Consider a box with sides x, y, and z, and fixed volume xyz = V.

(a) Without using Lagrange multipliers, find the minimum surface area (the sum of the areas of all 6 faces) of a box with volume V. What is the shape of this optimal box? Confirm that your solution is indeed a local minimum using the second derivative test.

(b) Find the minimum surface area and shape of a box that is open at the top (so only has 5 faces).

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Let the rectangular container be as seen in the Fig 1 and the lengths of the sides be x y and z Also ... View the full answer