*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 volume of the open box, explain whether the resultant function is a polynomial or any other.

Let's start by defining the dimensions of the cardboard and the cuts.

1. Let the width of the cardboard be \( w \) cm.
2. The length of the cardboard is \( 3w \) cm (since it is given to be thrice the width).
3. We cut 15 cm squares from each corner and fold up the edges to create an open box.

After cutting the corners and folding, the dimensions of the box will be:
- Width: \( w - 2 \times 15 \) cm = \( w - 30 \) cm
- Length: \( 3w - 2 \times 15 \) cm = \( 3w - 30 \) cm
- Height: 15 cm

Now, the volume \( V \) of the open box can be calculated using the formula for the volume of a rectangular prism, \( V = \text{length} \times \text{width} \times \text{height} \):

\[
V = (3w - 30)(w - 30)(15)
\]

Simplifying this:

\[
V = 15 (3w - 30)(w - 30)
\]

Next, we expand the expression inside the parentheses:

\[
(3w - 30)(w - 30) = 3w \cdot w - 3w \cdot 30 - 30 \cdot w + 30 \cdot 30
\]
\[
= 3w^2 - 90w - 30w + 900
\]
\[
= 3w^2 - 120w + 900
\]

Substitute back into the volume formula:

\[
V = 15(3w^2 - 120w + 900)
\]
\[
= 45w^2 - 1800w + 13500
\]

Therefore, the volume \( V \) of the open box is given by the polynomial:

\[
V = 45w^2 - 1800w + 13500
\]

To determine whether the resultant function is a polynomial or not, let's analyze its properties. A polynomial function is defined as a function that involves only non-negative integer powers of the variable. 

In this case, the volume function:

\[
V = 45w^2 - 1800w + 13500
\]

is a polynomial because it is a sum of terms involving \( w \) raised to non-negative integer powers (2 and 1), and constant terms. The coefficients (45, -1800, and 13500) are real numbers, which further supports it being a polynomial.

### Summary

The volume of the open box formed by cutting 15 cm squares from the corners of a cardboard of width \( w \) and length \( 3w \) is:

\[
V = 45w^2 - 1800w + 13500
\]

This function is a polynomial of degree 2.

Would you like more details or have any other questions?

Here are 5 questions you might want to ask:
1. How do we identify the degree of a polynomial?
2. Can you provide another example of a polynomial function?
3. What are the characteristics of polynomial functions?
4. How can we graph polynomial functions?
5. What is the significance of the coefficients in a polynomial function?

**Tip:** When dealing with polynomial functions, always pay attention to the degree and the leading coefficient, as they provide crucial information about the behavior of the function.

Learn how to calculate the volume of an open box formed by cutting corners from a cardboard with specific dimensions. This problem involves geometry and polynomial functions, with step-by-step solution and polynomial analysis. Suitable for students in Grades 9-12.

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