Question 1001621
x, side length of top and bottom;
y, height of the box;
Total area, {{{2x^2+4xy}}};
Volume, {{{x^2*y=25}}}.


The AREA is the function you want to study for its cost.
{{{C(xANDy)=1*2x^2+5*4xy}}} and {{{x^2*y=25}}}.


You want to make C(x)  OR  C(y), and you use the volume relationship to do this.  You can find the minimum cost from there.  Best might be to start with  {{{y=25/x^2}}}.  Substitute,....



{{{C(x)=2x^2+20x(25/x^2)}}}
{{{highlight(C(x)=2x^2+500/x)}}}-----Can you look for the minimum cost using this cost as a function of x?