The absolute maximum volume of the open box is 108 cubic inches, achieved with a depth of 1 inch and squares cut from each corner of size 1 inch. The absolute minimum volume is 120 cubic inches, achieved with a depth of 3 inches and squares cut from each corner of size 3 inches.
We are given that the cardboard is 11 inches wide and 14 inches high. We need to cut out squares of equal size from the four corners to create an open box. The depth of the box can vary between 1 inch and 3 inches. We want to find the absolute maximum and minimum volumes of the box within this depth range.
1. Define Variables:
Let's define the following variables:
* `width`: Width of the cardboard (11 inches)
* `height`: Height of the cardboard (14 inches)
* `depth`: Depth of the box (variable between 1 and 3 inches)
* `x`: Size of the square cut from each corner
2. Volume Formula:
The volume of the box can be calculated using the following formula:
Volume = depth * (width - 2x) * (height - 2x)
3. **Constraints:**
We are given two constraints:
* The depth (`depth`) must be between 1 inch and 3 inches (inclusive).
* All four squares cut from the corners must be of equal size (`x`).
4. Objective:
Our objective is to find the absolute maximum and minimum volumes of the box within the given depth range.
5. Finding the Maximum Volume:
To maximize the volume, we need to minimize the size of the squares cut from the corners (`x`). Since the depth (`depth`) can be at most 3 inches, the smallest possible value of `x` is 1 inch (which corresponds to a depth of 3 inches).
Therefore, the maximum volume can be achieved when:
* `depth = 1 inch`
* `x = 1 inch`
Substituting these values into the volume formula, we get the maximum volume:
Maximum Volume = 1 * (11 - 2 * 1) * (14 - 2 * 1) = 1 * 9 * 12 = 108 cubic inches
6. Finding the Minimum Volume:
To minimize the volume, we need to maximize the size of the squares cut from the corners (`x`). Since the depth (`depth`) can be at least 1 inch, the largest possible value of `x` is 3 inches (which corresponds to a depth of 1 inch).
Therefore, the minimum volume can be achieved when:
* `depth = 3 inches`
* `x = 3 inches`
Substituting these values into the volume formula, we get the minimum volume:
Minimum Volume = 3 * (11 - 2 * 3) * (14 - 2 * 3) = 3 * 5 * 8 = 120 cubic inches
7. Conclusion:
The absolute maximum volume of the box is 108 cubic inches, which occurs when the depth is 1 inch and the squares cut from the corners are 1 inch by 1 inch. The absolute minimum volume of the box is 120 cubic inches, which occurs when the depth is 3 inches and the squares cut from the corners are 3 inches by 3 inches.
Summary:
| Feature | Value |
|---|---|
| Maximum Volume | 108 cubic inches |
| Minimum Volume | 120 cubic inches |
| Depth for Maximum Volume | 1 inch |
| Depth for Minimum Volume | 3 inches |
| Size of Squares for Maximum Volume | 1 inch by 1 inch |
| Size of Squares for Minimum Volume | 3 inches by 3 inches |