Answer
Square base of side is $40$ cm and height is $20$ cm.
Work Step by Step
Volume of a box is given by $f(x,y,z)=V=xyz=32,000$ it yields $z=\dfrac{32,000}{xy}$
Since, $f(x,y,z)=xy+2xz+2yz$ yields $f_x=y-\dfrac{64,000}{x^2}$ and
$f_y=x-\dfrac{64,000}{y^2}$
After solving, we get $x=64,000y^{-2}, y=64,000x^{-2}$
or, $x=40,y=40 z=20$
Hence,
Square base of side is $40$ cm and height is $20$ cm.
