to nd the pattern of the sunrise time. It

is often diicult to predict the sunrise times in the southern

hemisphere due to the occasional climate change which could

often be aected by global warming, seasons and other external

factors. To accomplish this my aims will be to rstly design a

model for the sunrise times in order to then secondly use the

model(s) to help distinguish the sunrises for planning a daybrea

run and programming switching of the street lighting. In order to

determine my model I will be identifying two variables which are

!bservations (Independent "ariable) and Time of #unrise ($ependent

"ariable)

The following table gives the results of the times (in hours)

measured for %& observations of the sunrise'

Table 1 #unrise times for $unedin, ew ealand

Obser!ati"n Date Ti#e Obser!ati"n Date Ti#e* +an-* -'/* 0- +ul*- -1'*1

0 +an** -/'-0 0* +ul0- -1'**

% +an0* -/'*/ 00 +ul%- -1'-*

+an%* -/'%- 0% 2ug-3 -&'&

/ 4eb*- -/'/ 0 2ug*3 -&'%0

5 4eb0- -5'-- 0/ 2ug03 -&'*



& 6ar-0 -5'* 05 #ep-1 -5'/5

1 6ar*0 -5'01 0& #ep*1 -5'%5

3 6ar00 -5'0 01 #ep01 -5'*&

*- 2pr-* -5'// 03 !ct-1 -/'/1

** 2pr** -&'-1 %- !ct*1 -/'%3

*0 2pr0* -&'0* %* !ct01 -/'00

*% 6ay-* -&'% %0 ov-& -/'-&

* 6ay** -&'5 %% ov*& -'//

*/ 6ay0* -&'/1 % ov0& -'5

*5 6ay%* -1'-1 %/ $ec-& -'0

*& +un*- -1'*/ %5 $ec*& -'0

*1 +un0- -1'0- %& $ec0& -'&

*3 +un%- -1'0*

In table 0 below I have converted the times to hours (in decimals) for

%& observations of the sunrise time which will therefore allow me to

graph my data more easily.

Table 0

Obser!ati"ns Ti#e $in h"%rs& *3 1.%/

* .1/ 0- 1.%

0 /.-% 0* 1.*1

% /.0/ 00 1.-0

/./ 0% &.&1

/ /.&/ 0 &./%

5 5 0/ &.0%

& 5.0% 05 5.3%

1 5.& 0& 5.5

3 5.& 01 5.01

*- 5.30 03 /.3&

** &.*% %- /.5/

*0 &.%/ %* /.%&

*% &./& %0 /.*0

* &.&& %% .30



*/ &.3& % .&&

*5 1.*% %/ .&

*& 1.0/ %5 .&

*1 1.%% %& .&1

2fter I have clearly manipulated the time into hours (in decimals) I then

input my results into 7ogger 8ro and generated a #catter 8lot graph

allowing me to analyse the patterns and deduce generally the

appropriate model for the investigation. The graph consists of hours (y

axis) against observations (x axis).

9raph * #catter plot of #unrise times in $unedin



:y ;ualitatively observing my scatter plot, it had a general shape of a

periodical pattern which was closely related to a trigonometric

function. 6y assumptions for this model is that the #ine 4it would be

best for the 9raph as the data is periodic, I used a general sine function

because the trend shows that a maximum and a minimum value which

repeats itself in which it could probably be the amplitude the sine

function, hence I use the formula y < 2sin :( x = ) > $ to model the

investigation.

The general sine function consists of the e;uation y < 2sin :( x = ) > $therefore it is evident that the variables(x and y) are aected by theconstants of 2, :, = and $. To determine the best t I have calculated theparameters of (2,:,=,$) below

A ' a#(lit%)e ' Ma*i#%# !al%e Mini#%# !al%e+

2 < 1.%/ .&-0

2 < *.10/

,eri") '+B

8eriod < %&

: < 0%&

: < -.*53



- ' the graph had moved to the right with the value of *./ from the originalnon transformed sine graph, hence = is e;ual to *./--

y < *.10/sin -.*53( x =) > 5./0/

.1/ < *.10/sin -.*53( * =) > 5./0/

.1/ 5./0/ < *.10/sin -.*53( * =)

*.5&/ *.10/sin -.*53( *) < =

D ' Ma*i#%# !al%e . Mini#%# /al%e+

$ < 1.%/ > .&- 0

$ < 5./0/

2s a result, the function which I have closely calculated to represent the

best model t for my scatter plot graph is 0 ' 12+3sin 4156$ *

1344& . 53+3

The graph below is a #ine graph which was manually tted by the sineformula I obtained.

9raph 0 #unrise Times of $unedin (#ine 9raph 6anual 4it)



9raph % 2uto t



9raph



2s seen above it is evident that the residual for the 2uto t graph is not

evenly spread throughout the autot line therefore this could lead us to

believe that this is probably not the most reliable model to represent and

depict the #unrise Time 8attern in $unedin. Therefore the ?6#@ value is a

better statistic to measure and compare the reliability between the auto

(9raph %) and manual t (graph 0). :elow is the formula for the ?6#@

calculation.

?oot 6ean #;uare @rror < *n i<*(yiŷi)0



yi< observed value of the dependent value.ŷi< expected value of the dependent variable.

The ?6#@ "alue for the manual t is AA whereas the ?6#@ value of the

2uto t is AA. Bence this shows that the 2uto t graph is a more preferable

and reliable model to analyse the data.

#tatistic that supports the appropriate model for my investigation is ?oot

6ean #;uare @rror



9raph / #unset

:elow I have created a table where I calculated the length of daylight

received in all %& observations throughout the year. This is accomplished by

nding the dierence between the #unset and #unrise times which gives an

estimated value of the length of daylight. The length of the shortest day was

estimated on +une 0-th, as it only received 1.0% hours of sunlight, this was

probably due to the season of winter in the southern hemisphere which

usually has a shorter length of daylight. 4urthermore, there are *1 7engths

of daylight that exceed *0 hours which is in between +anuary * 6arch 00

and !ctober 1 $ecember 0&th



$ate #unset #unrise 7ength of daylight

+an-* 0-./ .1/ */.5/

+an**0-./

/.-%*/.0

+an0* 0-.% /.0/ */.-/

+an%* 0-.0 /./ *.&

4eb*- 0-.- /.&/ *.0/

4eb0- *3.&/ 5 *%.&/

6ar-0 *3./ 5.0% *%.00

6ar*0 *3.* 5.& *0.5%

6ar00 *1.&/ 5.& *0.-/

2pr-* *1./ 5.30 **./1

2pr** *1.0 &.*% **.-&

2pr0* *&.3 &.%/ *-.//

6ay-* *&.// &./& 3.31

6ay** *&.% &.&& 3./%

6ay0* *&.*/ &.3& 3.*1

6ay%* *&.-/ 1.*% 1.30

+un*- *& 1.0/ 1.&/

+un0- *& 1.%% 1.5&

+un%- *&.-/ 1.%/ 1.&

+ul*- *&.* 1.% 1.1

+ul0- *&.0 1.*1 3.-0

+ul%- *&.* 1.-0 3.%3

2ug-3 *&.5 &.&1 3.10

2ug*3 *&.1 &./% *-.0&



2ug03 *1.* &.0% *-.1&

#ep-1 *1.% 5.3% **.%&

#ep*1 *1./ 5.5 **.3

#ep01 *1.& 5.01 **.&%

!ct-1 *1.3 /.3& *0.3%

!ct*1 *3.* /.5/ *%./

!ct01 *3. /.%& *.-%

ov-& *3.5 /.*0 *.1

ov*& *3.1 .30 *.11

ov0& 0- .&& */.0%

$ec-& 0-.0 .& */./

$ec*& 0-. .& */.&

$ec0& 0-./ .&1 */.&0

To apply and incorporate this model into a real life situation it would

be useful to determine the time of someone planning daybrea run in which

the sunrise time is important in organising the event. !n the other hand, the

model would also be a vital tool to someone programming switching o the

street lighting. 4or the day brea run the organiCer would Dust need to use

the model by tting the xvariable (observation) which is a particular day in

determining the possible accurate time of the sunrise. This would allow the

organiCer to mae plans by determining the correct route, location and the



necessary arrangement of the event to run smoothly. 2s for someone

programming switching o the street lighting, it would be an eective tool

for programming a computeriCed automatic switch that could be tted into

the street lights which would wor by calculating the sunrise time

accordingly to whereby automatically informing the switch to switch o once

the sun has risen.

#unrise time would dier accordingly to the latitude of the earth. 2s

+apan is called as the country of sunrise because of itEs location at the

further east hence it gets the sunrise rst. 4or that we could conclude the

country which is more east on the world map which would have an early

sunrise than the east as it is scientically proven that the sun rises from the

east to the west. =onsidering from going north from south the time to

sunrise would be earlier as it approaches the e;uator and vise versa. If you

travel *---m north the sunrise would be earlier as it is approaching the

e;uator and it probably might have a smaller amplitude when observed in a

year period because as we move north the seasonal period would decrease

hence a less change in sunrise time due to its approach to the e;uator.

Therefore it would result in a dierent model then the model predicted.

9oing south the sunrise time would predictably be much later or much

earlier as it is closer to the south pole with an excessively longer summer

and winter season therefore itEs model would be less periodic. 4inally if you

approach *--- m to the west of dunedin this would result in a sunrise that

is later in the day and a sunset that is earlier, however the model could be

applied as the latitude does not change and the periodic pattern would

remain closely the same. In conclusion 6oving to the north or south would



alter the model to a certain extent, whereas approaching the west or even

the east would not aect the general periodic trend.

next you must use the data from this website to strengthen the facts of (question 1000km to

north,southand west)