#### Answer

Dimensions of a box are: $x=\sqrt[3] {\dfrac{2}{5}}V$,$y=\sqrt[3] {\dfrac{2}{5}}V$ and $z=\dfrac{5}{2}\sqrt[3] {\dfrac{2}{5}}V$

#### Work Step by Step

Need to apply Lagrange Multipliers Method to determine the dimensions of a rectangular box of maximum volume.
we have $\nabla f=\lambda \nabla g$
The volume of a box is $f=V=xyz$ and
From the given question let us consider $\nabla f=\lt 10x+4z,4x \gt$ and $\lambda \nabla g=\lambda \lt 2xz, x^2 \gt$
Using the constraint condition we get,
$10x+4z=\lambda 2xz$
$4x=\lambda x^2;$
$V=x^2z$
Simplify to get the value of $z$.
we get $z=\dfrac{5x}{2}$
and $V=x^2z$
or, $x=\sqrt[3] {\dfrac{2}{5}}V$
Dimensions of a box are: $x=\sqrt[3] {\dfrac{2}{5}}V$,$y=\sqrt[3] {\dfrac{2}{5}}V$ and $z=\dfrac{5}{2}\sqrt[3] {\dfrac{2}{5}}V$