Introduction:

Beverages industry is a billion dollar industry with popular brands such as pepsico , coca cola

provide packaged soft drinks and other items in a tin can or sealed plastic bottles. Millions of

them sold every month a year which peaks in summer season across the globe. Different

shapes of bottles and cans with 200ml, 300ml ,500 ml etc. variant are available in the market

which is a scope to study for about how the optimization for a fixed volume can be achieved

so as to reduce the cost of manufacturing the bottle by using minimum material and minimum

packaging part.

Thus it becomes an area of research to look into solving both the economic and

environmental challenge as the wastage can be reduced which will affect the structure overall

and may lead to better working conditions for both the company and society.

Aim:
The aim of this assignment is to design and optimize the cylindrical and a rectangular prism

can shape packed items.

Rationale:
I have an enthusiasm over knowing how things are working in real world and mathematical

functions relate me into this about how this function can design a product. Thus in a real

world application based product, and applying this mathematical concept help me to

understand and correlate what all factors are involved in the designing of it. So I have chosen

this topic of optimization of cans as it will help me to understand how the most selling

product designs can be optimized by using mathematics and a little knowledge of shapes.
Methodology:
In this assignment I am investigating into the usual shapes which are used in day to day

products and its optimized condition for which the wastage can be minimized by comparing

the actual surface area to the optimized surface area using the calculus method and then

subtracting both the areas to know the amount of saving in packaging can be made if we

follow the set optimized parameters for the product specifications required.

Cylindrical Cans:

The most common shape used by majority of packaging is the cylindrical shape. This shape is

being chosen as it can carry larger volume with minimum surface area and is easy to carry.

No. symbol Description

1 r Radius of cylindrical can

2 h Height of can

Optimization using a Fixed volume:

2
Volume of cylinder =π r h

V
h= 2
πr
 2
Surface area of both side closed cylinder =2 π r +2 πrh

Using the height ‘h’ equation in surface area we get,

2 V
SA=2 π r +2 πr 2
πr

To optimize the design, we need to minimize the surface area,

dSA
Thus, =0
dr

dSA
dr
−V
=4 πr +2 2 =0
r ( )
4 πr =2
( )V
r
2

3
V opt =2 π r

2
d SA V
2
=4 π +4 3
dr r

Using V =2 π r 3

( )
2 3
d SA 2π r
2
=4 π +4 3
dr r

2
d SA
2
=12 π >0
dr

Hence the optimum condition is true to achieve minimum surface area.

By comparing Volume equation, V =π r 2 h we get,

h
r opt =
2

3
∴ V opt =2 π r
 2 2
SAopt =2 π r +2 πrh=2 π r +2 πr ( 2 r )

2
SAopt =6 π r

Optimization of Hunts Tomato sauce can (8 oz)

I am going to optimize a popular brand Hunts Tomato sauce can of 8oz (236.60 ml) of

volume.

Description Dimensions
Radius (r) 6.8 cm
Height (h) 7.6 cm

For optimizing using the cylindrical optimization parameters for fixed volume 236.60 ml

3
V =2 π r opt

3
236.6=2 π r opt

( )
1
236.6 3
r opt = =3.34 cm
2π

h=2 r=6.68 cm

2 2
SAopt =6 π r opt =6 × π ×3.34

2
SAopt =210.85 cm
Actual Surface area of Hunt’s can is:

2
SA=2 π r +2 πrh

2
SA=2 π ( 6.8 ) +2 π × 6.8× 7.6

2
SA=235.72 cm

235.72−210.85
∴ Percentage saving∈surface area= ×100=10.72 %
235.72

A decrease in 10.72% in surface area can be achieved for the same volume if optimized

conditions are used for packaging of the cans instead of the regular dimension which can be

economical for the company too.

Cylindrical –cone Can

cylindrical function can. I am using Desmos tool to find the surface area of the can by

dividing the whole can into four functions and then applying the rotational theorem to

calculate the surface area.

Function1: