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4.17 Geometric example of a minimum

Example.

An open-topped rectangular box has sides x,yx,y and zz, and volume 32.32. For which choice of side lengths is the surface area smallest? (The idea is to make a box with volume 32 using a small amount of cardboard.)

Solution. The volume is given by V=x⁢y⁢zV=xyz. The surface area SS is given by contributions from the sides and the base, but there is nothing from the open top, so

S=x⁢y+2⁢x⁢z+2⁢y⁢z.S=xy+2xz+2yz.

Now since V=32V=32 by assumption, we have z=32x⁢y.z=\,{\frac{32}{xy}.} Thus we have to find the minimum of the function S⁢(x,y)=x⁢y+64y+64x.S(x,y)=\,{xy+\frac{64}{y}+\frac{64}{x}.}