Question 907313
x for length, y for width,
x=2y
volume is 10 cubic meters, so if for z height,
xyz=10


Finding z:
{{{(2y)yz=10}}}
{{{2y^2*z=10}}}
Still more information is needed to have a better relationship among the three dimensions.


These are the costs of the different rectangular parts altogether:
{{{xy*20+2*xz*12+2*yz*12}}}
{{{20xy+24xz+24yz}}}
and if you substitute for y,
{{{20(2y)y+24(2y)+24yz}}}
{{{40y^2+48y+24yz}}}-----This cost written in terms of y and z.


You can use the earlier found, {{{2y^2*z=10}}} to either substitute for y or for z in the cost function.
First, simplify the "10" equation to {{{z*y^2=5}}}.
{{{z=5/y^2}}}; substitute for z, done here.


cost, {{{40y^2+48y+24y(5/y^2)}}}
{{{highlight_green(c(y)=40y^2+48y+120/y)}}}.


This is probably meant as a calculus problem using the derivative {{{dc/dy}}}.
Form the derivative, and solve for y in {{{dc/dy=0}}} to find and check extremes (for a minimum value for c).