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Answer :
We are given the dimensions of the rectangular prism as follows:
- Length: [tex]$$4x$$[/tex]
- Width: [tex]$$2x$$[/tex]
- Height: [tex]$$x^3 + 3x + 6$$[/tex]
The formula for the volume [tex]$$V$$[/tex] of a rectangular prism is:
[tex]$$
V = \text{length} \times \text{width} \times \text{height}
$$[/tex]
Step 1: Multiply the Length and Width
First, we multiply the length and width:
[tex]$$
\text{Length} \times \text{Width} = (4x) \times (2x) = 8x^2
$$[/tex]
Step 2: Multiply by the Height
Next, we multiply the result from Step 1 by the height:
[tex]$$
V = 8x^2 \times \left( x^3 + 3x + 6 \right)
$$[/tex]
Step 3: Expand the Expression
Distribute [tex]$$8x^2$$[/tex] across the terms inside the parentheses:
- Multiply [tex]$$8x^2 \times x^3$$[/tex]:
[tex]$$
8x^2 \times x^3 = 8x^{2+3} = 8x^5
$$[/tex]
- Multiply [tex]$$8x^2 \times 3x$$[/tex]:
[tex]$$
8x^2 \times 3x = 24x^{2+1} = 24x^3
$$[/tex]
- Multiply [tex]$$8x^2 \times 6$$[/tex]:
[tex]$$
8x^2 \times 6 = 48x^2
$$[/tex]
Putting all these together, the volume is:
[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]
The formula for the volume [tex]$$V$$[/tex] of a rectangular prism is:
[tex]$$
V = \text{length} \times \text{width} \times \text{height}
$$[/tex]
Step 1: Multiply the Length and Width
First, we multiply the length and width:
[tex]$$
\text{Length} \times \text{Width} = (4x) \times (2x) = 8x^2
$$[/tex]
Step 2: Multiply by the Height
Next, we multiply the result from Step 1 by the height:
[tex]$$
V = 8x^2 \times \left( x^3 + 3x + 6 \right)
$$[/tex]
Step 3: Expand the Expression
Distribute [tex]$$8x^2$$[/tex] across the terms inside the parentheses:
- Multiply [tex]$$8x^2 \times x^3$$[/tex]:
[tex]$$
8x^2 \times x^3 = 8x^{2+3} = 8x^5
$$[/tex]
- Multiply [tex]$$8x^2 \times 3x$$[/tex]:
[tex]$$
8x^2 \times 3x = 24x^{2+1} = 24x^3
$$[/tex]
- Multiply [tex]$$8x^2 \times 6$$[/tex]:
[tex]$$
8x^2 \times 6 = 48x^2
$$[/tex]
Putting all these together, the volume is:
[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]
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