We are tasked with finding the dimensions of the rectangular box of maximum volume that can be inscribed inside the sphere described by the equation $x^2 + y^2 + z^2 = 4$. The volume of the box is given by the product of its dimensions, i.e., $V = x \cdot y \cdot z$. To maximize this volume subject to the constraint $x^2 + y^2 + z^2 = 4$, we can use the method of Lagrange multipliers.

Step 1: Set up the Lagrange multiplier problem

Let the objective function be the volume of the box: $f(x, y, z) = x \cdot y \cdot z$ And the constraint is given by: $g(x, y, z) = x^2 + y^2 + z^2 - 4 = 0$

Step 2: Compute the gradients

The method of Lagrange multipliers tells us to solve the system: $\nabla f = \lambda \nabla g$ where $\nabla f$ and $\nabla g$ are the gradients of $f$ and $g$, respectively, and $\lambda$ is the Lagrange multiplier.

First, compute the gradients: $\nabla f = \left( \frac{\partial}{\partial x}(x \cdot y \cdot z), \frac{\partial}{\partial y}(x \cdot y \cdot z), \frac{\partial}{\partial z}(x \cdot y \cdot z) \right) = (y \cdot z, x \cdot z, x \cdot y)$ $\nabla g = \left( \frac{\partial}{\partial x}(x^2 + y^2 + z^2 - 4), \frac{\partial}{\partial y}(x^2 + y^2 + z^2 - 4), \frac{\partial}{\partial z}(x^2 + y^2 + z^2 - 4) \right) = (2x, 2y, 2z)$

Step 3: Set up the system of equations

We now have the system: $(yz, xz, xy) = \lambda (2x, 2y, 2z)$ This results in the following three equations:

1. $yz = 2\lambda x$
2. $xz = 2\lambda y$
3. $xy = 2\lambda z$

Additionally, the constraint is: $x^2 + y^2 + z^2 = 4$

Step 4: Solve the system of equations

From the first equation $yz = 2\lambda x$, the second equation $xz = 2\lambda y$, and the third equation $xy = 2\lambda z$, we can solve for $\lambda$ and look for relationships between $x, y, z$.

By dividing the first equation by the second: $\frac{yz}{xz} = \frac{2\lambda x}{2\lambda y} \quad \Rightarrow \quad \frac{y}{x} = \frac{x}{y}$ This simplifies to $y = x$.

Next, divide the second equation by the third: $\frac{xz}{xy} = \frac{2\lambda y}{2\lambda z} \quad \Rightarrow \quad \frac{z}{y} = \frac{y}{z}$ This simplifies to $z = y$.

Thus, we have $x = y = z$.

Step 5: Substitute into the constraint

Substitute $x = y = z$ into the constraint equation $x^2 + y^2 + z^2 = 4$: $3x^2 = 4 \quad \Rightarrow \quad x^2 = \frac{4}{3} \quad \Rightarrow \quad x = \frac{2}{\sqrt{3}}$

Thus, $x = y = z = \frac{2}{\sqrt{3}}$.

Step 6: Calculate the maximum volume

The volume of the box is: $V = x \cdot y \cdot z = \left( \frac{2}{\sqrt{3}} \right)^3 = \frac{8}{3\sqrt{3}}$

Thus, the dimensions of the box that maximize the volume are $x = y = z = \frac{2}{\sqrt{3}}$, and the maximum volume is $\frac{8}{3\sqrt{3}}$.

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Would you like more details or have any questions about this solution?

Here are some follow-up questions for you:

1. How would you solve the same problem using a different method, such as by substitution or directly differentiating the volume with respect to the constraint?
2. What is the geometric interpretation of the Lagrange multiplier in this problem?
3. How can you apply the method of Lagrange multipliers to optimize other geometric shapes inside constraints?
4. How do the results change if the constraint were a different shape, such as an ellipsoid?
5. Can you extend this method to problems involving multiple constraints?

Tip: When using the method of Lagrange multipliers, always ensure that the gradient of the constraint is not the zero vector, as it would not provide useful information.