John Paul College Mathematics B: Biorhythms Extended Modelling Task

This project, completed for Mathematics B at John Paul College, delves into the topic of biorhythms using periodic functions. It covers the definition of biorhythms, including physical, emotional, and intellectual cycles, and uses trigonometric functions to model these cycles. The project includes tasks such as graphing the biorhythms, calculating intersection points, and analyzing a summed "Luck Cycle." The student identifies their best and worst days based on the model and discusses the advantages and limitations of the biorhythm theory. The project concludes with a critical analysis of the theory's validity, assumptions, and potential effects, ultimately arguing against its predictive power. References to relevant academic sources are also provided. Desklib offers a range of similar solved assignments and past papers for students.

John Paul College
Senior School Mathematics
Subject: 12 Mathematics B
Assessment Piece: Extended Modelling and Problem Solving Task (AAS3)
Topics Covered: Periodic Functions and Applications
Student’s Name:
Teacher’s Name Mrs L Blood Mrs K. Bowness Mrs Charlesworth Miss Chen Mr P. Gallagher
Dates: Released: 20/2/19 Draft Due: 7/3/19 Submitted: 14/3/19
Equipment Allowed: Scientific Calculator, Graphics Calculator, Notebook Computer
Declaration:
I declare that this assignment is all my own work:
.................................................................................. ...................................
Student Signature Date

Question Definition
Biorhythms
Biorhythm theory states that a person’s biological functioning is controlled by three
phenomena which are emotional, physical and intellectual, that vary trigonometrically with time.
It uses the graphs of three simple sine functions to make predictions about an individual’s
physical, emotional and intellectual potential for a particular day.
The graphs are given by:
y= Asin Bx( i)
x=0, corresponds to date of birth of a person and where A=0, denotes 100% potential. The
theory states that when the physical cycle is near a high point, the person can do well in physical
activities, and so forth.
Task 1 (K1, M2)
The physical cycle has a period of 23 days, from equation (i), 2 π
B =23, thus, B= 2 π
23 . The
physical biorhythm can be described by equation (ii) below:
P ( x )=sin ( 2 π
23 x )(ii)
The emotional cycle has a period of 28 days, from equation (i), 2 π
B =28, thus, B= 2 π
28 . The
physical biorhythm can be described by equation (iii) below:
P ( x ) =sin ( 2 π
28 x ) (iii)
The intellectual cycle has a period of 33 days, from equation (i), 2 π
B =33, thus, B= 2 π
33 . The
physical biorhythm can be described by equation (ii) below:
P ( x )=sin ( 2 π
33 x )(ii)
Now graphing the equation over the range (−1 ≤ x ≤1) and domain (6153 ≤ x ≤ 6184), is shown in
figure 1 below:

6155 6160 6165 6170 6175 6180
Time [Days]
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
A
Individual’s physical, emotional and intellectual biorhythms
Physical
Emotional
Intellectual
Figure 1: Individual’s physical, emotional and intellectual biorhythms
Task 2 (K2)
The three functions described are periodic, to calculate how old I was the first time any two of
the functions intersected at y=0 can be given as:
i) For physical and emotional intersecting when
sin ( 2 π
23 x )=sin ( 2 π
28 x )⟹ x=23 ×14=322 for x=322
I was 322 days old when physical and emotional intersected
ii) For physical and intellectual intersecting at y=0 or
sin ( 2 π
23 x )=sin ( 2 π
33 x )⟹ x=23 ×33=759I was 759 days old when physical and
intellectual intersected
iii) For emotional and intellectual intersecting at y=0
sin ( 2 π
28 x )=sin ( 2 π
33 x )⟹ x=33 ×14=462

I was 462 days old when emotional and intersexual intersects at y=0
When any of the two function intersect is min ( 322 ,759,462 ) =322days
To calculate how old I was the three functions intersected can be given as:
sin ( 2 π
28 x )=sin ( 2 π
28 x )=sin ( 2 π
33 x ) x=23 ×14 × 33=10626 days
1.0625 1.0626 1.0627 1.0628 1.0629 1.063
Time [Days] 104
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
A Physical
Emotional
Intellectual
Figure 2: Time when all the three functions intersected at y=0
Graphically, the three intersect at 10626 days. Therefore, I was 10626 days old when physical,
emotional and intellectual intersected.
Since at the end of March, there is only one day when all the three functions intersected at y=0,
which is when I was 10626 days old, as shown in figure above.:
Task 3 (M2)
The summed function is defined as the addition of two or more periodic functions, which gives a
single periodic function. A special summed function, called the “Luck Cycle”. The ‘luck’
function of the biorhythms is given as:
Y ( x )=sin ( 2 π
23 x )+sin ( 2 π
28 x )+sin ( 2 π
33 x )
Figure 3 below shows the “Luck” function in terms of the physical, emotional and intellectual
functions.

6150 6155 6160 6165 6170 6175 6180 6185
Time [Days]
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
A
“Luck” function in terms of the physical, emotional and intellectual functions
Luck Cycle
X: 6170
Y: 1.589
X: 6158
Y: -2.046
Figure 3: The “Luck” function in terms of the physical, emotional and intellectual functions
The domain of the ‘Luck’ function similar as task 1, (6153 , ≤ x ≤6184 )
The range of the ‘Luck’ function is: (−3 ≤ x ≤ 3), since when all are maximum then,
Amax=1+1+1=3 Amin 1= ( −1 ) + ( −1 ) + ( −1 ) =−3
Amplitude is 3
Period is LCM ( 23,28,33 )=21252
Task 4 (M3)
As the domain plotted on figure 2, is March 1, 2019 to March 31,2019. From the figure,
on March 5th will be my worst day since the luck cycle will be at minimum with 204.6 %. On the
other1 hand my lucky cycle will be at maximum on March 17TH, 2019 with a 158.9%. I can
conclude that, I will experience bad day’s more than good days as seen from figure 2.
Advantage of the model
I) The advantage of the above, indicates the overall emotional, physical and intellectual
reactions of a person.

II) The model could be effective in reducing the unsafe behaviors and accidents.
Limitation of the model
III) The model does not indicate individual percentage aspects of the day, that is, does
not indicate which of the three biorhythm will be good or bad.
Task 5 (M3)
Based on these three cycles, it is believed that one can predict their luck now and in the
future. But on my view I don’t validate the information valid. For instance, one cannot wake up
one day and hope to win a lottery based on their life cycle. Luck is a random chance of success
or failure. If somebody has succeeded that may be termed as good luck. On the other hand, if one
fails to succeed, then they may be said to have bad luck.
Assumptions of biorhythms
i) Biorhythm Theory assumes that every cycle begins at a zero point and then it starts to
rise in a positive phase when the energies and abilities are high (O'Neil & Phillips
(2012). The cycle started to decline after reaching the peak until it reaches the
minimum point when the abilities are reducing slowly (Benca et al., 2009). The
process is assumed to continue until the time when one dies.
ii) Also the model assumes that psychological and physical state is not a steady flow, but
suffer up’s and down’s as time goes by and human voluntary acts cannot do anything
to avoid it.
iii) The physical condition of individual remains fairly constant throughout his/her life.
Effect of the model:
i) The physical, emotional, and intellectual cycles have a significant effect on the
frequencies of accidents and the accidents are occurring not by chance alone.
ii) Reducing accidents that might occur due to physical, emotional, and intellectual
behaviors (Sharma & Singh, 2009)
According to this information, there is no scientific proof about this theory since the
assumptions are very unrealistic. More also, according to Charles Darwin, more the animals and
plants are born and will live to maturity. Many of them will fall victim to predators due to the
law of survival for the fittest. However, this conclusion may not be true about the life cycles

which is a continuation of the Biorhythm theory. From the mentioned arguments, this theory is
not valid and can never predict luck both now and the future.

References
Benca, R., Duncan, M. J., Frank, E., McClung, C., Nelson, R. J., & Vicentic, A. (2009).
Biological rhythms higher brain function, and behaviour: Gaps, opportunities, and
challenges. Brain research reviews, 62(1), 57-70.
O'Neil, B., & Phillips, R. V. (2012). Biorhythms: how to live with your life cycles. Ward Ritchie
Press.
Sharma, R., & Singh, R. (2009). Research Article Open Access: Critical Analysis of Biorhythms
and Their Effect on Industrial Accidents in Agra Casting Manufacturing Units.
International Journal of Advancements in Technology, 1-3