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Dimensions of a Beverage Can Presented by: Tan Chee Meng Ahmad Tajuddin

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Our role 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.

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Our Goals 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

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Our Plan 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

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Our Plan 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

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Our Plan 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=400cm 3, 375cm 3 etc.

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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

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Find general equation relating variables Implement the Strategy

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Formulate equation of A in terms of r where V is a constant Formulate equation of A in terms of r where V is a constant Find the value of r

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When V= 400 cm 3 Find the values of r, h and minimum surface area A

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When V= 375 cm 3

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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):

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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 cm 2 12.9 cm 2 13.3 cm 2 13.6 cm 2

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Graphical Method: Using Geometer’s Sketchpad

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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 cm 3 Minimum value of surface area of can, A=300.48 cm 3 Graph 1 f(r)=0 Value of r when surface area is minimum When V= 400 cm 3

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(3.91, 287.83) (3.91, 0.00) f(r) r Minimum value of surface area of can, A=287.83 cm 3 Minimum value of surface area of can, A=287.83 cm 3 Find the coordinates of intersection between the graph f’(r) and the x axis Graph 2 When V= 375 cm 3

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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

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Algebra method (differential) VS Graphical method (GSP) 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 1 st and 2 nd 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)

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(cm 2 ) Minimum surface area =300.5 cm 2 When radius =4.0 cm Minimum surface area =300.5 cm 2 When radius =4.0 cm Comparing with the minimum surface (300.5 cm 3 )as r changes Percentage increase in the surface area as compared to the minimum surface area of 300.5 cm 2 Find the surface area using spreadsheet Table 1 Minimum surface area =310.8 cm 2 When radius =3.3 cm Minimum surface area =310.8 cm 2 When radius =3.3 cm When V= 400 cm 3

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(cm 2 ) Minimum surface area =287.8 cm 2 When radius =3.9 cm Minimum surface area =287.8 cm 2 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 cm 3

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Data indicate that the preferred volumes of drink in the can are 325cm 3 and 350 cm 3 among customers. We narrow down the choice of volumes of drink to 350cm 3 or 325cm 3. Data Analysis Table 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

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Data Analysis Table 4 shows the preferred dimensions of the can of volume 350 cm 3 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

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Data Analysis Table 5 shows the preferred dimensions of the can of volume 325 cm 3 Data indicate that the preferred choice of diameter and height of the can is 6.6 cm and 9.5 cm respectively. Table 5

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Potential Customers' Comments “ 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”

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Potential Customers' Comments “ 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 cm 3 will be sufficient to quench my thirst.

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Discussions The preferred choice of dimension for the diameter of the can is 6.6cm with a height of 10.2 cm (for Volume of 350cm 3 ) and 9.5 cm (for volume of 325cm 3 ). We need to do a comparison between the surface areas of the can of volume 350 cm 3 and 325 cm 3 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.

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Difference between surface areas V=350cm 3 and V=325cm 3 Table 6 Percentage difference between surface areas V=350cm 3 and V=325cm 3 When diameter of the can is 6.6 cm 5.4% Compare the Surface Area

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Discussions 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 cm 3 to 325 cm 3. 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 ??

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Current News Think about its Implications..?

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Think… 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?

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Conclusion 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 cm 3 Radius = 6.6 cm Height =9.5 cm

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