Answer :
Sure, let's find the volume of the rectangular prism given the dimensions:
- Length (L) = [tex]\(4x\)[/tex]
- Width (W) = [tex]\(2x\)[/tex]
- Height (H) = [tex]\(x^3 + 3x + 6\)[/tex]
The formula for the volume [tex]\(V\)[/tex] of a rectangular prism is:
[tex]\[ V = L \times W \times H \][/tex]
1. Substitute the given expressions for length, width, and height into the volume formula:
[tex]\[ V = (4x) \times (2x) \times (x^3 + 3x + 6) \][/tex]
2. Multiply the length and the width first:
[tex]\[ 4x \times 2x = 8x^2 \][/tex]
3. Now, multiply this result by the height:
[tex]\[ V = 8x^2 \times (x^3 + 3x + 6) \][/tex]
4. Distribute [tex]\(8x^2\)[/tex] through the terms inside the parentheses:
[tex]\[ 8x^2 \times x^3 + 8x^2 \times 3x + 8x^2 \times 6 \][/tex]
5. Multiply term by term:
[tex]\[ 8x^2 \times x^3 = 8x^{2+3} = 8x^5 \][/tex]
[tex]\[ 8x^2 \times 3x = 8 \times 3 \times x^{2+1} = 24x^3 \][/tex]
[tex]\[ 8x^2 \times 6 = 8 \times 6 \times x^2 = 48x^2 \][/tex]
6. Combine all the terms to get the volume:
[tex]\[ V = 8x^5 + 24x^3 + 48x^2 \][/tex]
So, the volume of the rectangular prism is:
[tex]\[ 8x^5 + 24x^3 + 48x^2 \][/tex]
Looking at the options provided:
- [tex]\(6x^5 + 18x^3 + 36x^2\)[/tex]
- [tex]\(6x^4 + 18x^3 + 36x^2\)[/tex]
- [tex]\(8x^5 + 24x^3 + 48x^2\)[/tex]
- [tex]\(8x^4 + 24x^3 + 48x^2\)[/tex]
The correct answer is:
[tex]\[ 8x^5 + 24x^3 + 48x^2 \][/tex]
- Length (L) = [tex]\(4x\)[/tex]
- Width (W) = [tex]\(2x\)[/tex]
- Height (H) = [tex]\(x^3 + 3x + 6\)[/tex]
The formula for the volume [tex]\(V\)[/tex] of a rectangular prism is:
[tex]\[ V = L \times W \times H \][/tex]
1. Substitute the given expressions for length, width, and height into the volume formula:
[tex]\[ V = (4x) \times (2x) \times (x^3 + 3x + 6) \][/tex]
2. Multiply the length and the width first:
[tex]\[ 4x \times 2x = 8x^2 \][/tex]
3. Now, multiply this result by the height:
[tex]\[ V = 8x^2 \times (x^3 + 3x + 6) \][/tex]
4. Distribute [tex]\(8x^2\)[/tex] through the terms inside the parentheses:
[tex]\[ 8x^2 \times x^3 + 8x^2 \times 3x + 8x^2 \times 6 \][/tex]
5. Multiply term by term:
[tex]\[ 8x^2 \times x^3 = 8x^{2+3} = 8x^5 \][/tex]
[tex]\[ 8x^2 \times 3x = 8 \times 3 \times x^{2+1} = 24x^3 \][/tex]
[tex]\[ 8x^2 \times 6 = 8 \times 6 \times x^2 = 48x^2 \][/tex]
6. Combine all the terms to get the volume:
[tex]\[ V = 8x^5 + 24x^3 + 48x^2 \][/tex]
So, the volume of the rectangular prism is:
[tex]\[ 8x^5 + 24x^3 + 48x^2 \][/tex]
Looking at the options provided:
- [tex]\(6x^5 + 18x^3 + 36x^2\)[/tex]
- [tex]\(6x^4 + 18x^3 + 36x^2\)[/tex]
- [tex]\(8x^5 + 24x^3 + 48x^2\)[/tex]
- [tex]\(8x^4 + 24x^3 + 48x^2\)[/tex]
The correct answer is:
[tex]\[ 8x^5 + 24x^3 + 48x^2 \][/tex]
