Answer
Dimensions: $x=\sqrt[3] {\dfrac{2}{5}}V$,$y=\sqrt[3] {\dfrac{2}{5}}V$ and $z=\dfrac{5}{2}\sqrt[3] {\dfrac{2}{5}}V$
or,
Dimensions: $\sqrt[3] {\dfrac{2}{5}}V \times \sqrt[3] {\dfrac{2}{5}}V \times \dfrac{5}{2}\sqrt[3] {\dfrac{2}{5}}V$
Work Step by Step
Use Lagrange Multipliers Method:
$\nabla f=\lambda \nabla g$
Volume of a box is given by $V=xyz$
This yields $\nabla f=\lt 10x+4z,4x \gt$ and $\lambda \nabla g=\lambda \lt 2xz, x^2 \gt$
Using the constraint condition we get, $10x+4z=\lambda 2xz, 4x=\lambda x^2,V=x^2z$
After solving, we get $z=\dfrac{5x}{2}$
and $V=x^2z$ yields $x=\sqrt[3] {\dfrac{2}{5}}V$
Hence,
Dimensions: $x=\sqrt[3] {\dfrac{2}{5}}V$,$y=\sqrt[3] {\dfrac{2}{5}}V$ and $z=\dfrac{5}{2}\sqrt[3] {\dfrac{2}{5}}V$
or,
Dimensions: $\sqrt[3] {\dfrac{2}{5}}V \times \sqrt[3] {\dfrac{2}{5}}V \times \dfrac{5}{2}\sqrt[3] {\dfrac{2}{5}}V$