To solve this problem, let’s analyze the information provided and derive the function for the volume of the rectangular prism.

Problem Breakdown

1. Dimensions of the Paper: The piece of construction paper has an initial size of $11$ inches by $8$ inches.
2. Cutting Square Corners: Allie will cut squares of side length $x$ inches from each corner of the paper. After cutting these squares, she will fold the paper to form an open rectangular prism.
3. Adjusted Dimensions:
   • The length of the paper after cutting the squares will be reduced by $2x$, so the new length becomes $11 - 2x$.
   • The width of the paper after cutting the squares will be reduced by $2x$, so the new width becomes $8 - 2x$.
   • The height of the box will be $x$, since this is the height formed when the sides are folded up.

Volume Function

The volume $V$ of a rectangular prism is given by: $V = \text{length} \times \text{width} \times \text{height}$

Using the dimensions above:

• Length = $11 - 2x$
• Width = $8 - 2x$
• Height = $x$

Thus, the volume function $f(x)$ is: $f(x) = (11 - 2x)(8 - 2x)(x)$

Simplification

Now, let’s simplify $f(x)$:

1. Expand $(11 - 2x)(8 - 2x)$: = 88 - 38x + 4x^2$$
2. Multiply this result by $x$: = 4x^3 - 38x^2 + 88x$$

Final Answer

The function that models the volume of the rectangular prism is: $f(x) = 4x^3 - 38x^2 + 88x$

Would you like further explanation on any of these steps, or assistance with other parts of the problem?

Here are five related questions to expand your understanding:

1. What are the domain restrictions for the function $f(x)$ in this context?
2. How would you find the maximum volume of this rectangular prism?
3. What happens to the volume as $x$ approaches the limits of the domain?
4. How does changing the initial dimensions of the paper affect the volume function?
5. Can we use calculus to find the value of $x$ that maximizes the volume?

Tip: Always verify the domain for variables in a geometric problem, as physical constraints limit possible values.