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This problem involves finding the volume of a box formed by cutting squares from the corners of a 12 cm by 12 cm cardboard and folding up the sides.

### a) Express the volume of the box as a function of the side \( x \) of the square cut-out.
When you cut squares of side length \( x \) from each corner of the 12 cm by 12 cm square cardboard, the resulting box will have:
- **Length**: \( 12 - 2x \)
- **Width**: \( 12 - 2x \)
- **Height**: \( x \)

The volume \( V(x) \) of the box is the product of the length, width, and height:
\[
V(x) = (12 - 2x)(12 - 2x)(x) = x(12 - 2x)^2
\]
This is the function for the volume of the box in terms of \( x \).

### b) Find the domain of the function.
The domain of \( V(x) \) is determined by the physical constraints of the problem:
- \( x \) must be greater than 0 (because it represents the side length of the square cut-out).
- \( 12 - 2x > 0 \) (because the remaining length and width of the box must be positive).

Solving \( 12 - 2x > 0 \):
\[
12 > 2x \quad \Rightarrow \quad x < 6
\]
Thus, \( x \) must satisfy:
\[
0 < x < 6
\]
So, the domain of the function is \( (0, 6) \).

### c) Determine the dimensions that yield the maximum volume using the graph.
From the graph, we can see that the volume reaches a maximum when \( x = 2 \) cm, with a corresponding volume of 128 cm³. Therefore, when \( x = 2 \) cm:
- **Length**: \( 12 - 2(2) = 8 \) cm
- **Width**: \( 12 - 2(2) = 8 \) cm
- **Height**: \( 2 \) cm

Thus, the dimensions of the box that yield the maximum volume are 8 cm by 8 cm by 2 cm.

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Would you like further details on any part of the explanation?

Here are 5 related questions:
1. How would the volume formula change if the cardboard was a rectangle, not a square?
2. How can we find the maximum volume using calculus instead of a graph?
3. What happens to the volume function if \( x = 0 \) or \( x = 6 \)?
4. How would the domain change if the cardboard had a different initial size?
5. What would be the dimensions of the box if the side cut-out were \( x = 3 \)?

**Tip**: For problems like this, the volume of a shape often requires careful attention to the dimensions that are affected by the cut. Always write the expressions for the remaining length, width, and height clearly.

From a 12-cm by 12-cm piece of cardboard, square corners are cut out so that the sides can be folded up to make a box. Express the volume of the box as a function of the side x in centimeters of the cut-out square. Find the domain of the function. Using the graph, determine the dimensions that yield the maximum volume.

Learn how to express the volume of a box as a function of the side length x of a square cut-out. The problem covers the volume formula, domain restrictions, and determining the dimensions that yield maximum volume using a graph. Suitable for students in Grades 9-12.

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