Question 399060
Suppose the dimensions of the prism are L, W, H. Then,


{{{2(LW + WH + LH) = 256}}} --> {{{LW + WH + LH = 128}}}


By the AM-GM inequality (arithmetic mean - geometric mean inequality),


{{{(LW + WH + LH)/3 >= root(3, L^2W^2H^2)}}}


{{{128/3 >= root(3, L^2W^2H^2)}}} Cubing both sides,


{{{128^3/3^3 >= L^2W^2H^2}}}


{{{sqrt(128^3/3^3) >= LWH}}}, this value is the maximum volume of the prism, which occurs when the solid is a cube.