https://file.aiquickdraw.com/mathbot/file/17265517713479ri9jgfb.jpg

To solve the problem and complete the table, we will break it down step by step:

### Given:
- A sheet of paper has dimensions 7 inches (width) and 9 inches (length).
- Squares with side length \( x \) are cut from each corner.
- The dimensions of the box after folding are:
  - Width \( w = 7 - 2x \)
  - Length \( l = 9 - 2x \)
  - Height \( h = x \)

### Formula for Volume:
The volume \( V \) of the box is calculated as:
\[
V = w \times l \times h = (7 - 2x)(9 - 2x)(x)
\]

### Steps to Complete the Table:
1. Use the given values of \( x \) (0, 0.5, 3.5) to calculate \( w \), \( l \), \( h \), and \( V \).
2. If the calculations are invalid (e.g., the dimensions are negative), mark the entry as "DNE".

Let's compute the values for each row.

#### For \( x = 0 \):
- \( w = 7 - 2(0) = 7 \)
- \( l = 9 - 2(0) = 9 \)
- \( h = 0 \)
- \( V = 7 \times 9 \times 0 = 0 \)

This row is already filled.

#### For \( x = 0.5 \):
- \( w = 7 - 2(0.5) = 7 - 1 = 6 \)
- \( l = 9 - 2(0.5) = 9 - 1 = 8 \)
- \( h = 0.5 \)
- \( V = 6 \times 8 \times 0.5 = 24 \)

So for \( x = 0.5 \), \( w = 6 \), \( l = 8 \), \( h = 0.5 \), and \( V = 24 \).

#### For \( x = 3.5 \):
- \( w = 7 - 2(3.5) = 7 - 7 = 0 \)
- \( l = 9 - 2(3.5) = 9 - 7 = 2 \)
- \( h = 3.5 \)
- \( V = 0 \times 2 \times 3.5 = 0 \)

So for \( x = 3.5 \), \( w = 0 \), \( l = 2 \), \( h = 3.5 \), and \( V = 0 \).

### Final Table:
| \( x \) | \( w \) | \( l \) | \( h \) | \( V \)  |
|:-------:|:-------:|:-------:|:-------:|:--------:|
| 0       | 7       | 9       | 0       | 0        |
| 0.5     | 6       | 8       | 0.5     | 24       |
| 3.5     | 0       | 2       | 3.5     | 0        |

Would you like further explanation or details on these calculations? Here are five related questions for further exploration:

1. How does the volume of the box change as \( x \) increases?
2. What is the maximum possible value of \( x \) before the box cannot be formed?
3. How would the table change if the original dimensions of the paper were different?
4. Can you derive a formula for the maximum volume of the box?
5. What happens if you cut rectangles instead of squares from the corners?

**Tip:** When optimizing volume problems like this, it's useful to differentiate the volume function to find the maximum!

A box is formed by cutting squares from the four corners of a 7"-wide by 9"-long sheet of paper and folding up the sides. Complete the table of values based on the relationship between the side length of the square cutout and the resulting volume of the box.

Learn how to calculate the volume of a box formed by cutting squares from the corners of a 7"x9" sheet of paper. Use the formula V = (7 - 2x)(9 - 2x)(x) to complete the table of values and explore how the side length of the square cutout affects the box's dimensions and volume.

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