Answer :
To find the volume of a rectangular prism, we use the formula:
[tex]\[ V = l \cdot w \cdot h \][/tex]
where [tex]\( l \)[/tex] is the length, [tex]\( w \)[/tex] is the width, and [tex]\( h \)[/tex] is the height. In the problem, the dimensions are given as follows:
- Length [tex]\( l = 4x \)[/tex]
- Width [tex]\( w = 2x \)[/tex]
- Height [tex]\( h = x^3 + 3x + 6 \)[/tex]
To calculate the volume, multiply these expressions:
1. First, calculate the product of the length and the width:
[tex]\[ l \cdot w = (4x) \cdot (2x) = 8x^2 \][/tex]
2. Next, multiply this result by the height:
[tex]\[ 8x^2 \cdot (x^3 + 3x + 6) \][/tex]
3. Distribute [tex]\( 8x^2 \)[/tex] across the terms in the height expression:
[tex]\[
\begin{align*}
8x^2 \cdot x^3 & = 8x^{2+3} = 8x^5 \\
8x^2 \cdot 3x & = 24x^{2+1} = 24x^3 \\
8x^2 \cdot 6 & = 48x^{2} \\
\end{align*}
\][/tex]
4. Combining these results gives the volume:
[tex]\[ 8x^5 + 24x^3 + 48x^2 \][/tex]
Therefore, the volume of the rectangular prism is [tex]\( 8x^5 + 24x^3 + 48x^2 \)[/tex].
[tex]\[ V = l \cdot w \cdot h \][/tex]
where [tex]\( l \)[/tex] is the length, [tex]\( w \)[/tex] is the width, and [tex]\( h \)[/tex] is the height. In the problem, the dimensions are given as follows:
- Length [tex]\( l = 4x \)[/tex]
- Width [tex]\( w = 2x \)[/tex]
- Height [tex]\( h = x^3 + 3x + 6 \)[/tex]
To calculate the volume, multiply these expressions:
1. First, calculate the product of the length and the width:
[tex]\[ l \cdot w = (4x) \cdot (2x) = 8x^2 \][/tex]
2. Next, multiply this result by the height:
[tex]\[ 8x^2 \cdot (x^3 + 3x + 6) \][/tex]
3. Distribute [tex]\( 8x^2 \)[/tex] across the terms in the height expression:
[tex]\[
\begin{align*}
8x^2 \cdot x^3 & = 8x^{2+3} = 8x^5 \\
8x^2 \cdot 3x & = 24x^{2+1} = 24x^3 \\
8x^2 \cdot 6 & = 48x^{2} \\
\end{align*}
\][/tex]
4. Combining these results gives the volume:
[tex]\[ 8x^5 + 24x^3 + 48x^2 \][/tex]
Therefore, the volume of the rectangular prism is [tex]\( 8x^5 + 24x^3 + 48x^2 \)[/tex].
