dimensions of a beverage can presented by: tan chee meng ahmad tajuddin

Dimensions of a Beverage Can

Presented by:

Tan Chee MengAhmad Tajuddin

We are production managers in a beverage company. Our task is to determine the dimension of a can that is cost effective and satisfies our customer’s needs.

To determine the appropriate dimension of a cylindrical beverage can

To find the radius and the height of the cylindrical can when its total surface area is minimum

To find the minimum surface area of the can using various strategies

To find the relationship between the height and radius of the can

To investigate the preference of customer in terms of volume, dimension and aesthetic value

Explore & Compare Various Strategies

Strategy 1: Apply differential calculus Choose suitable symbols to represent the variables:

radius (r), height (h), surface area (A) and volume (V) Formulate an equation of surface area (A) in terms of

radius Differentiate A with respect to r Find the turning point when dA/dr =0 Substitute value of r to find A and h Repeat the above steps with different values of the

volume (V) Find the relationship between r and h

Explore & Compare Various Strategies

Strategy 2: Use Geometer’s Sketchpad Plot the graph of f(r) Find the first derivative of f(r) i.e. f’(r) Plot the graph of f’(r) Find the intersection of the graph f’(r) with the x-axis Find the value of r (x-coordinate) when surface area (A)

is minimum

Explore & Compare Various Strategies

Strategy 3: Make tables using spreadsheet Make a table to find the surface area of the can with

different values of r and h (write formula to enable Spreadsheet to calculate the required values automatically)

Make tables to show the value of A with different volume of the can e.g. V=400cm3, 375cm3 etc.

Make a survey to determine the preference and needs of the customers:

Collect data regarding the preferences and needs of the customer in terms of the can dimension, the volume and the appearance (aesthetic value) through survey and Internet research

Make a survey to determine the preference and needs of the customers:

Collect data regarding the preferences and needs of the customer in terms of the can dimension, the volume and the appearance (aesthetic value) through survey and Internet research

Find general equation relating variables

Implement the Strategy

Formulate equation of A in terms of r whereV is a constant

Formulate equation of A in terms of r whereV is a constant

Find the value of rFind the value of r

Find the values of r , h and minimum surface area A

Find the values of r , h and minimum surface area A

The height of the can is approximately equal to its base diameter when the surface area is minimum

The height of the can is approximately equal to its base diameter when the surface area is minimum

h = 2r= Diameter

The values of r and h based on different values of V are as follows:

Relationship between h and r (or D):

The table below shows the surface area of the can, its base radius and height

The surface area is reduced from 12.7 to 13.6 cm2 when the radius is decreased by 0.1 cm from 4 cm to 3.6cm

Difference in surface area

12.7 cm2 12.9 cm2 13.3 cm2 13.6 cm2

The function is minimum if f’(r)=0. Find the coordinates of intersection between the graph f’(r) and the x axis.

The function is minimum if f’(r)=0. Find the coordinates of intersection between the graph f’(r) and the x axis.

(3.99,300.48)

(3.99,0.00)

f(r)

r

Minimum value of surface area of can,A=300.48 cm3

Minimum value of surface area of can,A=300.48 cm3

Graph 1f(r)=0

Value of r when surface area is minimum

When V= 400 cm3

(3.91, 287.83)

(3.91, 0.00)

f(r)

r

Minimum value of surface area of can,A=287.83 cm3

Minimum value of surface area of can,A=287.83 cm3

Find the coordinates of intersection between the graph f’(r) and the x axis

Find the coordinates of intersection between the graph f’(r) and the x axis

Graph 2

When V= 375 cm3

Plot the graph of h vs r to find the values of h as r varies

r

h

Values of r

Move the point along the line to determine values of h and r (coordinates)

Values of h

ALGEBRA METHOD (DIFFERENTIAL)

• Need to carry out tedious calculations to determine each value of h and A

• Need to use scientific calculator to calculate the values

GRAPHICAL METHOD (GSP)

• Can use GSP to plot complicated graph of function, its 1st and 2nd derivatives.

• Able to determine the minimum / maximum value from the graph with ease

• From the graph of function relating h and r, we can determine the value of h for any value of r by moving the point along the graph (determine the coordinates)

(cm2)

Minimum surface area=300.5 cm2

When radius =4.0 cm

Minimum surface area=300.5 cm2

When radius =4.0 cm

Comparing with the minimum surface (300.5 cm3 )as r changes

Comparing with the minimum surface (300.5 cm3 )as r changes

Percentage increase in the surface area as compared to the minimum surface area of 300.5 cm2

Percentage increase in the surface area as compared to the minimum surface area of 300.5 cm2

Find the surface area using spreadsheet

Table 1

Minimum surface area=310.8 cm2

When radius =3.3 cm

Minimum surface area=310.8 cm2

When radius =3.3 cm

When V= 400 cm3

(cm2)

Minimum surface area=287.8 cm2

When radius =3.9 cm

Minimum surface area=287.8 cm2

When radius =3.9 cm

Table 2

When the radius decreases, the total surface area of the can increases significantly from 0.1 % to 17.9 %

When V= 375 cm3

Data indicate that the preferred volumes of drink in the can are 325cm3 and 350 cm3 among customers. We narrow down the choice of volumes of drink to 350cm3 or 325cm3.

Data AnalysisTable 3 shows the preferred choice of the volume of the drink in the can. Total number of people participated in the survey is 128.

Table 3

Data AnalysisTable 4 shows the preferred dimensions of the can of volume 350 cm3

Data indicate that the preferred choice of diameter and height of the can is 6.6 cm and 10.2 cm respectively.

The elevation of the cylindrical can is a rectangle of sides 6.6 cm x 10.2 cm. The ratio 10.2/6.6 = 1.55 is very close to golden ratio which is aesthetically pleasing to the eye.

Table 4

Data AnalysisTable 5 shows the preferred dimensions of the can of volume 325 cm3

Data indicate that the preferred choice of diameter and height of the can is 6.6 cm and 9.5 cm respectively.

Table 5

“ The shape when r=6.6cm and 5 cm is pleasing to the eye”

“It looks ugly if the height of the can is much bigger than the diameter of the can. Even though the surface area is minimum when diameter equals height of can, the side elevation of the can is a square. This shape is not interesting”

“ The can is nice to hold when its diameter is 6.6cm. It is difficult to have a good grip of the can if the diameter is too large”

“For a carbonated drink, about 325 cm3 will be sufficient to quench my thirst.

The preferred choice of dimension for the diameter of the can is 6.6cm with a height of 10.2 cm (for Volume of 350cm3) and 9.5 cm (for volume of 325cm3).

We need to do a comparison between the surface areas of the can of volume 350 cm3 and 325 cm3 as shown in table 6 so as to choose the most suitable dimension that reduces the surface area and hence the cost of aluminum to make the can.

Difference between surface areas V=350cm3 and V=325cm3

Difference between surface areas V=350cm3 and V=325cm3

Table 6

Percentage difference between surface areas V=350cm3 and V=325cm3

Percentage difference between surface areas V=350cm3 and V=325cm3

When diameter

of the can is 6.6 cm

5.4%5.4%

Compare the Surface Area

From table 6, when the diameter of the can is 6.6cm, the surface area of the can is decreased by 5.4 % when its volume decreases from 350 cm3 to 325 cm3. The percentage of reduction is quite significant

Assume for a beverage company, the total cost for the production of the cans is RM5 million, then reduction in cost will amount to RM 270,000 i.e. (5.4/100) x 5 000 000

What will be the amount if the cost is RM 100 million ??

Think about its Implications..?Think about its Implications..?

Can we use 3-D shape such as cuboid, sphere or prism for packaging carbonated soft drinks? Why?

Can we recycle the aluminum cans? What is the cost of recycling the cans?

Can we use other cheaper materials other than aluminum?

What is the actual dimensions of the beverage can available at the local supermarket?

What is the possible future design and dimensions of the can?

After considering customers’ perception and needs, cost of production and the rising price of aluminum and theoretical calculation, we decided that the dimensions of the can are as follows:

Volume of can = 325 cm3

Radius = 6.6 cm

Height =9.5 cm