Differentiation

1
Differentiation

• Dimensions of a Beverage Can

Presented by Tan Chee Meng Ahmad Tajuddin
2
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
  customers needs.

3
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

4
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

5
Our Plan

• Explore Compare Various Strategies
• Strategy 2 Use Geometers 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

6
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. V400cm3, 375cm3 etc.

7

• 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

8
Implement the Strategy
Find general equation relating variables
9
Formulate equation of A in terms of r where V is
a constant
Find the value of r
10
When V 400 cm3
Find the values of r , h and minimum surface area
A
11
When V 375 cm3
12

• Repeat the procedures to calculate the value of r
  , h and A when the values of V are 350 cm3,325
  cm3 and 300 cm3

13
The values of r and h based on different values
of V are as follows
Relationship between h and r (or D)
h 2r Diameter
The height of the can is approximately equal to
its base diameter when the surface area is minimum
14
(No Transcript)
15
Graphical Method Using Geometers Sketchpad
16
f(r)
When V 400 cm3
Minimum value of surface area of can, A300.48 cm3
(3.99,300.48)
The function is minimum if f(r)0. Find the
coordinates of intersection between the graph
f(r) and the x axis.
r
(3.99,0.00)
Value of r when surface area is minimum
f(r)0
Graph 1
17
When V 375 cm3
f(r)
Minimum value of surface area of can, A287.83 cm3
(3.91, 287.83)
Find the coordinates of intersection between the
graph f(r) and the x axis
r
(3.91, 0.00)
Graph 2
18

• Repeat the procedures to plot the graph of the
  function and its first derivatives when the
  values of V are 350 cm3,325 cm3 and 300 cm3

19
Plot the graph of h vs r to find the values of h
as r varies
h
Move the point along the line to determine values
of h and r (coordinates)
Values of r
Values of h
r
20
Algebra method (differential) VS Graphical method
(GSP)

• ALGEBRA METHOD (DIFFERENTIAL)

• GRAPHICAL METHOD
• (GSP)

• Need to carry out tedious calculations to
  determine each value of h and A
• Need to use scientific calculator to calculate
  the values

• 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)

21
Find the surface area using spreadsheet
Comparing with the minimum surface (300.5 cm3 )as
r changes
When V 400 cm3
Minimum surface area 310.8 cm2 When radius 3.3
cm
Percentage increase in the surface area as
compared to the minimum surface area of 300.5 cm2
Minimum surface area 300.5 cm2 When radius 4.0
cm
Table 1
22
When V 375 cm3
When the radius decreases, the total surface area
of the can increases significantly from 0.1 to
17.9
Minimum surface area 287.8 cm2 When radius 3.9
cm
Table 2
23

• Repeat the procedures to draw the table of values
  for A and h when the values of V are 350 cm3,325
  cm3 and 300 cm3
• The values of A and h obtained using
  differentiation, GSP and Spreadsheet are very
  close.

24
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
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.
25
Data Analysis Table 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.
26
Data Analysis Table 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.
27
Potential Customers' Comments

• The shape when r6.6cm 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

28
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 cm3 will be
  sufficient to quench my thirst.

29
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 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.

30
Compare the Surface Area
31
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 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 ??

32
Current News
Think about its Implications..?
33
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?

34
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 cm3
• Radius 6.6 cm
• Height 9.5 cm

35
Thank You