On a rectangular piece of cardboard with perimeter inches, three parallel and equally spaced creases are made (see Figure 1). The cardboard is then folded along the creases to make a rectangular box with open ends (see Figure 2). Letting represent the distance (in inches) between the creases, use the ALEKS graphing calculator to find the value of that maximizes the volume enclosed by this box. Then give the maximum volume. Round your responses to two decimal places.

Lots of dimensions missing.

To find the value of that maximizes the volume enclosed by the box, we need to first understand the relationship between and the volume of the box.

Let's start by determining the dimensions of the rectangular box after it is folded along the creases.

Since there are three equally spaced creases on a rectangular piece of cardboard, there will be four equal segments. The length of each segment can be calculated as follows:

Length of each segment = (total perimeter) / 4
= (perimeter) / 4
= inches / 4
= inches

When the cardboard is folded along the creases, the resulting box will have a height and a width.

The width of the box will be two times the distance between the creases, which is inches. Therefore, the width of the box is given by:

Width of the box = 2 *
= 2 * inches
= inches

The height of the box will be equal to the length of the segments, which is inches.

Now, we can express the volume of the rectangular box in terms of :

Volume of the box = (width) * (height) * (length)
= * *

To find the value of that maximizes the volume, we can take the derivative of the volume function with respect to and set it equal to zero. We can then solve for to find the critical point.

Let's use the ALEKS graphing calculator to find this critical point and the corresponding maximum volume.

1. Open the ALEKS graphing calculator.
2. Enter the function for the volume of the box: V() = * *
3. Find the derivative of the volume function with respect to : V'() = * ^2
4. Set the derivative equal to zero and solve for : * ^2 = 0
5. Solve for by taking the square root of both sides: = 0
6. Calculate the maximum volume by substituting the value of into the volume function: V() = * *

The ALEKS graphing calculator will provide the value of that maximizes the volume and the corresponding maximum volume rounded to two decimal places.

It's important to note that this explanation assumes you have access to the ALEKS graphing calculator. If you don't have access to it, you can use other graphing calculators or software, or perform the calculations manually to find the critical point and the maximum volume.