Answer :
To find the volume of a rectangular prism, we need to use the formula for volume:
[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.
Given:
- Length ([tex]\( l \)[/tex]) = [tex]\( 4x \)[/tex]
- Width ([tex]\( w \)[/tex]) = [tex]\( 2x \)[/tex]
- Height ([tex]\( h \)[/tex]) = [tex]\( x^3 + 3x + 6 \)[/tex]
Let's find the volume:
1. Multiply the length and the width:
[tex]\[ l \cdot w = 4x \cdot 2x = 8x^2 \][/tex]
2. Now multiply the result by the height:
[tex]\[ V = 8x^2 \cdot (x^3 + 3x + 6) \][/tex]
We will distribute [tex]\( 8x^2 \)[/tex] to each term in the height expression:
- [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]
3. Combine these results to get the expression for the volume:
[tex]\[ V = 8x^5 + 24x^3 + 48x^2 \][/tex]
Now let's match this expression to the given multiple-choice options:
- [tex]\( 6x^5+18x^3+36x^2 \)[/tex]
- [tex]\( 6x^6+18x^3+36x^2 \)[/tex]
- [tex]\( 8x^5+24x^3+48x^2 \)[/tex]
- [tex]\( 8x^6+24x^3+48x^2 \)[/tex]
The correct volume expression is:
[tex]\[ 8x^5 + 24x^3 + 48x^2 \][/tex]
Therefore, the correct answer is:
[tex]\( 8x^5 + 24x^3 + 48x^2 \)[/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.
Given:
- Length ([tex]\( l \)[/tex]) = [tex]\( 4x \)[/tex]
- Width ([tex]\( w \)[/tex]) = [tex]\( 2x \)[/tex]
- Height ([tex]\( h \)[/tex]) = [tex]\( x^3 + 3x + 6 \)[/tex]
Let's find the volume:
1. Multiply the length and the width:
[tex]\[ l \cdot w = 4x \cdot 2x = 8x^2 \][/tex]
2. Now multiply the result by the height:
[tex]\[ V = 8x^2 \cdot (x^3 + 3x + 6) \][/tex]
We will distribute [tex]\( 8x^2 \)[/tex] to each term in the height expression:
- [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]
3. Combine these results to get the expression for the volume:
[tex]\[ V = 8x^5 + 24x^3 + 48x^2 \][/tex]
Now let's match this expression to the given multiple-choice options:
- [tex]\( 6x^5+18x^3+36x^2 \)[/tex]
- [tex]\( 6x^6+18x^3+36x^2 \)[/tex]
- [tex]\( 8x^5+24x^3+48x^2 \)[/tex]
- [tex]\( 8x^6+24x^3+48x^2 \)[/tex]
The correct volume expression is:
[tex]\[ 8x^5 + 24x^3 + 48x^2 \][/tex]
Therefore, the correct answer is:
[tex]\( 8x^5 + 24x^3 + 48x^2 \)[/tex].
