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Let's work through the problem in parts:

**(a) Drawing and labeling the diagram:**

You start with a square piece of material, 24 cm on each side. From each corner of the square, you cut out equal squares with side lengths \( x \), and fold up the edges to create an open-top box. When you remove the corners and fold up the sides, the new dimensions of the box will be:
- Length: \( 24 - 2x \)
- Width: \( 24 - 2x \)
- Height: \( x \)

The box is formed by folding up the flaps along the sides created by cutting out the squares. I'll describe this:
1. Draw a square representing the original material with side 24 cm.
2. Draw smaller squares of side length \( x \) cut out from each corner.
3. After folding, label the height of the box as \( x \), the new length as \( 24 - 2x \), and the width as \( 24 - 2x \).

**(b) Verifying the volume formula \( V = 4x(12 - x)^2 \):**

The volume \( V \) of a box is given by the formula for the volume of a rectangular prism: \( V = \text{Length} \times \text{Width} \times \text{Height} \).

- Length = \( 24 - 2x \)
- Width = \( 24 - 2x \)
- Height = \( x \)

So,
\[
V = (24 - 2x)(24 - 2x)(x)
\]

Simplify \( (24 - 2x)(24 - 2x) \):
\[
(24 - 2x)^2 = 576 - 96x + 4x^2
\]

Now, substitute into the volume equation:
\[
V = x(576 - 96x + 4x^2)
\]
\[
V = 576x - 96x^2 + 4x^3
\]

To match the given formula, express this in factored form:
\[
V = 4x(12 - x)^2
\]

This confirms that the formula given is correct.

**(c) Completing the table:**

To observe the behavior of the function \( V(x) = 4x(12 - x)^2 \), we need to calculate the volume for each value of \( x \) given in the table:

| \( x \)   | Volume \( V(x) \) |
|----------|-------------------|
| 3.5      | ?                 |
| 3.9      | ?                 |
| 4        | ?                 |
| 4.1      | ?                 |
| 4.5      | ?                 |
| 5        | ?                 |

I will now calculate these values. Let's compute the volumes for each value of \( x \).Here are the computed volumes for the values of \( x \) from the table:

| \( x \)   | Volume \( V(x) \)  |
|----------|--------------------|
| 3.5      | 1011.5 cm³         |
| 3.9      | 1023.516 cm³       |
| 4        | 1024 cm³           |
| 4.1      | 1023.524 cm³       |
| 4.5      | 1012.5 cm³         |
| 5        | 980 cm³            |

You can observe that the maximum volume occurs when \( x = 4 \), which results in a volume of \( 1024 \, \text{cm}^3 \). This matches the function's maximum value near \( x = 4 \).

Let me know if you have any questions or if you'd like further explanation on any part.

Here are 5 related questions to consider:
1. What happens to the volume as \( x \) approaches the extremes of 0 and 12?
2. Why does the volume decrease after \( x = 4 \)?
3. How would the problem change if the side length of the original square was different?
4. What is the significance of factoring the volume formula in part (b)?
5. How could you find the maximum volume using calculus instead of a graphing utility?

**Tip:** When finding maximum or minimum values, use calculus to check critical points by setting the derivative to zero.

You create an open box from a square piece of material, 24 cm on a side. Equal squares are cut from the corners and the sides are turned up to form the box. (a) Draw and label a diagram that represents the box. (b) Verify the volume formula V = 4x(12 - x)^2. (c) Use a graphing utility to complete the table and find the maximum volume.

Solve a problem involving creating an open-top box by cutting squares from a 24 cm square material. Verify the volume formula V = 4x(12 - x)^2, and calculate the maximum volume using algebra and graphing tools. Ideal for Grades 10-12 geometry and algebra students.

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