Answer :
We are given a rectangular prism with the following dimensions:
- Length: [tex]\( 4x \)[/tex]
- Width: [tex]\( 2x \)[/tex]
- Height: [tex]\( x^3 + 3x + 6 \)[/tex]
To find the volume of a rectangular prism, we use the formula
[tex]$$
V = \text{length} \times \text{width} \times \text{height}.
$$[/tex]
Step 1: Multiply the Length and Width
First, multiply the length and width:
[tex]$$
4x \times 2x = 8x^2.
$$[/tex]
Step 2: Multiply by the Height
Now, multiply the result from Step 1 by the height:
[tex]$$
8x^2 \times (x^3 + 3x + 6).
$$[/tex]
Step 3: Distribute the Multiplication
Distribute [tex]\( 8x^2 \)[/tex] to each term inside the parentheses:
[tex]\[
\begin{aligned}
8x^2 \times x^3 &= 8x^{2+3} = 8x^5, \\
8x^2 \times 3x &= 24x^{2+1} = 24x^3, \\
8x^2 \times 6 &= 48x^2.
\end{aligned}
\][/tex]
Step 4: Write the Final Expression for the Volume
Combine all the terms:
[tex]$$
V = 8x^5 + 24x^3 + 48x^2.
$$[/tex]
Thus, the volume of the rectangular prism is
[tex]$$
\boxed{8x^5+24x^3+48x^2}.
$$[/tex]
- Length: [tex]\( 4x \)[/tex]
- Width: [tex]\( 2x \)[/tex]
- Height: [tex]\( x^3 + 3x + 6 \)[/tex]
To find the volume of a rectangular prism, we use the formula
[tex]$$
V = \text{length} \times \text{width} \times \text{height}.
$$[/tex]
Step 1: Multiply the Length and Width
First, multiply the length and width:
[tex]$$
4x \times 2x = 8x^2.
$$[/tex]
Step 2: Multiply by the Height
Now, multiply the result from Step 1 by the height:
[tex]$$
8x^2 \times (x^3 + 3x + 6).
$$[/tex]
Step 3: Distribute the Multiplication
Distribute [tex]\( 8x^2 \)[/tex] to each term inside the parentheses:
[tex]\[
\begin{aligned}
8x^2 \times x^3 &= 8x^{2+3} = 8x^5, \\
8x^2 \times 3x &= 24x^{2+1} = 24x^3, \\
8x^2 \times 6 &= 48x^2.
\end{aligned}
\][/tex]
Step 4: Write the Final Expression for the Volume
Combine all the terms:
[tex]$$
V = 8x^5 + 24x^3 + 48x^2.
$$[/tex]
Thus, the volume of the rectangular prism is
[tex]$$
\boxed{8x^5+24x^3+48x^2}.
$$[/tex]
