Answer
square base of side is $40$ cm and Height is $20$ cm.
Work Step by Step
Let us consider $V$ be the volume of a box.
and $V=xyz$
From the given question: $xyz=32,000$
or, $z=\dfrac{32,000}{xy}$
Consider $f(x,y,z)=xy+2xz+2yz$
and $f_x=y-\dfrac{64,000}{x^2};f_y=x-\dfrac{64,000}{y^2}$
Simplify to get the values of $x$ and $y$
we have $x=64,000y^{-2}, y=64,000x^{-2}$
Solve for $x,y$, we get $x=40,y=40 z=20$
Thus, the square base of side is $40$ cm and Height is $20$ cm.
