This problem involves maximizing the volume of a rectangular box without a lid, given that the total surface area is $A$. Let's break down the solution:

Step 1: Define Variables

Let:

• $l$ be the length of the box,
• $w$ be the width of the box, and
• $h$ be the height of the box.

The goal is to maximize the volume of the box while keeping the surface area constant.

Step 2: Formulate the Volume and Surface Area

The volume $V$ of the box is given by:

$V = l \times w \times h$

The surface area (without a lid) consists of:

• The bottom of the box: $l \times w$,
• Two sides of the box: $2 \times (l \times h)$,
• Two other sides: $2 \times (w \times h)$.

Thus, the total surface area is:

$A = l \times w + 2 \times (l \times h) + 2 \times (w \times h)$

Step 3: Solve the Problem Using Lagrange Multipliers

To maximize the volume while maintaining a constant surface area, we can use the method of Lagrange multipliers, or set up the system to find the critical points. Here's the process:

1. Express $h$ in terms of $l$, $w$, and the surface area $A$: $A = lw + 2lh + 2wh \quad \Rightarrow \quad h = \frac{A - lw}{2(l + w)}$

2. Substitute this expression for $h$ into the volume formula: $V = l \times w \times \frac{A - lw}{2(l + w)}$

3. Maximize $V$ by taking partial derivatives with respect to $l$ and $w$, and solve the resulting system of equations. This process would provide the optimal dimensions of the box.

Step 4: Solution

After solving the system, it turns out that for the given surface area $A$, the optimal dimensions that maximize the volume occur when the base is a square. That is, $l = w$. Substituting $l = w$ into the surface area and volume expressions leads to the following:

• The height $h$ is equal to half the base side length.

Thus, the dimensions are:

• $l = w = \frac{\sqrt{A}}{2}$,
• $h = \frac{\sqrt{A}}{4}$.

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

Here are 5 related questions you might explore:

1. How does the method of Lagrange multipliers apply to other optimization problems with constraints?
2. What would the dimensions be if the box had a lid (changing the surface area constraint)?
3. How does maximizing volume differ when working with cylindrical shapes instead of rectangular ones?
4. What if the material for the bottom of the box was more expensive and had a different cost per unit area?
5. Can the optimization process be applied to non-rectangular shapes like pyramids?

Tip: Always check if symmetry (like $l = w$) simplifies the problem, as in this case, where a square base maximizes the volume.