Question 1137049
<pre>If the box is flattened it will look like this:
A square base whose dimensions are x inches by x inches, and
4 equal flaps each of whose dimensions are x inches by h inches:

{{{drawing(400,400,-3.5,9.5,-3.5,9.5,
locate(3,0,x), locate(6.1,3,x), locate(7.5,0,h),
line(-3,0,9,0),line(-3,0,-3,6),line(-3,6,9,6),line(9,0,9,6),
line(0,-3,0,9),line(0,-3,6,-3),line(0,9,6,9),line(6,9,6,-3) )}}}

The formula for the volume is:

{{{V = l*w*h}}} with V = 108 in³, l = length = x, and width = w = x, and height = x

{{{108 = x*x*h}}}
{{{108 = x^2h}}}
{{{108/x^2=h}}}

The formula for the surface area, the square base whose dimensions are x
inches by x inches, plus 4 equal flaps each of whose dimensions are x inches
by h inches is:

{{{S = x^2+4hx}}} and we substitute {{{108/x^2}}} for h

{{{S = x^2+4(108/x^2)x}}}
{{{S = x^2+432/x}}}
{{{S = x^2+432x^(-1)}}}
{{{(dS)/(dx)=2x-432x^(-2)}}}
Set that = 0
{{{2x-432x^(-2)=0}}}
Divide through by 2:
{{{x-216x^(-2)=0}}}
Rewrite x<sup>-2</sup> as x² in the denominator
{{{x-216/x^2=0}}}
Multiply through by LCD = x²
{{{x^3-216=0}}}
{{{x^3=216}}}
{{{x=6}}} in

Then use:

{{{108^""/x^2=h}}}
{{{108^""/x^2=h}}} 
{{{108^""/6^2=h}}}
{{{108/36=h}}}
{{{3=h}}}

Edwin</pre>