Question 889186
Volume, {{{h*x^2=256}}}.


Total one-sided area, {{{x^2+4xh}}}, represents the amount of material making the box.  A is the amount of material.


{{{A=x^2+4xh}}}


Solve the volume equation for h.
{{{h=256/x^2}}} and substitute into A equation.
{{{A=x^2+4x(256/x^2)}}}
{{{highlight(A=x^2+4*256/x)}}}.


The derivative way to continue for the minimum is {{{dA/dx=0}}}.  Not sure if you are studying Calculus so here is a try with just arithmetic.


{{{A=(x*x^2+4*256)/x}}}-----------No.  I am uncomfortable with this method.


DIFFENTIATING A:
{{{dA/dx=2x+(-1)4*256*x^(-2)}}}
{{{dA/dx=2x-4*256/x^2}}}
{{{(2x^3-4*256)/x^2}}}
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{{{dA/dx=0=2x^3-4*256}}}, we only need to focus on the numerator to be zero.
{{{2x^3=4*256}}}
{{{x^3=2*256}}}
{{{highlight(x=8)}}}----------assuming that this extreme will be for the minimum area.
-
{{{h=256/64}}}
{{{highlight(h=4)}}}-----------the height dimension for this expected minimum material area.