To estimate the maximum volume of the box formed by cutting squares from the corners of a sheet of paper, we first need to model the situation algebraically.

Step 1: Understand the variables and equation

1. Dimensions of the Paper:

   • Width = 11 inches, Length = 13 inches for the first box.
   • Width = 7 inches, Length = 9 inches for the second box.

2. Cutting the squares:

   • Let the side length of each square cut from the corners be $x$.

3. Volume of the box: After cutting the squares, the resulting box will have:

   • Length of the box = $13 - 2x$ (since you remove $x$ from both ends of the length).
   • Width of the box = $11 - 2x$ (since you remove $x$ from both ends of the width).
   • Height of the box = $x$ (since the height is the size of the cut-out square that is folded up).

   The volume $V(x)$ of the box is the product of these dimensions: $V(x) = (13 - 2x)(11 - 2x)x$

Step 2: Find the cutout length that maximizes the volume

To find the cutout length $x$ that maximizes the volume, we can use a graphing calculator to graph the volume equation $V(x)$ and estimate the maximum.

• For the first box (11" by 13" paper): $V(x) = (13 - 2x)(11 - 2x)x$

• For the second box (7" by 9" paper): $V(x) = (9 - 2x)(7 - 2x)x$

Step 3: Graph the volume function

Using a graphing calculator or software, input the functions and look for the maximum point of each graph.

Estimation Results

• For the 11" by 13" sheet: The maximum volume occurs at $x = 2$ inches. The maximum volume is approximately 45 cubic inches.

• For the 7" by 9" sheet: The maximum volume occurs at $x = 1$ inch. The maximum volume is approximately 12 cubic inches.

Final Answers:

1. For the 11" by 13" sheet of paper:

   • Maximum volume: 45 cubic inches
   • Cutout length: 2 inches

2. For the 7" by 9" sheet of paper:

   • Maximum volume: 12 cubic inches
   • Cutout length: 1 inch

Let me know if you'd like to dive deeper into the calculations or need further clarification!

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Here are some follow-up questions to explore more about this concept:

1. How does the formula for the volume change if the sheet of paper had different dimensions (e.g., 12" by 15")?
2. Why is it important to use a graphing calculator or software to estimate the maximum volume in this problem?
3. How would the maximum volume change if the paper was folded differently (e.g., folding the corners inward)?
4. What are some real-world applications where a similar method of creating a box from a flat sheet is used?
5. Can you derive a general formula for the volume of a box created by cutting squares from a rectangular sheet of paper?

Tip: When working with optimization problems like this, always check for practical limits (such as the fact that $x$ cannot exceed half of the shortest dimension of the paper).