Sure, let's solve this step-by-step.
To find the volume of the trapezoidal prism, we use the fact that the volume [tex]\( V \)[/tex] is given by
[tex]\[ V = \text{Base Area} \times \text{Height} \][/tex]
From the problem, we have already derived the expression for the area [tex]\( A \)[/tex] of the trapezoidal base as follows:
[tex]\[
A = \frac{1}{2}((x+4)+(x+2))x
\][/tex]
[tex]\[
A = \frac{1}{2}(2x + 6)x
\][/tex]
[tex]\[
A = (x + 3)x
\][/tex]
[tex]\[
A = x^2 + 3x
\][/tex]
Let's assume the height (which I'll denote as [tex]\( H \)[/tex]) of the prism is [tex]\( x \)[/tex]. Thus, the volume [tex]\( V \)[/tex] can be computed as:
[tex]\[
V = A \times H
\][/tex]
Substitute [tex]\( A = x^2 + 3x \)[/tex]:
[tex]\[
V = (x^2 + 3x) \times x
\][/tex]
Simplify this expression:
[tex]\[
V = x^3 + 3x^2
\][/tex]
Therefore, the expression that represents the volume of the trapezoidal prism is:
[tex]\[
x^3 + 3x^2
\][/tex]
Now, let's match this resulting volume with one of the answer choices:
1. [tex]\( 2x^3 + 6x^2 \)[/tex]
2. [tex]\( x^3 + 6x^2 \)[/tex]
3. [tex]\( x^3 + 3x^2 \)[/tex]
4. [tex]\( 2x^3 + 3x^2 \)[/tex]
The correct answer is:
[tex]\[
x^3 + 3x^2
\][/tex]
So, the correct choice is:
[tex]\( x^3 + 3x^2 \)[/tex]
