Let's analyze and solve the problem step by step.

1. Understanding the Problem

We are tasked with minimizing the cost of materials for a box that has:

• An open top (so no top surface),
• A square base of dimensions $(L - 2x) \times (L - 2x)$, where $x$ is the height of the box,
• A fixed volume of 1,096 cubic centimeters.

The cost per square centimeter is:

• $\text{Php2 for the base}$,
• $\text{Php2 for the sides}$.

We need to determine the height $x$ that minimizes the total cost of materials.

2. Defining Variables

• Let $L$ be the length of the original square cardboard.
• Let $x$ be the height of the box (cut-out size).

After folding, the dimensions are:

• Base: $(L - 2x) \times (L - 2x)$,
• Height: $x$.

The volume of the box is: $\text{Volume} = \text{Base Area} \times \text{Height} = (L - 2x)^2 \cdot x.$

We know the volume is $1,096$ cm³: $(L - 2x)^2 \cdot x = 1,096.$

3. Cost Function

The total cost of materials includes:

1. Base cost: $\text{Base Cost} = 2 \cdot (\text{Base Area}) = 2 \cdot (L - 2x)^2.$
2. Side cost (4 rectangular sides):
   • Each side has an area of $(L - 2x) \cdot x$,
   • Total side area: $4 \cdot (L - 2x) \cdot x$,
   • Cost of sides: $2 \cdot \text{Total Side Area}$.

Thus: $\text{Side Cost} = 2 \cdot 4 \cdot (L - 2x) \cdot x = 8x(L - 2x).$

The total cost function $C$ is: $C(x) = 2(L - 2x)^2 + 8x(L - 2x).$

4. Solving for $L$ in Terms of $x$

From the volume constraint: $(L - 2x)^2 \cdot x = 1,096.$

Solve for $L$ in terms of $x$: $L - 2x = \sqrt{\frac{1,096}{x}}.$

Thus: $L = 2x + \sqrt{\frac{1,096}{x}}.$

5. Substituting into the Cost Function

Substitute $L$ into the cost function $C(x)$: $C(x) = 2\left(\sqrt{\frac{1,096}{x}}\right)^2 + 8x\left(\sqrt{\frac{1,096}{x}}\right) - 2(2x)^2 + ...$