Soft Drink Package Efficiency
Adapted from Mathematics: Modeling Our World by COMAP.
Soft drinks are often sold in packages of twelve. The package is
usually made of paperboard. This package is not 100% efficient
because the cans do not fill all the space that is available in
the package. (Note: A standard soft drink can has a radius
of approximately 3.2 cm and a height of approximately 12
- What percentage of the package space is filled by the cans?
- Does a 9-pack with a square cross-section use package
space more or less efficiently? That is, does this package
use a higher percentage of the available package space than
- Which package is more efficient if the criterion is package
material used per can? That is, which package uses less
paperboard per can?
- Experiment with other packages that have a rectangular
cross-section and in which the cans are not stacked in more
than one layer. What can be concluded
- about the efficiency of package space use?
- about the efficiency of package material use?
Hint to problem 1. What is the volume of the package? What is
the volume of the twelve cans?
Hint to problem 2. What is the volume of the package? What is
the volume of the nine cans?
Hint to problem 3. What is the surface area of each package?
- Approximately 78.5%.
- The efficiency is the same.
- The 12-pack is more efficient than the 9-pack.
- In terms of package space used by the cans, the efficiency is
always about 78.5%. In terms of package material used per
can, the more cans in the package, the better the efficiency,
given that cans are arranged in a package in a rectangle of
a close-to-square form.
- The volume of the twelve cans is 12(×3.22×12) 4632.5
cm3. The volume of the package is 12(6 × 3.2)(8 × 3.2)
5898.2cm3. The cans use × 100 78.5% of the
- The volume of the nine cans is 9( × 3.22 × 12) 3474.4
cm3. The volume of the package is 12(6 × 3.2)(6 × 3.2)
4423.7 cm3. The cans use × 100 78.5% of the
- The surface area of the 12-pack is 2 × 12(6 × 3.2) + 2 ×
12(8 × 3.2) + 2(8 × 3.2)(6 × 3.2) 2058 cm2. The 12-pack
contains approximately 171.5 cm2 of paperboard per can.
The surface area of the 9-pack is 2×12(6×3.2)+2×12(6×
3.2) + 2(6 × 3.2)(6 × 3.2) 1659 cm2. The 9-pack contains
approximately 184.3 cm2 of paperboard per can.
- Consider a package of m by n cans, each can of a radius r and
- The total volume of cans is m × n( × r2 × h). The
volume of the package is h(m × 2r)(n × 2r). The
cans use × 100 = × 100 78.5%. So,
regardless of the number of cans in a package as well
as the size of cans, the efficiency of package space used
by cans is the same.
- The surface area of an m × n package is 2(m × 2rh +
n × 2rh + 2mr × 2nr) = 4rh(m + n) + 8r2mn. There
are m × n cans in a package, so a pack contains
= + 8r2 paperboard per can.
The second addend in this sum does not depend on the
number of cans at all, but the first one does. To make
the efficiency better, mn (number of cans in a package)
must be big, while m+n (the sum of dimensions of the
package) must be as small as possible. For example, a
package of 100 cans is more efficient when the cans are
arranged in 10 rows of 10, not in 20 rows of 5. At the
same time, it is more efficient in terms of paperboard
amount to make 1 package of 100 (10 × 10) cans than
to make 4 packages of 25 (5 × 5) cans.