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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,
- [tex]\( h \)[/tex] is the height.
For this problem:
- The length ([tex]\( l \)[/tex]) is [tex]\( 4x \)[/tex],
- The width ([tex]\( w \)[/tex]) is [tex]\( 2x \)[/tex],
- The height ([tex]\( h \)[/tex]) is [tex]\( x^3 + 3x + 6 \)[/tex].
To find the volume, we multiply these expressions together:
[tex]\[ V = (4x) \cdot (2x) \cdot (x^3 + 3x + 6) \][/tex]
First, multiply the length and width:
[tex]\[ 4x \times 2x = 8x^2 \][/tex]
Now, multiply this result by the height:
[tex]\[ V = 8x^2 \cdot (x^3 + 3x + 6) \][/tex]
Distribute [tex]\( 8x^2 \)[/tex] to each term inside the parentheses:
1. [tex]\( 8x^2 \cdot x^3 = 8x^{5} \)[/tex]
2. [tex]\( 8x^2 \cdot 3x = 24x^{3} \)[/tex]
3. [tex]\( 8x^2 \cdot 6 = 48x^{2} \)[/tex]
Combine all the terms to get the volume:
[tex]\[ V = 8x^5 + 24x^3 + 48x^2 \][/tex]
Therefore, the volume of the rectangular prism is:
[tex]\[ \boxed{8x^5 + 24x^3 + 48x^2} \][/tex]
[tex]\[ V = l \cdot w \cdot h \][/tex]
where:
- [tex]\( l \)[/tex] is the length,
- [tex]\( w \)[/tex] is the width,
- [tex]\( h \)[/tex] is the height.
For this problem:
- The length ([tex]\( l \)[/tex]) is [tex]\( 4x \)[/tex],
- The width ([tex]\( w \)[/tex]) is [tex]\( 2x \)[/tex],
- The height ([tex]\( h \)[/tex]) is [tex]\( x^3 + 3x + 6 \)[/tex].
To find the volume, we multiply these expressions together:
[tex]\[ V = (4x) \cdot (2x) \cdot (x^3 + 3x + 6) \][/tex]
First, multiply the length and width:
[tex]\[ 4x \times 2x = 8x^2 \][/tex]
Now, multiply this result by the height:
[tex]\[ V = 8x^2 \cdot (x^3 + 3x + 6) \][/tex]
Distribute [tex]\( 8x^2 \)[/tex] to each term inside the parentheses:
1. [tex]\( 8x^2 \cdot x^3 = 8x^{5} \)[/tex]
2. [tex]\( 8x^2 \cdot 3x = 24x^{3} \)[/tex]
3. [tex]\( 8x^2 \cdot 6 = 48x^{2} \)[/tex]
Combine all the terms to get the volume:
[tex]\[ V = 8x^5 + 24x^3 + 48x^2 \][/tex]
Therefore, the volume of the rectangular prism is:
[tex]\[ \boxed{8x^5 + 24x^3 + 48x^2} \][/tex]
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