Answer :
To solve the problem of finding the new volume of a cube after its side lengths have been reduced, we can use the Binomial Theorem to expand the expression for the new side length.
1. Original Side Length: The original side length of the cube is [tex]\(2x\)[/tex].
2. Reduction: We are told that each side length is reduced by 3 units. Therefore, the new side length of the cube is [tex]\((2x - 3)\)[/tex].
3. Expression for New Volume: The volume of a cube is given by the cube of its side length. So, the volume of the cube with the reduced side length is [tex]\((2x - 3)^3\)[/tex].
4. Use the Binomial Theorem to Expand [tex]\((2x - 3)^3\)[/tex]:
The Binomial Theorem states:
[tex]\[
(a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3
\][/tex]
Applying this formula to our expression:
- Let [tex]\(a = 2x\)[/tex] and [tex]\(b = 3\)[/tex].
- [tex]\(a^3 = (2x)^3 = 8x^3\)[/tex]
- [tex]\(-3a^2b = -3(2x)^2(3) = -3 \cdot 4x^2 \cdot 3 = -36x^2\)[/tex]
- [tex]\(3ab^2 = 3(2x)(3)^2 = 3 \cdot 2x \cdot 9 = 54x\)[/tex]
- [tex]\(-b^3 = -(3)^3 = -27\)[/tex]
5. Combine Terms:
Combine all the terms obtained from the expansion:
[tex]\[
(2x - 3)^3 = 8x^3 - 36x^2 + 54x - 27
\][/tex]
The expanded expression for the new volume of the cube is:
[tex]\[ 8x^3 - 36x^2 + 54x - 27 \][/tex]
Thus, the correct expression for the new volume of the cube is [tex]\(\boxed{8x^3 - 36x^2 + 54x - 27}\)[/tex].
1. Original Side Length: The original side length of the cube is [tex]\(2x\)[/tex].
2. Reduction: We are told that each side length is reduced by 3 units. Therefore, the new side length of the cube is [tex]\((2x - 3)\)[/tex].
3. Expression for New Volume: The volume of a cube is given by the cube of its side length. So, the volume of the cube with the reduced side length is [tex]\((2x - 3)^3\)[/tex].
4. Use the Binomial Theorem to Expand [tex]\((2x - 3)^3\)[/tex]:
The Binomial Theorem states:
[tex]\[
(a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3
\][/tex]
Applying this formula to our expression:
- Let [tex]\(a = 2x\)[/tex] and [tex]\(b = 3\)[/tex].
- [tex]\(a^3 = (2x)^3 = 8x^3\)[/tex]
- [tex]\(-3a^2b = -3(2x)^2(3) = -3 \cdot 4x^2 \cdot 3 = -36x^2\)[/tex]
- [tex]\(3ab^2 = 3(2x)(3)^2 = 3 \cdot 2x \cdot 9 = 54x\)[/tex]
- [tex]\(-b^3 = -(3)^3 = -27\)[/tex]
5. Combine Terms:
Combine all the terms obtained from the expansion:
[tex]\[
(2x - 3)^3 = 8x^3 - 36x^2 + 54x - 27
\][/tex]
The expanded expression for the new volume of the cube is:
[tex]\[ 8x^3 - 36x^2 + 54x - 27 \][/tex]
Thus, the correct expression for the new volume of the cube is [tex]\(\boxed{8x^3 - 36x^2 + 54x - 27}\)[/tex].
