Question 1065987
The base is x^2 and the height y.  It has a volume=32, so 32=x^2y
The area of the box is x^2 (bottom) + 4xy (the sides).
Therefore, the area is x^2+4x*32/x^2 by substitution. 
Take the derivative and set it equal to 0.
x^2+4*32/x is the function; the derivative is 2x-128/x^2=0
x=64/x^2
x^3=64
x=4 units
y=2 units
area is 16+4*8=48 square units
Can try x=5 and y=1.3.  This gives a volume of 32.25 cubic units but area is 25+26=51 square units.
Can try x=2 and y=8.  This gives a volume of 32 cubic units but area is 4+64=68.
{{{graph(300,300,0,6,-10,40,x^2+(128/x)-32)}}}