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Complimentary samples

Advertising is the very soul of trade, but it has costs! Your advice is needed again to choose a nice (and cheap!) packaging for some complimentary boxes, containing a smaller version of the pasta that you have created.

Option no. 1 - Cylindrical boxes

These complimentary boxes will be shaped as small cylinders, having a volume of cm3. The top and the bottom will be transparent, made with a plastic material that costs €2/m2, and the side is made with fine cardboard, that costs €10/m2.

Find the radius r of the base and the height h of the box that minimize its cost.

Option no. 2 - Rectangular parallelepiped

These classic complimentary boxes will be shaped as rectangular parallelepipeds, having the same volume of the cylindrical ones, and height 4 cm. They will be built using the same materials for the top and the side as the cylindrical ones.

Find the length l and width w of the base of the box that minimize its cost. What do you observe?

Compare costs, and decide

How much does a cylindrical box cost? And the "classic" one? (Round costs to 2 decimal places) So the cheapest complimentary box is...