The problem involves constructing a box from a rectangular piece of cardboard, which measures 19 inches in length and 11 inches in width. Squares with side lengths of $x$ inches are cut from each corner, and then the sides are folded up to create the box.

We are asked to write expressions for the dimensions of the box (length, width, and height) using $x$ as the variable.

Length of the Box:

• The original length of the cardboard is 19 inches.
• After cutting out squares with side length $x$ from each end, the length will be reduced by $2x$ (since squares are cut from both sides).
• Expression for the length: $\text{Length} = 19 - 2x$

Width of the Box:

• The original width of the cardboard is 11 inches.
• Similarly, after cutting out squares with side length $x$, the width is reduced by $2x$.
• Expression for the width: $\text{Width} = 11 - 2x$

Height of the Box:

• The height of the box will be the side length of the square that was cut out, which is $x$.
• Expression for the height: $\text{Height} = x$

Final Expressions:

• Length: $19 - 2x$
• Width: $11 - 2x$
• Height: $x$

Would you like more details or have any questions?

Here are 5 related questions to explore:

1. How would you calculate the volume of the box using the expressions for length, width, and height?
2. What happens to the box's volume if the value of $x$ becomes very small?
3. What value of $x$ will maximize the volume of the box?
4. How would you find the surface area of the constructed box?
5. How does changing the dimensions of the cardboard affect the size of the box?

Tip: When writing expressions for dimensions, always subtract twice the cut length (2x) since cuts occur on both sides of the cardboard.