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Answer :
To solve the problem of finding the volume of the box after each dimension is increased by 2, let's go through the steps one by one.
1. Initial Dimensions of the Box:
The original dimensions of the box are given as:
- Length: [tex]\( x \)[/tex]
- Width: [tex]\( 2x \)[/tex]
- Height: [tex]\( 3x \)[/tex]
2. Increased Dimensions:
Each dimension is increased by 2:
- New Length: [tex]\( x + 2 \)[/tex]
- New Width: [tex]\( 2x + 2 \)[/tex]
- New Height: [tex]\( 3x + 2 \)[/tex]
3. Volume of the Box with New Dimensions:
The volume [tex]\( V \)[/tex] of a box is given by the product of its length, width, and height. Therefore, the volume of the box with the new dimensions is:
[tex]\[
V = (x + 2) \times (2x + 2) \times (3x + 2)
\][/tex]
4. Expanding the Expression:
First, we expand [tex]\( (x + 2) \times (2x + 2) \)[/tex]:
[tex]\[
(x + 2)(2x + 2) = x \cdot 2x + x \cdot 2 + 2 \cdot 2x + 2 \cdot 2 = 2x^2 + 2x + 4x + 4 = 2x^2 + 6x + 4
\][/tex]
Next, multiply this result by [tex]\( (3x + 2) \)[/tex]:
[tex]\[
(2x^2 + 6x + 4)(3x + 2)
\][/tex]
This requires distributing each term in [tex]\( 2x^2 + 6x + 4 \)[/tex] across [tex]\( 3x + 2 \)[/tex]:
[tex]\[
2x^2 \times 3x = 6x^3
\][/tex]
[tex]\[
2x^2 \times 2 = 4x^2
\][/tex]
[tex]\[
6x \times 3x = 18x^2
\][/tex]
[tex]\[
6x \times 2 = 12x
\][/tex]
[tex]\[
4 \times 3x = 12x
\][/tex]
[tex]\[
4 \times 2 = 8
\][/tex]
5. Combining Like Terms:
Summing up all these terms, we get:
[tex]\[
6x^3 + 4x^2 + 18x^2 + 12x + 12x + 8
\][/tex]
Combine the like terms:
[tex]\[
6x^3 + (4x^2 + 18x^2) + (12x + 12x) + 8 = 6x^3 + 22x^2 + 24x + 8
\][/tex]
So, the volume of the box after each dimension is increased by 2 is:
[tex]\[
\boxed{6x^3 + 22x^2 + 24x + 8}
\][/tex]
1. Initial Dimensions of the Box:
The original dimensions of the box are given as:
- Length: [tex]\( x \)[/tex]
- Width: [tex]\( 2x \)[/tex]
- Height: [tex]\( 3x \)[/tex]
2. Increased Dimensions:
Each dimension is increased by 2:
- New Length: [tex]\( x + 2 \)[/tex]
- New Width: [tex]\( 2x + 2 \)[/tex]
- New Height: [tex]\( 3x + 2 \)[/tex]
3. Volume of the Box with New Dimensions:
The volume [tex]\( V \)[/tex] of a box is given by the product of its length, width, and height. Therefore, the volume of the box with the new dimensions is:
[tex]\[
V = (x + 2) \times (2x + 2) \times (3x + 2)
\][/tex]
4. Expanding the Expression:
First, we expand [tex]\( (x + 2) \times (2x + 2) \)[/tex]:
[tex]\[
(x + 2)(2x + 2) = x \cdot 2x + x \cdot 2 + 2 \cdot 2x + 2 \cdot 2 = 2x^2 + 2x + 4x + 4 = 2x^2 + 6x + 4
\][/tex]
Next, multiply this result by [tex]\( (3x + 2) \)[/tex]:
[tex]\[
(2x^2 + 6x + 4)(3x + 2)
\][/tex]
This requires distributing each term in [tex]\( 2x^2 + 6x + 4 \)[/tex] across [tex]\( 3x + 2 \)[/tex]:
[tex]\[
2x^2 \times 3x = 6x^3
\][/tex]
[tex]\[
2x^2 \times 2 = 4x^2
\][/tex]
[tex]\[
6x \times 3x = 18x^2
\][/tex]
[tex]\[
6x \times 2 = 12x
\][/tex]
[tex]\[
4 \times 3x = 12x
\][/tex]
[tex]\[
4 \times 2 = 8
\][/tex]
5. Combining Like Terms:
Summing up all these terms, we get:
[tex]\[
6x^3 + 4x^2 + 18x^2 + 12x + 12x + 8
\][/tex]
Combine the like terms:
[tex]\[
6x^3 + (4x^2 + 18x^2) + (12x + 12x) + 8 = 6x^3 + 22x^2 + 24x + 8
\][/tex]
So, the volume of the box after each dimension is increased by 2 is:
[tex]\[
\boxed{6x^3 + 22x^2 + 24x + 8}
\][/tex]
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