Answer
$\dfrac{c}{12}$
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 $g=4x+4y+4z=c$
Now, $\lt yz,xz,xy \gt =\lambda \lt 4,4,4 \gt$
Simplify to get the values of x,y and z. $y=x,z=x$
$4x+4y+4z=c$
$4x+4y+4z=c$
$ x=\dfrac{c}{12}$
Thus, Box is a cube with edge length: $c=$ $\dfrac{c}{12}$
