MATHS IA: Optimization of Packaging

Mathematics Standard Level

Internal Assessment

Title: Optimisation of packaging of commercial products


All the products used/included are only for academic purposes, the aim of this research does not criticize any particular firm.


My interest in Calculus made me explore all the real-life situations where Calculus is applicable. In that exploration, I found an article which talks about how calculus can be used by firms/businesses. Businesses can use calculus for many things like for, knowing the output at which they receive maximum profit or minimum cost, for sales forecasting, etc… out of all those I found that firms use calculus for packaging, this attracted me and made me know everything about it.

There comes the topic of my Maths I.A. optimization in packaging. I found that many firms are using packaging as their major marketing strategy mainly the firms that are dealing in the food and beverage industry. Firms use packaging for their product differentiation.

Even after five years of the adoption of The Sustainable Development Goals, many countries are said to be unsustainable, this is due to the increasing levels Solid Domestic Waste.

Solid Domestic Waste Management is the major problem that is being faced by many of the developing and developed countries of the world. And through surveys, it is clear that more than 80% of this waste is being generated through packaging. Therefore, I wish to give a solution to this environmental and economic problem by using Mathematics, i.e. by using calculus to find the minimum optimal packaging for different shapes like cylindrical, cuboidal, cube-shaped products.

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This exploration is applicable only to cylindrical and cuboidal products. The real products used are Pepsi 250ml can( cylinder ), Kellogg’s Cornflakes( cuboid ).


Being a Business Management student, I know the significance of packaging for commercial firms. It is one of the most important components of firm marketing mix. Many M.N.C.s and branded companies like, coca cola, Kellogg’s etc use intensive packaging. They try to make their product unique by different packaging styles. Contradicting to the above argument, I, being an E.S.S. student also know that waste disposal is one of the major problems faced by majority of the developed and developing countries. As the world is developing day by day, the volume of the waste generated is also increasing. Asking the firms not to use packaging will not be a feasible solution for this environmental problem. Optimal minimum packaging of different products is the solution to this issue.

With my knowledge in Mathematics, especially in calculus, finding the optimal minimum Total Surface Area of different products(cylindrical, cuboidal) for a constant or given volume is the purpose of this exploration. This would provide an economically and environmentally sustainable solution for the issue.


3276600517525Equation 1

020000Equation 1

Volume of the cylinder = v



4943474761999004038600771525004552950666750 742950120650here, v = volume of the cylinder



h = height of the cylinder

405765015176500 r = radius of the cylinder.

5302885223520Figure 1

00Figure 1

3933825483235Equation 2

020000Equation 2

Total Surface Area(T.S.A.) of cylinder:


From Equation 1,

10001251409700036576008255Equation 3

020000Equation 3

By substituting equation 3 in equation 2:




Now, we have an equation for T.S.A. in terms of ‘r’ and ‘v’.

In order to get the optimal minimum T.S.A. of cylinder, assuming ‘v’ as a constant, we need to derive T.S.A. in terms of ‘r’.

4086225526415Equation 4

00Equation 4


Now, equate the equation 4 to zero to get the stationary points, i.e. to get the value of ‘v’ at which T.S.A. is maximum or minimum.

4324350643890Equation 5

00Equation 5


So, when , the T.S.A. of cylinder is minimum.

To confirm this, we should do the second derivative test.

4095750836295Equation 6

00Equation 6


Now substitute equation 5 in equation 6


Therefore at , T.S.A. of cylinder is minimum.

In order to apply this in Real life it is better to get it in terms of ‘h’, so,

By equating equation 1 to equation 5

3638550198120Equation 7

00Equation 7


With the above calculations we can say that and cylindrical shaped product of constant volume ‘v’ will have minimum surface area when the height ‘h’ of the cylinder is equal to the diameter of the cylinder(2r).

Real life application:

Pepsi 250ml can,

Pepsi co sells millions of its products daily and one of their famous products is Pepsi Height of this can = 5.2 inches Radius of this can = 2.0 inches

Volume of drink = 250ml

By putting these values in Equation 7,

h = 2r

given that radius of the can = 2 inches

h = 4 inches.

This is the optimal minimum height of the Pepsi tin for volume of 250ml.

Actual Total Surface Area of Pepsi tin:

3819525885190Three significant values

00Three significant values


So, the actual T.S.A. of Pepsi tin(250ml) is

Now, the optimal minimum T.S.A. od Pepsi tin(250ml) is,

Waste from packaging = actual T.S.A.-optimal T.S.A.

Therefore, if the firm use this optimal minimum packaging for their product, of waste is saved/reduced per single piece.


l= length of the cuboid

w= width of the cuboid

x= height of the cuboid

v= volume of the cuboid

4581525466090Equation 8

00Equation 8


Now in order to get the minimum T.S.A. of the cuboid we need to differentiate V in terms of x, let l and w be constants.

5029200507365Equation 9

00Equation 9


Now equate the equation 9 to zero in order to get the stationary points i.e. the value of ‘x’ at which volume ‘v’ is maximum and minimum.

By second derivative test.

436245055880Equation 10

00Equation 10


Now put the ‘x’ values that we have got in equation 10

Therefore, when ‘v’ is maximum at

With the calculations above we have determined a formula for a value of ‘x’ that leads to the same volume ‘v’, length ‘l’, and width ‘w’, but uses optimal minimum packaging materials in cuboidal shaped products.

Real life application


This exploration has broadened the explorer’s knowledge in mathematics especially in calculus. This exploration also proves that applications of calculus are not just kinematics and marginal costs, but also for optimal packaging. The results of the exploration are mailed to different companies like, Kellogg’s etc some of them have not responded to my suggestion, which will be one of the possible limitations of this exploration, some other limitations and merits of the exploration are given below:

Limitations of the exploration:

This might affect the packaging strategies of businesses like, Kellogg’s, etc.

Due to the reduced surface area of the product, the firms might not be able to provide more details about the product.

Firms might use their uniqueness, due to the change in this packaging strategies.

Merits od the exploration:

Society as a whole gets benefited because of less amount of waste generated.

If this strategy is adopted by the firms, then this add on to the Corporate Social Responsibility of the firm. And therefore, increase in its brand loyalty.

Now it might be easy to achieve sustainable development for countries.


Sustainable development goals. (n.d.). Retrieved September 17, 2019, from

B. (n.d.). Packaging Retrieved September 17, 2019; from Retrieved October 29, 2019, from

(n.d.). Retrieved October 29, 2019, from Retrieved October 29, 2019, from

Retrieved October 30, 2019 from

(n.d.). Retrieved October 30, 2019 from

P., Kadelburg, V., Woolley, B., & Ward, S. (2013). p. 527-566. Mathematics standard level. Cambridge: Cambridge University Press. [Textbook]

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MATHS IA: Optimization of Packaging
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