ight hours in
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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 -&'*
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& 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
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*/ &.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
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: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
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- ' 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)
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9raph % 2uto t
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9raph
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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
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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
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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 $ecember 0&th
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$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&
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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
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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
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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)