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Answer :
Sure, let's solve this problem step-by-step using the Binomial Theorem.
We start with the initial side length of the cube, which is [tex]\(2x\)[/tex].
When the side length is shrunk by 3 units, the new side length becomes:
[tex]\[ 2x - 3 \][/tex]
The volume of a cube is given by the cube of its side length. So, we need to find the new volume, which is:
[tex]\[ (2x - 3)^3 \][/tex]
To expand [tex]\((2x - 3)^3\)[/tex] using the Binomial Theorem, we recognize that:
[tex]\[ (a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3 \][/tex]
Here, [tex]\(a\)[/tex] is [tex]\(2x\)[/tex] and [tex]\(b\)[/tex] is 3. Now we substitute these values into the binomial expansion formula:
[tex]\[
(2x - 3)^3 = (2x)^3 - 3(2x)^2(3) + 3(2x)(3)^2 - (3)^3
\][/tex]
Next, let’s compute each term:
1. [tex]\( (2x)^3 = 8x^3 \)[/tex]
2. [tex]\( 3(2x)^2(3) = 3(4x^2)(3) = 36x^2 \)[/tex]
3. [tex]\( 3(2x)(3)^2 = 3(2x)(9) = 54x \)[/tex]
4. [tex]\( (3)^3 = 27 \)[/tex]
Putting it all together, we get:
[tex]\[
(2x - 3)^3 = 8x^3 - 36x^2 + 54x - 27
\][/tex]
Thus, the correct expression for the new volume of the cube is:
[tex]\[ 8x^3 - 36x^2 + 54x - 27 \][/tex]
So the correct answer is:
[tex]\[ \boxed{8x^3 - 36x^2 + 54x - 27} \][/tex]
We start with the initial side length of the cube, which is [tex]\(2x\)[/tex].
When the side length is shrunk by 3 units, the new side length becomes:
[tex]\[ 2x - 3 \][/tex]
The volume of a cube is given by the cube of its side length. So, we need to find the new volume, which is:
[tex]\[ (2x - 3)^3 \][/tex]
To expand [tex]\((2x - 3)^3\)[/tex] using the Binomial Theorem, we recognize that:
[tex]\[ (a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3 \][/tex]
Here, [tex]\(a\)[/tex] is [tex]\(2x\)[/tex] and [tex]\(b\)[/tex] is 3. Now we substitute these values into the binomial expansion formula:
[tex]\[
(2x - 3)^3 = (2x)^3 - 3(2x)^2(3) + 3(2x)(3)^2 - (3)^3
\][/tex]
Next, let’s compute each term:
1. [tex]\( (2x)^3 = 8x^3 \)[/tex]
2. [tex]\( 3(2x)^2(3) = 3(4x^2)(3) = 36x^2 \)[/tex]
3. [tex]\( 3(2x)(3)^2 = 3(2x)(9) = 54x \)[/tex]
4. [tex]\( (3)^3 = 27 \)[/tex]
Putting it all together, we get:
[tex]\[
(2x - 3)^3 = 8x^3 - 36x^2 + 54x - 27
\][/tex]
Thus, the correct expression for the new volume of the cube is:
[tex]\[ 8x^3 - 36x^2 + 54x - 27 \][/tex]
So the correct answer is:
[tex]\[ \boxed{8x^3 - 36x^2 + 54x - 27} \][/tex]
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