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 $C = 6.5$ cents.
The manufacturing cost of the lid consists of a fixed cost of $M_{L} = 4.24$ cents per lid, and an additional cost that is proportional to the area of the lid. Thus the total cost is $$C_{L} = M_{L} + \kappa_{L} \pi \frac{d^2}{4}$$ cents per lid, for some constant $\kappa_L$.
Similarly, the manufacturing cost of the shell consists of a fixed cost of $M_{S} = 1.185$ cents per shell, and an additional cost that is proportional to the surface area of the shell. Thus the total cost is $$C_{S} = M_{S} + \kappa_{S} [\pi h d + \pi \frac{d^2}{4}]$$ cents per shell, for some constant $\kappa_{S}$.
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?
$\kappa_{L} =$ cents per square inch,
$\kappa_{S} =$ cents per square inch.