*An online courier service is ready to transport a diverse range of items to ensure efficient delivery. The agency requires boxes of various dimensions. Let's now focus on creating open boxes that have fixed height for storing these items. Take a cardboard of length thrice of the width and cut the edge of all 4 corners with 15cms, then fold the cardboard to get an open box*.   Find the possible domain for the volume function

To find the possible domain for the volume function of the open box, we need to consider the physical constraints of the cardboard and the resulting box.

### Constraints

1. **Cardboard dimensions:**
   - The width of the cardboard is \( w \) cm.
   - The length of the cardboard is \( 3w \) cm.

2. **Corner cuts:**
   - We cut squares with a side length of 15 cm from each corner.

3. **Resulting box dimensions:**
   - After cutting and folding, the dimensions of the box will be:
     - Width: \( w - 30 \) cm (since 15 cm is cut from each side)
     - Length: \( 3w - 30 \) cm
     - Height: 15 cm

### Domain Considerations

To ensure the open box is physically possible:

1. The width after cutting should be positive:
   \[
   w - 30 > 0 \implies w > 30
   \]

2. The length after cutting should be positive:
   \[
   3w - 30 > 0 \implies 3w > 30 \implies w > 10
   \]

Since \( w \) must satisfy both conditions, the more restrictive condition is \( w > 30 \).

### Summary

The possible domain for the volume function \( V(w) = 45w^2 - 1800w + 13500 \) is:

\[
w > 30
\]

This means that \( w \) must be greater than 30 cm for the open box to be feasible.

Would you like more details or have any other questions?

Here are 5 questions you might want to ask:
1. How do we graph the volume function over its domain?
2. Can you explain why the domain is restricted to \( w > 30 \)?
3. What happens to the volume function as \( w \) approaches 30?
4. How do we find the maximum volume of the box within the domain?
5. Can you show how to find the critical points of the volume function?

**Tip:** Always check the physical constraints and dimensions when determining the domain of functions related to real-world applications.

Explore the domain of the volume function for open boxes created from cardboard with specific dimensions. Understand the constraints and calculations involved in determining the feasible domain. Suitable for advanced high school students and those interested in geometry and algebra.

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