Answer
Dimensions of a box: $x=y=z=\sqrt{\dfrac{32}{3}}$
Work Step by Step
Given: Surface area =64 $cm^2$
Need to apply Lagrange Multipliers Method to determine the maximum volume of a rectangular box.
we have $\nabla f=\lambda \nabla g$
The volume of a box is $V=xyz$
Surface area, $S=2xy+2yz+2zx$
From the given question, $S=64 cm^2$
Now, $\nabla V=\lambda \nabla S$
$\lt yz,xz,xy \gt =\lambda \lt 2(y+z), 2(x+z),2(x+y) \gt$
and $xyz=2 \lambda (xz+yz)$
Simplify to get the values of x,y and z.
We have $x=y=z=\sqrt{\dfrac{32}{3}}$
Thus,
The Dimensions of a box: $x=y=z=\sqrt{\dfrac{32}{3}}$