In the preceding two problems we define a can to be optimal if it has the
least surface area for a given volume. In this problem we assume
the real issue in can manufacturing is cost, and the industry has
figured out which shape minimizes the cost of making the cans.
(Of course there are other issues, like how the can fits into one's
hand and whatever marketing appeal a given shape might have. Also
consider that once the country is saturated with vending machines it
would be very expensive to make substantial changes in the shape of
the can.)
The familiar 12oz aluminum can is made of two parts, a shell and a lid.
Assume that the total manufacturing cost for the lid and the shell is
cents.
The manufacturing cost of the lid consists of a fixed
cost of cents per lid, and an additional cost that
is proportional to the area of the lid. Thus the total cost is
cents per lid, for some
constant .
Similarly, the
manufacturing cost of the shell consists of a fixed cost
of cents per shell, and an additional cost that is
proportional to the surface area of the shell. Thus the total cost is
cents per
shell, for some constant .
Assuming that the total manufacturing cost of the can is minimized
for a diameter of 2.75 inches and a height of 5.0 inches, what are
the variable costs involved with the manufacture of the cans? cents per square inch, cents per square inch.