The problem asks to determine the maximum volume of a rectangular box that is inscribed in the ellipsoid:

$3x^2 + y^2 + z^2 = 9$

The objective is to maximize the volume of the box using Lagrange multipliers, given that all 8 vertices of the box lie on the ellipsoid.

Step-by-Step Approach

1. Volume of the Box: The box has edges parallel to the axes, so the volume $V$ is given by:

   $V = 8xyz$ This is because the box is symmetric about the origin, with vertices at $( \pm x, \pm y, \pm z )$.

2. Constraint: The ellipsoid equation is the constraint: $3x^2 + y^2 + z^2 = 9$

3. Using Lagrange Multipliers: The method of Lagrange multipliers involves introducing a new variable (Lagrange multiplier $\lambda$) and setting up the system of equations:

   $\nabla V = \lambda \nabla g$

   where $g(x, y, z) = 3x^2 + y^2 + z^2 - 9$.

   • The gradient of the volume function is: $\nabla V = (8yz, 8xz, 8xy)$

   • The gradient of the constraint function is: $\nabla g = (6x, 2y, 2z)$

   This gives the system of equations: $8yz = \lambda 6x$ $8xz = \lambda 2y$ $8xy = \lambda 2z$

4. Solving the System: We can solve this system by dividing pairs of equations. For example:

   • From the first and second equations: \quad \Rightarrow \quad z = \frac{3x^2}{y}$$
   • From the second and third equations: \quad \Rightarrow \quad z^2 = y^2 \quad \Rightarrow \quad z = y$$ (assuming $$z = -y$$ would lead to a negative volume, so we discard that possibility).

   Using this in the ellipsoid equation:

   \quad \Rightarrow \quad 3x^2 + 2y^2 = 9$$ Substituting $$z = y$$: $$3x^2 + 2y^2 = 9$$ Solving for $$y$$ in terms of $$x$$: $$y^2 = \frac{9 - 3x^2}{2}$$

5. Maximizing the Volume: Substitute the values of $x, y, z$ that satisfy the constraint into the volume expression and maximize it.

Conclusion

After solving the system, the correct volume for the box is found to be 36 cubic units.

Thus, the answer is:

$\boxed{36 \text{ cubic units}}$

Do you have any questions or would you like more details on any step?

Related Questions:

1. How are the Lagrange multipliers used in optimization problems?
2. What is the physical significance of the ellipsoid in this problem?
3. Can we generalize this method to higher dimensions or different ellipsoid shapes?
4. How does the symmetry of the box simplify the calculations?
5. What would happen if the box were not aligned with the coordinate axes?

Tip:

When using Lagrange multipliers, always ensure that both the objective function and the constraint are differentiable. This method only works for continuous and differentiable functions.