Maximum volume using Lagrange multipliers

Question

I need to determine the maximum volume of a rectangular box with these side conditions: its surface has 2m² and the sum of all its edges = 8 m of length.

How do I do that ?

Answer 1

Hints: The volume of rectangular box is: $\space V=V(x,y,z) = x \space y \space z$.

Side conditions:

$2xy + 2xz + 2yz = 2 \iff \varphi (x,y,z) = xy + xz + yz - 1 =0$;

$x+y+z=8 \iff \psi (x, y, z) = x+y+z -8 =0$.

Now compose the Lagrange function: $\Phi(x, y, z, \lambda, \mu) =V(x,y,z)+\lambda \varphi(x,y,z)+\mu \psi(x,y,z)$, the extreme value of $\space \Phi \space$ will be an extreme value also for $\space V$.