The least area and most volume occur with the same dimensions, those that make it a cube. It will be a cube with edge length 36^(1/3) ≈ 3.30192725.
The surface area will be 6*36^(2/3) = 6^(7/3) ≈ 65.4163413 square units.
The volume is defined by the problem statement as 36 cubic units.
We can write an expression for the area in terms of volume, width, and height. It is
area = 2(v/h + v/w + hw)
The partial derivative of this with respect to h is
∂(area)/∂h = 2(w-v/h^2)
If we recognize that v = L*w*h, then setting this to zero will show that L=h. By the symmetry of the equations, all edge dimensions must be equal. In other words, the prism must be a cube.
The surface area will be 6*36^(2/3) = 6^(7/3) ≈ 65.4163413 square units.
The volume is defined by the problem statement as 36 cubic units.
We can write an expression for the area in terms of volume, width, and height. It is
area = 2(v/h + v/w + hw)
The partial derivative of this with respect to h is
∂(area)/∂h = 2(w-v/h^2)
If we recognize that v = L*w*h, then setting this to zero will show that L=h. By the symmetry of the equations, all edge dimensions must be equal. In other words, the prism must be a cube.
