CAPE PURE MATHEMATICS UNIT 1 INTERNAL ASSESSMENT
CARIBBEAN ADVANCED PROFICIENCY EXAMINATION
PURE MATHEMATICS UNIT 1
INTERNAL ASSESSMENT
Candidate’s Name: Daynea Lewis
Candidate’s No.: 10013112144
Subject: CAPE Pure Mathematics Unit 1
Teacher: Mrs. Lashaun Lugg-McKenzie
School No.: 100131
School Name: York Castle High School
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CAPE PURE MATHEMATICS UNIT 1 INTERNAL ASSESSMENT
PROJECT TITLE
The Exponential Growth and Decay of Jamaica’s population over the past 5 years and the effects
it has had on the per capita income of Jamaicans.
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CAPE PURE MATHEMATICS UNIT 1 INTERNAL ASSESSMENT
TABLE OF CONTENTS
Table of Contents
TITLE ................................................................................................................................................. 4
PROBLEM STATEMENT ................................................................................................................... 4
MATHEMATICAL FORMULATION ....................................................................................................... 5
Diagrams: .................................................................................................................................. 5
Variables: ................................................................................................................................. 5
(i)
EXPONENTIAL GROWTH FORMULA METHOD .......................................................................... 6
(ii)
EXPONENTIAL FUNCTION (GRAPHICAL METHOD) ................................................................... 6
PROBLEM SOLUTION .......................................................................................................................... 7
Assumptions:............................................................................................................................. 7
EXPONENTIAL FORMULA CALCULATIONS: ........................................................................................ 8
GRAPHS .......................................................................................................................................... 9
APPLICATION OF SOLUTION .............................................................................................................. 13
DISCUSSION OF FINDINGS ................................................................................................................ 14
CONCLUSION .................................................................................................................................. 15
SUGGESTIONS FOR FUTURE ANALYSIS............................................................................................... 16
REFERENCES .................................................................................................................................... 17
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CAPE PURE MATHEMATICS UNIT 1 INTERNAL ASSESSMENT
TITLE
The Exponential Growth and Decay of Jamaica’s population over the past 5 years and the effects
it has had on the per capita income of Jamaicans
PROBLEM STATEMENT
In Jamaica the issues of underemployment, unemployment, income inequality, education system
strain and environmental degradation have all been major factors that have plagued the Jamaican
society. To shed light on the problem, a group of researchers decided to investigate the factors
that could have contributed to the current issues of the country, such as the country’s population;
in order to observe and decipher methods that could reduce the impact of these societal
problems. As such the research is aimed at investigating and gathering information on the
exponential growth and decay of Jamaica’s population over the past 5 years and the effects it has
had on the per capita income of Jamaicans.
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CAPE PURE MATHEMATICS UNIT 1 INTERNAL ASSESSMENT
MATHEMATICAL FORMULATION
Diagrams:
Variables:




A- the population at the beginning of the experiment.
e- Euler's number (approximately 2.71828)
P(t)- the population after t units of time
ℷ- the growth factor/ rate of growth
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CAPE PURE MATHEMATICS UNIT 1 INTERNAL ASSESSMENT
EXPONENTIAL GROWTH METHOD
(i)
To obtain the exponential growth or decay of the population and per capita income over the past
5 years in Jamaica:




Firstly, an equation of P(t) = A₀eλt will be used to determine the yearly data for the
population and per capita income.
From the equation, transpose to find λ, the growth factor or yearly rate of growth.
Similarly, the method used to determine the growth rate of the population will be
used to determine the growth rate of the per capita income.
From this the exponential growth of the population and per capita income can be
found and estimated patterns of growth determined.
GRAPHICAL METHOD
(ii)
To obtain the exponential growth or decay of the population over the past 5 years in Jamaica:







Firstly, an equation of f(x) = A₀eλx will be used to determine the yearly data for the
population and per capita income.
From the equation, transpose to find λ, the growth factor or yearly rate of growth.
As such the graph displaying the exponential growth and decay of the population will
show population(y-axis) against time(x-axis).
In contrast to this, the yearly growth rate will be plotted in a graph of growth rate versus
time for the population size.
Besides from this, a graph will be plotted with per capita income against time for the
exponential growth of the per capita income.
Lastly, a graph of the yearly growth rate will be plotted against time.
From this the exponential growth of the population and per capita income can be found
and estimated patterns of growth determined.
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CAPE PURE MATHEMATICS UNIT 1 INTERNAL ASSESSMENT
PROBLEM SOLUTION
Assumption:
A) The exponential growth and decay of the population influences the per capita
income of a country.
B) The growth factor of each variable will allow accurate estimations on future
exponential growth or decay.
EXPONENTIAL GROWTH FORMULA CALCULATIONS:
a) For Population
P(t) = A₀eλt
A₀ = 2,813,773
P(t) = 2,827,377
t=4
Therefore,
(2,827,377 / 2,813,773) = (2,813,773e4ℷ/ 2,813,773)
1.00= e4ℷ
ln (1.00) = ln e4ℷ
ln (1.00) = 4ℷ
ln (1.00) / 4 = ℷ
0 =ℷ
b) For Per Capita Income:
P(t) = A₀eλt
A₀ = 5500
P(t) = 4900
t=2
Therefore;
(4900 / 5500) = (5500eλt / 5500)
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CAPE PURE MATHEMATICS UNIT 1 INTERNAL ASSESSMENT
0.89 = eλt
ln (0.89) = lne2ℷ
ln (0.89) = 2ℷ
ln (0.89) / 2 = ℷ
-0.06 = ℷ
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CAPE PURE MATHEMATICS UNIT 1 INTERNAL ASSESSMENT
GRAPHS:
a) For the population:
LINEAR GRAPH DEPICTING POPULATION AGAINST TIME FOR
THE EXPONENTIAL GROWTH AND DECAY OF THE POPULATION
OVER THE LAST 5 YEARS
2 830 000
Population
2 825 000
2 827 695
2 827 377
2 825 544
2 820 000
2 820 436
2 815 000
2 810 000
2 813 773
2 805 000
2019
2020
2021
2022
2023
Years
TABLE SHOWING THE POPULATION SIZE FOR THE LAST 5 YEARS
Population Size
2,813,773
2,820,436
2,827,695
2,827,377
2,825,544
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Year
2019
2020
2021
2022
2023
CAPE PURE MATHEMATICS UNIT 1 INTERNAL ASSESSMENT
LINEAR GRAPH DEPICTING THE YEARLY GROWTH RATE
AGAINST TIME FOR THE EXPONENTIAL GROWTH AND DECAY
OF THE POPULATION THE LAST 5 YEARS
0,30%
Yearly Growth Rate
0,25%
2021; 0,26%
2020; 0,24%
0,20%
0,15%
0,10%
0,05%
2019; 0,07%
0,00%
-0,05%
2019
-0,10%
2020
2021
2022; -0,01%
2022
2023
2023; -0,06%
Year
TABLE SHOWING THE YEARLY GROWTH RATE OF THE POPULATION
SIZE FOR THE LAST 5YEARS
Yearly Growth Rate of the Population Size (%)
0.07
0.24
0.26
-0.01
-0.06
b) For Per Capita Income:
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Year
2019
2020
2021
2022
2023
CAPE PURE MATHEMATICS UNIT 1 INTERNAL ASSESSMENT
LINEAR GRAPH DISPLAYING THE PER CAPITA AGAINST
TIME FOR THE EXPONENTIAL GROWTH AND DECAY
FOR THE LAST 5YEARS
$6 000
PER CAPITA INCOME
$5 800
2022; $5 760
$5 600
2019; $5 500
$5 400
$5 200
2021; $5 190
$5 000
2020; $4 900
$4 800
$4 600
$4 400
2019
2020
2021
YEAR
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2022
CAPE PURE MATHEMATICS UNIT 1 INTERNAL ASSESSMENT
TABLE SHOWING THE PERCAPITA INCOME OF JAMAICANS FOR 4
CONSECUTIVE YEARS
Per Capita Income ($)
5500
4900
5190
5,760
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Year
2019
2020
2021
2022
CAPE PURE MATHEMATICS UNIT 1 INTERNAL ASSESSMENT
APPLICATION OF SOLUTION
Exponential function was used to determine the growth factor for the variables. As such an
equation of
𝑃(𝑡)
𝐴0
= eℷℷ was used. After which, both sides of the equation were logged naturally
resulting in the equation ℷ=
ln(𝑃(𝑡)/𝐴₀)
𝑡
which equated to zero as the growth factor of the
population and -0.06 as the growth factor for the per capita income. Moreover, from the graphs,
the effects of growth factor on the exponential growth and decay were noticed.
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CAPE PURE MATHEMATICS UNIT 1 INTERNAL ASSESSMENT
DISCUSSION OF FINDINGS
The objectives of this research were achieved via the exponential formula method and the
exponential function (graphical) method. The exponential formula method was used to determine
the growth factor for both variables (population and per capita income). As such the equation
P(t) = A₀eλt was used and transposed to give the equation ℷ=
factor and ℷ=
ln(0.89)
2
ln(1.00)
4
for the population’s growth
for the per capita income’s growth factor as observed in Mathematical
Formulation. From the calculations done, the corresponding values of 0 and -0.06 were obtained
as the growth factor for the respective variables. Furthermore, the exponential function
(graphical) method was another alternative to determining the growth factor of the variables. As
noted in the formula f(x) = A₀eλx. Besides from this the graphical method allowed for the
verification of the exponential growth and decay of the variables as observed in the decrease and
increase of variables for the last 5 years.
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CAPE PURE MATHEMATICS UNIT 1 INTERNAL ASSESSMENT
CONCLUSION
In accordance with the results obtained from the research, it can be surmised that the exponential
growth and decay of the country’s population would not have had a great influence on the per
capita income of the citizens has observed in the inconsistent and fluctuating increase and
decrease of the variables throughout the time span of 5 years which could have been affected by
other factors such as inflation, unemployment rate and others.
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CAPE PURE MATHEMATICS UNIT 1 INTERNAL ASSESSMENT
SUGGESTIONS FOR FUTURE ANALYSIS
The uses of the exponential function are not only limited and associated with determining the
growth rate of the population or the per capita income but can also be associated with medical
researches such as determining the exponential growth of tumors in cancer patients and the
spread of epidemics. Additionally, exponential functions can play a key role in identifying the
behavior of systems involving feedback loops as in the case of control systems and signal
processing. Besides from this, exponential functions can also play a crucial part in identifying
and obtaining information related to factors that affect the environment as in the case of the
depletion of natural resources. Lastly, in relation to the research conducted, it was observed that
economic indicators such as the GDP growth, inflation rate, unemployment rate and others could
have been examined over the same period to identify how they correlate with population growth
and per capita income changes.
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CAPE PURE MATHEMATICS UNIT 1 INTERNAL ASSESSMENT
REFERENCES
References
Statista. (2022). National Income per capita Jamaica. Retrieved from Statista.com:
https://www.statista.com/statista/1066979/gross-national-income-per-capita-jamaica/
Worldometer. (2024, March 11). Jamaica Population(2024). Retrieved from worldometer.info:
https:www.worldometers.info/world-population/jamaica-population/
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