College

Find the volume of a rectangular prism if the length is [tex]4x[/tex], the width is [tex]2x[/tex], and the height is [tex]x^3 + 3x + 6[/tex].

Use the formula [tex]V = l \cdot w \cdot h[/tex], where [tex]l[/tex] is length, [tex]w[/tex] is width, and [tex]h[/tex] is height, to find the volume.

A. [tex]6x^5 + 18x^3 + 36x^2[/tex]

B. [tex]6x^6 + 18x^3 + 36x^2[/tex]

C. [tex]8x^5 + 24x^3 + 48x^2[/tex]

D. [tex]8x^6 + 24x^3 + 48x^2[/tex]

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]