Answer :
Sure! Let's work through how to find the new volume of the cube when its side length is reduced.
1. Initial Side Length of the Cube:
- The side length of the cube is initially [tex]\(2x\)[/tex].
2. Reduction in Side Length:
- We reduce the side length by 3 units. So, the new side length becomes [tex]\(2x - 3\)[/tex].
3. New Volume of the Cube:
- The volume of a cube is given by the formula: [tex]\(\text{Volume} = \text{side length}^3\)[/tex].
- Substitute the new side length into this formula:
[tex]\[
\text{New Volume} = (2x - 3)^3
\][/tex]
4. Using the Binomial Theorem:
- Apply 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].
Applying this, we get:
- Let [tex]\(a = 2x\)[/tex] and [tex]\(b = 3\)[/tex].
- [tex]\((2x - 3)^3 = (2x)^3 - 3(2x)^2(3) + 3(2x)(3)^2 - 3^3\)[/tex].
Calculate each term:
- [tex]\((2x)^3 = 8x^3\)[/tex]
- [tex]\(3(2x)^2(3) = 36x^2\)[/tex]
- [tex]\(3(2x)(9) = 54x\)[/tex]
- [tex]\(3^3 = 27\)[/tex]
5. Combine the Terms:
- [tex]\((2x - 3)^3 = 8x^3 - 36x^2 + 54x - 27\)[/tex]
This expanded expression for the new volume matches the first option:
[tex]\[
8x^3 - 36x^2 + 54x - 27
\][/tex]
Therefore, the correct answer is:
[tex]\(8x^3 - 36x^2 + 54x - 27\)[/tex].
1. Initial Side Length of the Cube:
- The side length of the cube is initially [tex]\(2x\)[/tex].
2. Reduction in Side Length:
- We reduce the side length by 3 units. So, the new side length becomes [tex]\(2x - 3\)[/tex].
3. New Volume of the Cube:
- The volume of a cube is given by the formula: [tex]\(\text{Volume} = \text{side length}^3\)[/tex].
- Substitute the new side length into this formula:
[tex]\[
\text{New Volume} = (2x - 3)^3
\][/tex]
4. Using the Binomial Theorem:
- Apply 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].
Applying this, we get:
- Let [tex]\(a = 2x\)[/tex] and [tex]\(b = 3\)[/tex].
- [tex]\((2x - 3)^3 = (2x)^3 - 3(2x)^2(3) + 3(2x)(3)^2 - 3^3\)[/tex].
Calculate each term:
- [tex]\((2x)^3 = 8x^3\)[/tex]
- [tex]\(3(2x)^2(3) = 36x^2\)[/tex]
- [tex]\(3(2x)(9) = 54x\)[/tex]
- [tex]\(3^3 = 27\)[/tex]
5. Combine the Terms:
- [tex]\((2x - 3)^3 = 8x^3 - 36x^2 + 54x - 27\)[/tex]
This expanded expression for the new volume matches the first option:
[tex]\[
8x^3 - 36x^2 + 54x - 27
\][/tex]
Therefore, the correct answer is:
[tex]\(8x^3 - 36x^2 + 54x - 27\)[/tex].
