If we take the dimensions of the rectangular box as \(x = 3 \, \text{cm}\), \(y = 3 \, \text{cm}\), and \(z = 2 \, \text{cm}\), let's verify if these dimensions satisfy the given constraints and compute the volume of the box.

### Given:
- \(x = 3 \, \text{cm}\)
- \(y = 3 \, \text{cm}\)
- \(z = 2 \, \text{cm}\)

### 1. Perimeter Constraint Check
The total length of the wire required is:

\[
4(x + y + z) = 4(3 + 3 + 2) = 4 \times 8 = 32 \, \text{cm}
\]

This is **less than** the available 40 cm of wire, so it meets the perimeter constraint.

### 2. Surface Area Constraint Check
The surface area of the rectangular box is:

\[
2(xy + yz + zx) = 2(3 \times 3 + 3 \times 2 + 2 \times 3) = 2(9 + 6 + 6) = 2 \times 21 = 42 \, \text{cm}^2
\]

This is **less than** the available 45 cm² of metal sheet, so it also meets the surface area constraint.

### 3. Volume Calculation
Now, let's calculate the volume \(V\) of the box:

\[
V = x \cdot y \cdot z = 3 \times 3 \times 2 = 18 \, \text{cm}^3
\]

### Conclusion
- The dimensions \(x = 3 \, \text{cm}\), \(y = 3 \, \text{cm}\), \(z = 2 \, \text{cm}\) are valid under the given constraints.
- The volume of the box with these dimensions is **18 cm³**, which is less than the maximum volume of approximately 37.04 cm³ that we calculated earlier.

Would you like to explore different dimensions or need further clarification?

### Related Questions:
1. How can we determine the optimal dimensions for different shapes to maximize volume?
2. What other possible combinations of \(x, y, z\) could provide a higher volume than 18 cm³ but still meet the constraints?
3. How would the volume change if the perimeter or surface area constraints were modified?
4. What role does symmetry play in maximizing the volume of geometric shapes?
5. What are some real-world applications of optimization problems like this one?

**Tip:** In optimization problems, testing several feasible solutions helps to understand how constraints impact the objective function.

Explore the dimensions of a rectangular box with x = 3cm, y = 3cm, z = 2cm. Verify if these dimensions satisfy perimeter and surface area constraints. Calculate the volume and learn about geometric optimization. Suitable for Grades 6-8.

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