High School

The side length, [tex]s[/tex], of a cube is [tex]4x^2 + 3[/tex]. If [tex]V = s^3[/tex], what is the volume of the cube?

A. [tex]64x^6 + 144x^4 + 108x^2 + 27[/tex]

B. [tex]64x^6 + 48x^4 + 12x^2 + 1[/tex]

C. [tex]4x^6 + 36x^4 + 108x^2 + 27[/tex]

D. [tex]4x^6 + 12x^4 + 12x^2 + 7[/tex]

Answer :

To find the volume of a cube when the side length [tex]\( s \)[/tex] is given, we simply use the formula for the volume of a cube, which is [tex]\( V = s^3 \)[/tex].

Here, the side length [tex]\( s \)[/tex] is given by [tex]\( 4x^2 + 3 \)[/tex]. To find the volume, we need to raise this expression to the third power:

[tex]\[
V = (4x^2 + 3)^3
\][/tex]

Now, we'll expand this expression:

1. Use the binomial expansion formula for cube [tex]\((a + b)^3\)[/tex], which is given by:

[tex]\[
(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3
\][/tex]

Here, [tex]\( a = 4x^2 \)[/tex] and [tex]\( b = 3 \)[/tex].

2. Calculate each term:

- [tex]\( a^3 = (4x^2)^3 = 64x^6 \)[/tex]
- [tex]\( 3a^2b = 3(4x^2)^2 \cdot 3 = 3 \cdot 16x^4 \cdot 3 = 144x^4 \)[/tex]
- [tex]\( 3ab^2 = 3 \cdot 4x^2 \cdot 3^2 = 3 \cdot 4x^2 \cdot 9 = 108x^2 \)[/tex]
- [tex]\( b^3 = 3^3 = 27 \)[/tex]

3. Add all these terms together:

[tex]\[
V = 64x^6 + 144x^4 + 108x^2 + 27
\][/tex]

So, the volume of the cube is [tex]\( 64x^6 + 144x^4 + 108x^2 + 27 \)[/tex].

From the given options, the correct answer is:
[tex]\( 64x^6 + 144x^4 + 108x^2 + 27 \)[/tex].

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