Answer :
Of course! Let's solve the problem step-by-step to find the new volume of the cube after its side lengths are reduced.
1. Initial Side Length of the Cube:
The initial side length of the cube is [tex]\(2x\)[/tex].
2. Reduced Side Length:
The side length is reduced by 3 units. So, the new side length is:
[tex]\[
2x - 3
\][/tex]
3. New Volume of the Cube:
The volume of a cube is given by the cube of its side length. Therefore, the new volume will be:
[tex]\[
(2x - 3)^3
\][/tex]
4. Expanding [tex]\( (2x - 3)^3 \)[/tex] Using the Binomial Theorem:
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:
[tex]\[
(2x - 3)^3 = (2x)^3 - 3(2x)^2(3) + 3(2x)(3)^2 - (3)^3
\][/tex]
5. Calculating Each Term:
[tex]\[
(2x)^3 = 8x^3
\][/tex]
[tex]\[
-3(2x)^2(3) = -3(4x^2)(3) = -36x^2
\][/tex]
[tex]\[
3(2x)(3)^2 = 3(2x)(9) = 54x
\][/tex]
[tex]\[
-(3)^3 = -27
\][/tex]
6. Combining All Terms:
[tex]\[
(2x - 3)^3 = 8x^3 - 36x^2 + 54x - 27
\][/tex]
So, the new volume of the cube, after reducing its side lengths by 3 units, is:
[tex]\[
8x^3 - 36x^2 + 54x - 27
\][/tex]
Thus, the correct expression for the new volume of the cube is:
[tex]\[
8x^3 - 36x^2 + 54x - 27
\][/tex]
Therefore, the correct answer is:
[tex]\[
\boxed{8x^3 - 36x^2 + 54x - 27}
\][/tex]
1. Initial Side Length of the Cube:
The initial side length of the cube is [tex]\(2x\)[/tex].
2. Reduced Side Length:
The side length is reduced by 3 units. So, the new side length is:
[tex]\[
2x - 3
\][/tex]
3. New Volume of the Cube:
The volume of a cube is given by the cube of its side length. Therefore, the new volume will be:
[tex]\[
(2x - 3)^3
\][/tex]
4. Expanding [tex]\( (2x - 3)^3 \)[/tex] Using the Binomial Theorem:
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:
[tex]\[
(2x - 3)^3 = (2x)^3 - 3(2x)^2(3) + 3(2x)(3)^2 - (3)^3
\][/tex]
5. Calculating Each Term:
[tex]\[
(2x)^3 = 8x^3
\][/tex]
[tex]\[
-3(2x)^2(3) = -3(4x^2)(3) = -36x^2
\][/tex]
[tex]\[
3(2x)(3)^2 = 3(2x)(9) = 54x
\][/tex]
[tex]\[
-(3)^3 = -27
\][/tex]
6. Combining All Terms:
[tex]\[
(2x - 3)^3 = 8x^3 - 36x^2 + 54x - 27
\][/tex]
So, the new volume of the cube, after reducing its side lengths by 3 units, is:
[tex]\[
8x^3 - 36x^2 + 54x - 27
\][/tex]
Thus, the correct expression for the new volume of the cube is:
[tex]\[
8x^3 - 36x^2 + 54x - 27
\][/tex]
Therefore, the correct answer is:
[tex]\[
\boxed{8x^3 - 36x^2 + 54x - 27}
\][/tex]
