Answer
$x=y=z=\sqrt{\dfrac{32}{3}}$
or, Dimensions of a box are: $\sqrt{\dfrac{32}{3}} cm \times \sqrt{\dfrac{32}{3}} cm \times \sqrt{\dfrac{32}{3}}$ cm.
Work Step by Step
Volume of a box is given by $V=xyz$
Surface area, $S=2xy+2yz+2zx=64 cm^2$
Use Lagrange Multipliers Method:
$\nabla f=\lambda \nabla g$
This yields $\lt yz,xz,xy \gt =\lambda \lt 2(y+z), 2(x+z),2(x+y) \gt$
and $xyz=2 \lambda (xz+yz)$
After solving we get, $x=y=z$
Since, $S=2xy+2yz+2zx=64 cm^2$
Thus, $x=y=z=\sqrt{\dfrac{32}{3}}$
or, Dimensions of a box are: $\sqrt{\dfrac{32}{3}} cm \times \sqrt{\dfrac{32}{3}} cm \times \sqrt{\dfrac{32}{3}}$ cm.
