Answer :
To find the new volume of the cube, follow these steps:
1. The side of the cube originally has length [tex]$2x$[/tex]. After reducing each side by 3 units, the new side length becomes
[tex]$$
2x - 3.
$$[/tex]
2. The volume of a cube with side length [tex]$s$[/tex] is given by [tex]$s^3$[/tex]. Therefore, the new volume is
[tex]$$
(2x - 3)^3.
$$[/tex]
3. To expand the cube of the binomial, use the Binomial Theorem:
[tex]$$
(a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3.
$$[/tex]
Here, let [tex]$a = 2x$[/tex] and [tex]$b = 3$[/tex]. Substitute these into the formula:
[tex]\[
\begin{aligned}
(2x - 3)^3 &= (2x)^3 - 3(2x)^2(3) + 3(2x)(3)^2 - (3)^3 \\
&= 8x^3 - 3(4x^2)(3) + 3(2x)(9) - 27 \\
&= 8x^3 - 36x^2 + 54x - 27.
\end{aligned}
\][/tex]
Thus, the new volume of the cube is
[tex]$$
8x^3 - 36x^2 + 54x - 27.
$$[/tex]
The correct expression for the new volume is:
[tex]$$\textbf{8x}^3 - 36\textbf{x}^2 + 54\textbf{x} - 27.$$[/tex]
1. The side of the cube originally has length [tex]$2x$[/tex]. After reducing each side by 3 units, the new side length becomes
[tex]$$
2x - 3.
$$[/tex]
2. The volume of a cube with side length [tex]$s$[/tex] is given by [tex]$s^3$[/tex]. Therefore, the new volume is
[tex]$$
(2x - 3)^3.
$$[/tex]
3. To expand the cube of the binomial, use the Binomial Theorem:
[tex]$$
(a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3.
$$[/tex]
Here, let [tex]$a = 2x$[/tex] and [tex]$b = 3$[/tex]. Substitute these into the formula:
[tex]\[
\begin{aligned}
(2x - 3)^3 &= (2x)^3 - 3(2x)^2(3) + 3(2x)(3)^2 - (3)^3 \\
&= 8x^3 - 3(4x^2)(3) + 3(2x)(9) - 27 \\
&= 8x^3 - 36x^2 + 54x - 27.
\end{aligned}
\][/tex]
Thus, the new volume of the cube is
[tex]$$
8x^3 - 36x^2 + 54x - 27.
$$[/tex]
The correct expression for the new volume is:
[tex]$$\textbf{8x}^3 - 36\textbf{x}^2 + 54\textbf{x} - 27.$$[/tex]
