Question

the volume of an open top box is created by a cutting square from each corner of a 10 in by 12 piece of paper the situation is represented by the function Give me the correct answer

Answer 1

To find the volume of an open-top box created by cutting squares from the corners of a rectangular piece of paper measuring 10 inches by 12 inches, follow these steps:

1. **Define the Variables**:
Let \( x \) be the length of the side of each square cut from the corners.

2. **Determine the Dimensions of the Box**:
After cutting out squares of size \( x \) from each corner, the new dimensions of the box when folded would be:
- Length: \( 12 - 2x \) (since you cut out \( x \) from both ends of the length)
- Width: \( 10 - 2x \) (since you cut out \( x \) from both ends of the width)
- Height: \( x \) (the height of the box is equal to the side length of the cut square)

3. **Volume Function**:
The volume \( V \) of the box can be expressed as:
\[
V(x) = (12 - 2x)(10 - 2x)x
\]

4. **Volume Function Simplification**:
Expanding this function:
\[
V(x) = x(12 - 2x)(10 - 2x)
\]
\[
V(x) = x(120 - 24x - 20x + 4x^2)
\]
\[
V(x) = x(120 - 44x + 4x^2)
\]
\[
V(x) = 120x - 44x^2 + 4x^3
\]

5. **Determine the Domain**:
For the volume function to make sense, \( x \) must be in the range where the lengths remain positive:
\[
0 < x < 5 \quad (\text{because } 10 - 2x > 0 \Rightarrow x < 5 \text{ and } 12 - 2x > 0 \Rightarrow x < 6)
\]

Now, we have the volume function \( V(x) = 4x^3 - 44x^2 + 120x \) for \( 0 < x < 5 \).

If you need further calculations, such as finding the maximum volume or evaluating at specific points, please let me know!