Problem 1: Box from 9"-wide by 12"-long sheet of paper

Step 1: Set up the dimensions of the box

Let the cutout length be $x$. The dimensions of the box after cutting and folding will be:

• Length = $12 - 2x$
• Width = $9 - 2x$
• Height = $x$

The volume $V$ of the box is given by the formula: $V(x) = (12 - 2x)(9 - 2x)(x)$

Step 2: Find the maximum volume using a graphing calculator

We need to find the value of $x$ that maximizes the volume. To do this:

1. Enter the volume function $V(x) = (12 - 2x)(9 - 2x)(x)$ into a graphing calculator.
2. Look for the maximum point of the graph within the valid domain $0 < x < 4.5$ (since the cutout length $x$ must be less than half the width or length).

After graphing, you can find that the maximum volume occurs at approximately $x = 2$ inches.

Step 3: Calculate the maximum volume

Substitute $x = 2$ into the volume formula: $V(2) = (12 - 2(2))(9 - 2(2))(2) = (8)(5)(2) = 80 \, \text{cubic inches}$

Thus, the maximum volume is 80 cubic inches.

Step 4: Correct cutout length for maximum volume

The cutout length that produces the maximum volume is 2 inches.

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Problem 2: Box from 5"-wide by 7"-long sheet of paper

Step 1: Set up the dimensions of the box

Let the cutout length be $x$. The dimensions of the box after cutting and folding will be:

• Length = $7 - 2x$
• Width = $5 - 2x$
• Height = $x$

The volume $V$ of the box is given by the formula: $V(x) = (7 - 2x)(5 - 2x)(x)$

Step 2: Find the maximum volume using a graphing calculator

Enter the volume function $V(x) = (7 - 2x)(5 - 2x)(x)$ into the graphing calculator, and look for the maximum point of the graph within the domain $0 < x < 2.5$.

After graphing, you will find that the maximum volume occurs at approximately $x = 1$ inch.

Step 3: Calculate the maximum volume

Substitute $x = 1$ into the volume formula: $V(1) = (7 - 2(1))(5 - 2(1))(1) = (5)(3)(1) = 15 \, \text{cubic inches}$

Thus, the maximum volume is 15 cubic inches.

Step 4: Correct cutout length for maximum volume

The cutout length that produces the maximum volume is 1 inch.

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Summary:

1. Maximum volume for 9" by 12" sheet: 80 cubic inches
2. Cutout length for 9" by 12" sheet: 2 inches
3. Maximum volume for 5" by 7" sheet: 15 cubic inches
4. Cutout length for 5" by 7" sheet: 1 inch

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Would you like me to provide further details on how to input this into a graphing calculator, or do you have any other questions?

Here are 5 related questions:

1. How does the shape of the volume function change as the dimensions of the paper change?
2. Why must the cutout length $x$ be less than half the width or length of the paper?
3. How can the derivative of the volume function be used to find the critical points?
4. How does the change in paper dimensions affect the maximum volume and cutout size?
5. Can this method be extended to non-rectangular shapes?

Tip: Maximizing volume often involves finding the critical points of a function, which can be done either graphically or by using calculus techniques.