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Answer :
To find the volume of a rectangular prism with the given dimensions, we will use the formula for the volume of a rectangular prism:
[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 of the prism. Let's plug in the given values:
- Length [tex]\( l = 4x \)[/tex]
- Width [tex]\( w = 2x \)[/tex]
- Height [tex]\( h = x^3 + 3x + 6 \)[/tex]
Now, we use the volume formula:
[tex]\[ V = (4x) \cdot (2x) \cdot (x^3 + 3x + 6) \][/tex]
First, we can simplify the multiplication of the length and the width:
[tex]\[ 4x \cdot 2x = 8x^2 \][/tex]
Now we multiply [tex]\( 8x^2 \)[/tex] by the height [tex]\( x^3 + 3x + 6 \)[/tex]:
[tex]\[ V = 8x^2 \cdot (x^3 + 3x + 6) \][/tex]
Next, we distribute [tex]\( 8x^2 \)[/tex] across each term inside the parentheses:
[tex]\[ V = 8x^2 \cdot x^3 + 8x^2 \cdot 3x + 8x^2 \cdot 6 \][/tex]
Now, we perform the multiplications:
[tex]\[ 8x^2 \cdot x^3 = 8x^{2+3} = 8x^5 \][/tex]
[tex]\[ 8x^2 \cdot 3x = 24x^{2+1} = 24x^3 \][/tex]
[tex]\[ 8x^2 \cdot 6 = 48x^2 \][/tex]
So, combining all the terms, we get:
[tex]\[ V = 8x^5 + 24x^3 + 48x^2 \][/tex]
Therefore, the volume of the rectangular prism is:
[tex]\[ V = 8x^5 + 24x^3 + 48x^2 \][/tex]
Comparing this result with the given options, the correct answer 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, and [tex]\( h \)[/tex] is the height of the prism. Let's plug in the given values:
- Length [tex]\( l = 4x \)[/tex]
- Width [tex]\( w = 2x \)[/tex]
- Height [tex]\( h = x^3 + 3x + 6 \)[/tex]
Now, we use the volume formula:
[tex]\[ V = (4x) \cdot (2x) \cdot (x^3 + 3x + 6) \][/tex]
First, we can simplify the multiplication of the length and the width:
[tex]\[ 4x \cdot 2x = 8x^2 \][/tex]
Now we multiply [tex]\( 8x^2 \)[/tex] by the height [tex]\( x^3 + 3x + 6 \)[/tex]:
[tex]\[ V = 8x^2 \cdot (x^3 + 3x + 6) \][/tex]
Next, we distribute [tex]\( 8x^2 \)[/tex] across each term inside the parentheses:
[tex]\[ V = 8x^2 \cdot x^3 + 8x^2 \cdot 3x + 8x^2 \cdot 6 \][/tex]
Now, we perform the multiplications:
[tex]\[ 8x^2 \cdot x^3 = 8x^{2+3} = 8x^5 \][/tex]
[tex]\[ 8x^2 \cdot 3x = 24x^{2+1} = 24x^3 \][/tex]
[tex]\[ 8x^2 \cdot 6 = 48x^2 \][/tex]
So, combining all the terms, we get:
[tex]\[ V = 8x^5 + 24x^3 + 48x^2 \][/tex]
Therefore, the volume of the rectangular prism is:
[tex]\[ V = 8x^5 + 24x^3 + 48x^2 \][/tex]
Comparing this result with the given options, the correct answer is:
[tex]\[ \boxed{8x^5 + 24x^3 + 48x^2} \][/tex]
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