Question 703484
This problem keeps coming up and I am always re-drawing the picture.
{{{drawing(300,300,-1,9,-1,9,
rectangle(0,0,8,8),
green(rectangle(2,2,6,6)),
red(rectangle(0.05,0.05,2,2)),red(rectangle(7.95,7.95,6,6)),
red(rectangle(0.05,6,2,7.95)),red(rectangle(7.98,2,6,0.05)),
locate(1,2,x),locate(7,2,x),locate(3.5,2,12-x),
locate(3.8,-0.3,12),arrow(3.5,-0.5,0,-0.5),arrow(4.5,-0.5,8,-0.5)
)}}}
As you cut squares out of each corner (I made them red),
each square measuring {{{x}}} inches to a side,
you get four {{{x}}}-inch wide flaps to fold up around the (green) square bottom.
The side of that green square is {{{12-2x}}} inches long.
The surface area of the green square is {{{(12-2x)^2=144-48x+4x^2}}} square inches.
The flaps/sides will be {{{x}}} inches by {{{12-2x}}} inches,
with a total surface area of {{{4x(12-2x)=48x-8x^2}}} square inches.
The surface area of the box (just four sides and bottom, because we have no top/lid) is
{{{A(x)=4x^2-48x+144+48x-8x^2}}} square inches, so
{{{A(x)=-4x^2+144}}}
The box will be {{{x}}}-inches high,
with a base surface of {{{144-24x+x^2}}} square inches (as calculated above),
and a volume of
{{{Volume =x*(144-48x+4x^2)}}} --> {{{Volume =4x^3-48x^2+144x}}}
 
NOTE:
I know I make stupid arithmetic mistakes, so don't trust my calculations.