Biorhythms and Luck Cycle: Extended Modelling and Problem Solving Task

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