Personal Code: jtk075

Topic:

Using optimization to see which dimensions minimizes the amount of

packaging materials by keeping the volume constant

IBDP Mathematics

Applications and Interpretations SL

Internal Assessment

May 2024 Session

Persona code: jtk075

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Personal Code: jtk075

Introduction:
Having been exposed to the alarming consequences of climate change in the subject of IB

Geography, I was intrigued by this phenomenon, and I decided to investigate ways to reduce

the negative impact this has on the environment. According to the UN1, as of 2021, climate

change is one of the most important global issues. One of the main issues that result in global

warming is the vast amounts of waste that comes from packaging, producing extra plastic

leads to needless emissions, which shifts a focus on plastic pollution2. In my mathematics

Applications and Interpretations SL course, I learnt how to use basic Calculus to solve simple

optimization problems and I decided to further investigate how this knowledge is applied to

reduce packaging material and waste.

This IA was chosen out of curiosity, to try and do my part for the environment and live an

eco- friendlier lifestyle as nowadays, climate change is one of the most threatening and

ravaging global issues ruining our society. These packages, which in this IA will be a milk

package, are expensive to dispose of in large quantities and are considerably unsustainable.

The bottles are mainly made from plastic high density polyethene (HDPE)3, a thermoplastic

polymer made from petroleum. The main way to dispose this waste is to incarcerate them,

producing excessive amounts of gas (CO2) emissions. Hence, optimizing the packages

presents great positive environmental, ecological, and financial outcome. Therefore, 1

decided to do my part of the change, and this is by starting to optimize packaging for drinks,
1
“Climate Change.” United Nations, United Nations, www.un.org/en/global-issues/climate-change.
Accessed 31 Mar. 2024.

2
“Microplastics, Microbeads and Single-use Plastics Poisoning Sea Life and Affecting Humans.” UN News, 11
Dec. 2019, news.un.org/en/story/2019/11/1050511.

3
“What Is HDPE?” ACME Plastics, Inc., www.acmeplastics.com/what-is-hdpe. Accessed 31 Mar. 2024.

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to minimize the amount of materials needed for each package whilst still being able to keep

the same amount of liquid within it. I hope to create a solution to improve the planet's

sustainability and that this investigation will bring us a step closer in that direction.

Aim:
In this investigation, I will determine the amount of materials used to make the Coop milk

bottle, one of the most popular brands consumed by the average Swiss household. In this

case, I will be solving this problem with the optimizing the shape ‘s surface area and

simultaneously substitution the other equation in place of a variable and equating to 0 t

Figure 1: milk bottle4

Background Theory/Information:
The method of calculus optimization starts with marking two equations from a surface area

equation . substitute one unknow variable with its equation and equate to 0 . find the value of

other unknown variable and input in the original function. 5

4
“Pasteurized Whole Milk.” coop.ch, www.coop.ch/en/food/dairy-products-eggs/milk/fresh-milk/pasteurized-
whole-milk/p/6638289.

5
The Organic Chemistry Tutor. “Optimization Problems - Calculus.” YouTube, 26 Apr. 2021,
www.youtube.com/watch?v=lx8RcYcYVuU.

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Table 1: Definition of the symbols used throughout the investigation(self made on word )
Symbol of Variable Representation
h Height of the cuboid (cm)
H Height of the pyramid (cm)
l Length (cm)
w Width (cm)
S Surface Area (cm2)
Stotal Total Surface Area (cm2)
Sopt Optimized Surface Area (cm2)
Vbottle Volume of the milk bottle (cm3)
V cuboid Volume of the cuboid (cm3)
Vpyramid Volume of the pyramid (cm3)
Swasted Wasted Surface Area (excess) (cm2)

Any subscript used with one of the same variable symbols shown above in Table 1 has the

same representation but for the subscript. (e.g., Scuboid is the surface area of a cuboid).

Methodology:
Apparatus and real-world considerations:

The only apparatus I will be using is a bottle of milk with a volume of 1L. For measuring, I

will be using a ruler which has an uncertainty of ±0.05 cm.. The bottle was selected very

carefully, the first step was to find IL bottles of milk in a supermarket. The second step was to

find the bottle made of plastic. The final step was to find a bottle that was not dented in any

way so that the results would be more precise and accurate. An empty milk bottle was used to

not let any milk go to waste.

The problem with the milk bottle that I selected was that the bottom part of the milk bottle

isn't a nice spherical shape, it has a slight indent with a dome shape. I could try to account for

this using the formulas for the surface area and the volume of a sphere (or dome) but this

might overly complicate the work, so I just chose to assume that the bottom is flat. These

limitations will mean that the exact surface area will differ however the data will be more

than sufficient. Therefore, to fix this, I have decided that the nearest millimeter is sufficient.

Due to this limitation, I will open the package and measure it, so it is more accurate.
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Mathematical Procedure:
First example Cylinder :

Volume of a cylinder = π r 2 h

Volume of a mug =350cm 2

2
π r h = 350

350
h= 2
πr h

surface area = 2 π r 2+ 2 πrh

350 2
a= 2 πr ( 2
)+2 π r
πr h

350 2
=2 × +2π r
r

700 2
a= +2 π r - equation 2
r

For minimizing surface area :

da
=0.
dr

Differentiate equation 2 with respect to “r ”

da d 700 2
= ( +2 π r )
dr dr r

2
d ( d 2π r
700 r ) +
−1
¿
dr dr
−2 1
¿ 700 (−1 ) r +2 π 2(r )

−2
¿−700 r +4 πr

da −700
= 2 + 4 πr
dr r

For minimum surface area:

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da
=0.
dr

−700
2
+ 4 πr=0
r
3
4 πr =700

3 700
r=
4π

r =√ 55.73
3

r =3.8

700 2
Area = +2 π r
r

700 2
¿ +2 ×3.14 ×(3.8)
3.8

¿ 184.21+90.68

2
A=274.8 cm
2
let the packaging cost for 1 cm =rs 0.2

Packaging cost for this area 274.8 ×0.2=rs 55

Verification : (self made on words )

r= 3.5cm

r = 3.8 cm ( actual answer )

r = 4.5 cm

700 2
Area = +2 π r
r

When r = 3.5

700 2
¿ +2 ×3.14 ×(3.5)
3.5

¿ 200+76.93

2
A=276.93 cm

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Packaging cost for this area 276.93 ×0.2=rs 56

When r = 4.5

700 2
¿ +2 ×3.14 ×(4.5)
4.5

¿ 155.55+127.17

2
A=282.7 cm

Packaging cost for this area 282.7 × 0.2=rs 56.55

Second example cuboid :

A square based cuboid volume = 350 cm3

Length= width = x

Height = h

Volume of a cuboid = l ×w × h

x × x × h=350

2
x × h=350

350
h= 2 - equation 1
x

surface area (A)=2(lw +wh+hl)

¿ 2((x × x )+(x ×h)+(h × x))

¿ 2 ( x 2+2 xh )

2
A=2 x +4 xh

2 350
A=2 x +4 x( 2
)
x

2 1400
A=2 x +
x

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For minimum surface area :

da
=0.
dx

da d 2 d 1400
= 2x +
dx dx dx x

d −1
¿ 4 x+ 1400 x
dx

−2
¿ 4 x−1400 x

da 1400
=4 x− 2
dx x

For minimum length

1400
4 x− 2
=0
x

1400
4 x= 2
x
3
4 x =1400

3 1400
x=
4

3
x =350

x=√ 350
3

x=7.03 cm

x = length = width = 7.03

350 350
H= height = 2
= 2 = 7.08cm
x 7.03

2 1400
Area¿ 2 x +
x

¿2¿

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¿ 98.94 +199.14

2
area=297.98 cm

Total packaging cost per cm is 0.2 rupees hence 297.98 ×0.2=rs 59.5

Verification : (self made on words )

X=6 cm

X=7.03 cm ( actual answer )

x= 8 cm

2 1400
Area¿ 2 x +
x

When x= 6

¿2¿

¿ 72+233.33

2
area=305.33 cm

Total packaging cost per cm is 0.2 rupees hence 305.33 ×0.2=rs 61

When x= 8

¿2¿

¿ 128+175

2
area=303 cm

Total packaging cost per cm is 0.2 rupees hence 303 ×0.2=rs 60.6

Third example conical shape :

1 2
Volume of a cone = πr h
3

Volume = 350 cm3

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1 2
π r h=350
3

350 × 3
h= 2
πr

1050
h= 2
πr

334
h= 2
r
2
surface area of a cone=π r +πrl

¿ π r + πr √ r +h
2 2 2

√
¿ π r 2+ πr r 2 +
334
r2

area=π r 2 + πr r 2+
√ 334
r2

diffrentiate with respect ¿ r on both sides

da d 334
= (π r 2+ πr r 2+ 2 )
dr dr r √
πr −3
(2 r −334 r )
da
dr √ 334
=2 πr + π r 2 + 2 +
r
2

√ 2
r+
334
r
2

Minimizing surface area

πr −3
(2 r −334 r )

√
2 πr + π r 2 +
334
r
2
+
2

√ 334
2
r+ 2
r
=0

√
−3
334 1.57 r (2 r−334 r )
2 πr + π r 2 + =
r2 √r 2 +334 r −2

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−3
1.57 r (2 r−334 r )
=0
√r 2 +334 r−2
1.57 r ( 2 r−334 r−3 )=0× √ r 2+ 334 r −2

1.57 r ( 2 r−334 r−3 )=0

Either :

−1.57 r =0

0
r= =0
−1.57

Or:

( 2 r−334 r−3 ) =0
−3
2 r=334 r
3
2 r ×r =334

1 3 334
r ×r =
2

4
r =167

r =√ 167
4

r =3.6 cm

√ 2
Surface area = π r + πr r +
334
r2

√
¿ π 3.62 +π 3.6 3.62 +
334
3.62

¿ 12.96 π +3.6 π √12.96+25.77

¿ 12.96 π +22.4 π

2
area=35.36 π cm

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Total cost for packaging is 0.2 RS per cm hence 35.36 × 0.2=22.20 rs

Verification : (self made on words )

r =3 cm

r =3.6 cm ( actual answer )

r = 4 cm

√2
Surface area = π r + πr r +
334
r2

When r = 3

√
¿ π 32+ π 3 32+
334
32

¿ 9 π +3 π √ 9+37.11

¿ 28.26+100.4

2
area=128.66 cm

Total cost for packaging is 0.2 RS per cm hence 128.66 × 0.2=25.732 rs

When r = 4

√
¿ π 4 2+ π 4 42 +
334
42

¿ 16 π +4 π √ 16+20.875

¿ 16 π +4 π + √ 36.875
2
area=40.28 π=126.5 cm

Total cost for packaging is 0.2 RS per cm hence 126.5 ×0.2=25.3 rs

Real-world application/impact:

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Following the results above, I decided to estimate how much waste is produced annually
considering that 4 billion kilograms of milk are produced annually in Switzerland, according
to the "Swiss Farmers"'.

4 billion kilograms of milk is equal to approximately 3.88 billion Liters of milk.

m(mass)
ρ ( Density )=
V (volume )

So,
9
m 4 × 10 9
ρ= = =3.88 ×1 0 L
V 1030

Assuming that half of that milk is sold every year in only milk bottles and not in any other

form of dairy product, around 1.94 billion bottles of 1L milk are sold. The wasted surface

area in this milk bottle was of 26.51cm2. This means that in 1 year, with 1.94 billion bottles of

1L milk, the wastage can be reduced by approximately 5.14 × 1010 cm2 or 5.14 million m2or

5.14 km2if we use this optimized milk bottle shape.

These bottles create an unnecessary amount of waste which could easily be reduced. These

brands could change their packaging with a view to reducing waste and the negative impact

this has on the environment. After making some research on why these companies have not

yet altered their packaging to significantly reduce wastage, the answers do not surprise me at

all. It turns out that these companies employ techniques for business and marketing, creating

a visual appeal in the minds of the consumers and to promote the product's consumers thus

boosting profits, maximizing sales and revenue.

Conclusion:
My IA has found that an optimized shaped bottle is one with the following dimensions:

h=15.96 cm

w=l=7.51 cm

H=5.31 cm

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This produces a total surface area of 660.025 cm2, means that a total of approximately

26.51 cm2 is wasted by the Coop bottle.

In conclusion, the aim of this investigation was met as the waste produced by the bottle was

successfully calculated.

This investigation has led me to realize the significance of mathematics in tackling real-world

problems to reduce the negative impact of the production of goods and to promote a more

sustainable approach for the future. I would be keen to present my findings to relevant companies.

In doing so, I would hope to balance the issues of excess waste calculated in this work with

commercial decisions to not be too negative when corresponding with these companies.

Evaluation/Reflection:
There are various positive impacts of optimizing packages to companies and society. The

positive impacts of optimized packages are that they minimize the space that the

packaging of these products occupies, hence saving space. Moreover, less material will be

used for each individual product, therefore less energy from fuels is used for the same

number of products, hence reducing the emissions of CO2 and other gases from the

atmosphere. Furthermore, the company will show their consumers and clients that they hold

strong ethics as well as social, environmental, and ecological responsibility which will

probably lead to more consumers buying products from their company.

However, there are also negative impacts of optimizing packaged to companies and society.

Firstly, the look and the design of the product will quite possibly be lost. Moreover, the

company will have minimal space on the product to advertise the brand and logo. Finally, the

product's information (e.g., content/ingredients on the back of the bottle) will possibly be lost

as there will be less space; hence preventing customers from seeing the content of the product

which may cause them to be uninterested in the consumption of the product.

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The strengths of this investigation are that the product used was carefully addressed and it

was found without any dents or cracks, therefore the accuracy and precision of the data is at a

better standard than if any product were randomly picked and used. Moreover, The

weaknesses of this investigation are that some assumptions were made to stay within a certain

page limit and a reasonable time frame. Another weakness is that the measurements were not

done with the most accurate tools,

Extension:
To further this research, different products can be used to see if different shapes result in the
wastage of materials. In addition, a comparison of different brands may be another good idea
to improve this investigation to see if different brands waste materials than others. Moreover,
calculating the exact surface area including the little dents, cracks and other small details
would positively add to this investigation. For example, instead of making assumptions of the
shape of the product, you could calculate it fully. Furthermore, using more accurate and
precise tools with lower/smaller uncertainties would help this investigation in being more
precise and to gather better data. It would also be interesting to see what results would be
achieved if using the same method in business and economics pov which can reduce waste on
earth.

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Bibliography:

 “What Is HDPE?” ACME Plastics, Inc., www.acmeplastics.com/what-is-hdpe.

Accessed 31 Jan. 2024.
 The Organic Chemistry Tutor. “Optimization Problems - Calculus.” YouTube, 26 Apr.
2021, www.youtube.com/watch?v=lx8RcYcYVuU. Accessed on 31 Jan 2024
 “Milk - Swiss Farmers.” Swiss Farmers, 11 Oct. 2023,
www.swissfarmers.ch/knowledge-and%20facts/food/milk/#:~:text=Swiss%20milk
%20%2%80%93%20. Accessed on 28 Jan 2024
 “Climate Change.” United Nations, United Nations,
www.un.org/en/global-issues/climate-change. Accessed 31 Jan. 2024.
 “Microplastics, Microbeads and Single-use Plastics Poisoning Sea Life and Affecting
Humans.” UN News, 11 Dec. 2019, news.un.org/en/story/2019/11/1050511.
Accessed on 28 Jan 2024

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