Question 913117
The beginning for both of the functions first attends volume.


Bottom is a square shape, so if x is the sidelength for the bottom (as well as top), and if y is height of box, then
{{{10000=yx^2}}}
and from it, {{{y=10000/x^2}}}.


TOTAL SURFACE AREA

Let A be variable for total surface area, to be formed into a function of x.
{{{A=2x^2+4xy}}}
{{{A=2x^2+4x(10000/x^2)}}}
{{{highlight_green(A=2x^2+40000/x)}}} which you might possibly want as a single numerator expression and a single denominator expression.


COST FUNCTION


1 cent per sq in. for walls and top; 2 cents per sq. in. for bottom.
Let C be the cost for the box.


{{{C=(x^2)(2)+(x^2)(1)+4xy*1}}}
{{{2x^2+x^2+4xy}}}
{{{3x^2+4x(10000/x^2)}}}
{{{highlight_green(C=3x^2+40000/x)}}}


The form of the area function and the cost function are the same.  Both are rational functions.  They show the same format.


The question to minimize and then compare the dimensions of these two functions seems outside of and beyond Algebra II; but might be used in College Algebra, but more likely fit in Calculus 1.  Very likely most Algebra II students are allowed, encourage, even TAUGHT TO USE a graphing calculator, so if this is something you know how to use, you can find useful results.  Decide on the x value or pick what seems to correspond to each minimum, and use the value to evaluate y.


...
AREA
{{{graph(350,350,-10,35,-150,5000,2x^2+40000/x)}}}


COST
{{{graph(350,350,-10,35,-150,5000,3x^2+40000/x)}}}