To solve this problem, let's follow these steps:
1. Determine the original dimensions of the box:
- Length: [tex]\(x\)[/tex]
- Width: [tex]\(2x\)[/tex]
- Height: [tex]\(3x\)[/tex]
2. Decrease each dimension by 2:
- New length: [tex]\(x - 2\)[/tex]
- New width: [tex]\(2x - 2\)[/tex]
- New height: [tex]\(3x - 2\)[/tex]
3. Calculate each new dimension:
- If [tex]\(x = 1\)[/tex] (assuming a value for calculation):
- New length: [tex]\(1 - 2 = -1\)[/tex]
- New width: [tex]\(2(1) - 2 = 0\)[/tex]
- New height: [tex]\(3(1) - 2 = 1\)[/tex]
4. Calculate the volume of the new box:
The volume [tex]\(V\)[/tex] is given by the product of the length, width, and height:
[tex]\[
V = (\text{new length}) \times (\text{new width}) \times (\text{new height})
\][/tex]
Substituting the values we have:
[tex]\[
V = (-1) \times 0 \times 1 = 0
\][/tex]
5. Conclusion:
The new dimensions of the box are [tex]\(-1, 0, 1\)[/tex], and the volume of the box with the new dimensions is [tex]\(0\)[/tex].
