Question 1055889
Bottom of x by x, and height of y.
S, the surface area for the five sides.


{{{system(S=x^2+4xy,x^2*y=62.5)}}}


{{{y=62.5/x^2}}}


Substitute into S.
{{{S=x^2+4x(62.5)/x^2}}}

{{{S=x^2+4*62.5/x}}}


{{{S=x^2+4*62.5*x^(-1)}}}


{{{dS/(dx)=2x+4*62.5(-1)x^(-2)}}}


{{{dS/dx=2x-250/x^2}}}


{{{dS/dx=(2x^3-250)/x^2}}}


{{{highlight_green((2x^3-250)/x^2=0)}}}


{{{2x^3-250=0}}}

{{{2x^3=250}}}

{{{x^3=125}}}

{{{highlight(x=5)}}}
Bottom square dimensions each are 5 units.
-
{{{y=62.5/5^2}}}
{{{highlight(y=2.5)}}}
The height is  2.5 units, to make box have its maximum surface area.