Candidate code: jkj004
IB Mathematics SL applications and interpretations
Internal Assessment
An Investigation to Find the Optimum
Dimensions of 1ℓ Milk Packaging
Contents
I. Introduction.......................................................................................................................................1
II. Aim.......................................................................................................................................................3
III. Measurements:...............................................................................................................................3
IV. Main findings:.................................................................................................................................4
V. Graphs...............................................................................................................................................16
VI. Conclusion and validity.............................................................................................................18
Works cited..........................................................................................................................................20
I. Introduction
1
,Candidate code: jkj004
If you have ever been to a supermarket and have taken a note of milk containers, you have
learnt that most have differently shaped packaging. Common milk containers in Cyprus
include Biorganic Almond milk, Elle & Vire whole milk as well as Alambra Organic semi-
skilled milk. It is important to note that although all three milk containers are shaped
differently, each has a fixed volume of 1ℓ. Intrigued by this weird observation, I started
taking note of the product packaging of everyday goods in my home. After I investigated
various packaging in my home, my interest took me to scholarly articles exploring calculus’
application to everyday life. A particular article that grabbed my attention was titled “Using
Optimization to Redesign a Cylindrical Can Product” ("Using Optimization to Redesign A
Cylindrical Can Product On JSTOR"). The article discusses how calculus is used by many
manufacturers and producers worldwide in redesigning products as well as in helping to
determine the output at which the company receives maximum profit while minimizing costs.
Moreover, through my reading and knowledge on Business Management, I realised that
calculus is a driving force in a company’s operations department, such as in product
differentiation. Product differentiation is vital for a business as it highlights the unique
features and benefits of a certain product and service, thus differentiating it from possible
competitors. Therefore, optimisation as a topic, in particular in product packaging, gripped
my attention, leading me to the topic of optimisation of 1ℓ milk packaging.
In this investigation, I plan to calculate and analyse the ideal (and optimal) shape of milk-
container packaging containing a fixed volume of 1ℓ. This will be conducted by calculating
the minimum total surface area that is required to package the container, whilst also
maintaining the given volume of 1ℓ, and then comparing the total surface area that the three
already existing containers mentioned above currently employ (measured using a ruler, then
recording the measurements on an excel sheet). Subsequently, I will calculate the percentage
difference for all three containers whereby the packaging with the smallest percentage
difference will be the packaging closest to the optimal value. To support the above
calculations, I will then construct a graph on ‘Desmos’ for each milk container by using the
equation of the total surface area, in which I will be able to determine and hopefully validate
the value of the minimum total surface area calculated.
The topic of the exploration is important to investigate as much of the worlds environmental
and economic problems are due to product packaging wastage. Although my investigation
specifically focuses on three milk-producing companies, my aim for this mathematical
2
, Candidate code: jkj004
research and analysis is to make us as a society more aware of the current worldwide issue in
which we are facing.
II. Aim
The aim of this investigation is to explore the minimum total surface area for 1ℓ milk
packaging and thus determine which packaging is the most cost-effective and
environmentally friendly.
III. Measurements:
*All measurements of the various dimensions of the milk packaging, namely height, length,
width and base radius of the three milk containers will be measured in centimetres (cm).*
a. Visual representation: The below diagrams were extracted via Microsoft Word
and aim to visually represent how the above dimensions were measured for each
carton recorded:
Height
Height
Radius
Length Width
b. Measurements: The below measurements are the recorded dimensions (in cm) of
the three milk containers. Important to note is that the various milk containers and
the way in which they were each specifically measured for use in this
investigation, can be found in Appendix 1.
Biorganic almond milk:
Width: 7cm
3
IB Mathematics SL applications and interpretations
Internal Assessment
An Investigation to Find the Optimum
Dimensions of 1ℓ Milk Packaging
Contents
I. Introduction.......................................................................................................................................1
II. Aim.......................................................................................................................................................3
III. Measurements:...............................................................................................................................3
IV. Main findings:.................................................................................................................................4
V. Graphs...............................................................................................................................................16
VI. Conclusion and validity.............................................................................................................18
Works cited..........................................................................................................................................20
I. Introduction
1
,Candidate code: jkj004
If you have ever been to a supermarket and have taken a note of milk containers, you have
learnt that most have differently shaped packaging. Common milk containers in Cyprus
include Biorganic Almond milk, Elle & Vire whole milk as well as Alambra Organic semi-
skilled milk. It is important to note that although all three milk containers are shaped
differently, each has a fixed volume of 1ℓ. Intrigued by this weird observation, I started
taking note of the product packaging of everyday goods in my home. After I investigated
various packaging in my home, my interest took me to scholarly articles exploring calculus’
application to everyday life. A particular article that grabbed my attention was titled “Using
Optimization to Redesign a Cylindrical Can Product” ("Using Optimization to Redesign A
Cylindrical Can Product On JSTOR"). The article discusses how calculus is used by many
manufacturers and producers worldwide in redesigning products as well as in helping to
determine the output at which the company receives maximum profit while minimizing costs.
Moreover, through my reading and knowledge on Business Management, I realised that
calculus is a driving force in a company’s operations department, such as in product
differentiation. Product differentiation is vital for a business as it highlights the unique
features and benefits of a certain product and service, thus differentiating it from possible
competitors. Therefore, optimisation as a topic, in particular in product packaging, gripped
my attention, leading me to the topic of optimisation of 1ℓ milk packaging.
In this investigation, I plan to calculate and analyse the ideal (and optimal) shape of milk-
container packaging containing a fixed volume of 1ℓ. This will be conducted by calculating
the minimum total surface area that is required to package the container, whilst also
maintaining the given volume of 1ℓ, and then comparing the total surface area that the three
already existing containers mentioned above currently employ (measured using a ruler, then
recording the measurements on an excel sheet). Subsequently, I will calculate the percentage
difference for all three containers whereby the packaging with the smallest percentage
difference will be the packaging closest to the optimal value. To support the above
calculations, I will then construct a graph on ‘Desmos’ for each milk container by using the
equation of the total surface area, in which I will be able to determine and hopefully validate
the value of the minimum total surface area calculated.
The topic of the exploration is important to investigate as much of the worlds environmental
and economic problems are due to product packaging wastage. Although my investigation
specifically focuses on three milk-producing companies, my aim for this mathematical
2
, Candidate code: jkj004
research and analysis is to make us as a society more aware of the current worldwide issue in
which we are facing.
II. Aim
The aim of this investigation is to explore the minimum total surface area for 1ℓ milk
packaging and thus determine which packaging is the most cost-effective and
environmentally friendly.
III. Measurements:
*All measurements of the various dimensions of the milk packaging, namely height, length,
width and base radius of the three milk containers will be measured in centimetres (cm).*
a. Visual representation: The below diagrams were extracted via Microsoft Word
and aim to visually represent how the above dimensions were measured for each
carton recorded:
Height
Height
Radius
Length Width
b. Measurements: The below measurements are the recorded dimensions (in cm) of
the three milk containers. Important to note is that the various milk containers and
the way in which they were each specifically measured for use in this
investigation, can be found in Appendix 1.
Biorganic almond milk:
Width: 7cm
3
