SOLUTION: A cereal box is designed to hold 3375 cubic cm of cereal. What dimensions for the box will minimize the cost of producing the box?

I don't know about the cost of producing the box, and I think that this passage is not relevant to the rest of the condition.


But I know which dimensions will minimize the surface area of the box (which directly relate to the cost of the material).


The dimensions what minimize the surface area are 15 x 15 x 15 centimeters:  the box must be a cube.


It is easy to get this result analytically, using Calculus.


The surface area of the (x,y,z)-box is  A(x,y,z) = 2*(xy + xz + yz).


The volume = xyz = 3375,  so  z = .


Therefore, A(x,y,z) =   at the given volume.


The conditions   =  = 0 give


     -  = 0  ====>  x^3 = 3375  ====>  x =  = 15,   and

     -  = 0  ====>  y^3 = 3375  ====>  y =  = 15.


And then  z =  =  = 15.