Answer :
To find the new volume of the cube after its side lengths are reduced, we start with the original side length of the cube, which is given as [tex]\(2x\)[/tex].
1. Determine the new side length:
- The side length of the cube is reduced by 3 units, so the new side length becomes:
[tex]\[
2x - 3
\][/tex]
2. Apply the formula for the volume of a cube:
- The volume [tex]\(V\)[/tex] of a cube with side length [tex]\(s\)[/tex] is given by:
[tex]\[
V = s^3
\][/tex]
- Here, the new side length is [tex]\(2x - 3\)[/tex], so the new volume is:
[tex]\[
(2x - 3)^3
\][/tex]
3. Use the Binomial Theorem to expand [tex]\((2x - 3)^3\)[/tex]:
- The Binomial Theorem states that [tex]\((a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3\)[/tex].
- Here, [tex]\(a = 2x\)[/tex] and [tex]\(b = 3\)[/tex].
- Substitute [tex]\(a\)[/tex] and [tex]\(b\)[/tex] into the formula to expand [tex]\((2x - 3)^3\)[/tex]:
[tex]\[
(2x)^3 - 3(2x)^2(3) + 3(2x)(3)^2 - 3^3
\][/tex]
4. Calculate each term:
- [tex]\( (2x)^3 = 8x^3 \)[/tex]
- [tex]\( -3(2x)^2(3) = -3 \times 4x^2 \times 3 = -36x^2 \)[/tex]
- [tex]\( 3(2x)(3)^2 = 3 \times 2x \times 9 = 54x \)[/tex]
- [tex]\( -3^3 = -27 \)[/tex]
5. Combine all the terms:
- New volume of the cube:
[tex]\[
8x^3 - 36x^2 + 54x - 27
\][/tex]
Therefore, the correct expression for the new volume of the cube is:
[tex]\[
8x^3 - 36x^2 + 54x - 27
\][/tex]
This corresponds to the option:
[tex]\[
8x^3 - 36x^2 + 54x - 27
\][/tex]
1. Determine the new side length:
- The side length of the cube is reduced by 3 units, so the new side length becomes:
[tex]\[
2x - 3
\][/tex]
2. Apply the formula for the volume of a cube:
- The volume [tex]\(V\)[/tex] of a cube with side length [tex]\(s\)[/tex] is given by:
[tex]\[
V = s^3
\][/tex]
- Here, the new side length is [tex]\(2x - 3\)[/tex], so the new volume is:
[tex]\[
(2x - 3)^3
\][/tex]
3. Use the Binomial Theorem to expand [tex]\((2x - 3)^3\)[/tex]:
- The Binomial Theorem states that [tex]\((a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3\)[/tex].
- Here, [tex]\(a = 2x\)[/tex] and [tex]\(b = 3\)[/tex].
- Substitute [tex]\(a\)[/tex] and [tex]\(b\)[/tex] into the formula to expand [tex]\((2x - 3)^3\)[/tex]:
[tex]\[
(2x)^3 - 3(2x)^2(3) + 3(2x)(3)^2 - 3^3
\][/tex]
4. Calculate each term:
- [tex]\( (2x)^3 = 8x^3 \)[/tex]
- [tex]\( -3(2x)^2(3) = -3 \times 4x^2 \times 3 = -36x^2 \)[/tex]
- [tex]\( 3(2x)(3)^2 = 3 \times 2x \times 9 = 54x \)[/tex]
- [tex]\( -3^3 = -27 \)[/tex]
5. Combine all the terms:
- New volume of the cube:
[tex]\[
8x^3 - 36x^2 + 54x - 27
\][/tex]
Therefore, the correct expression for the new volume of the cube is:
[tex]\[
8x^3 - 36x^2 + 54x - 27
\][/tex]
This corresponds to the option:
[tex]\[
8x^3 - 36x^2 + 54x - 27
\][/tex]
