To solve this problem, we need to find the length of the rectangular prism given its volume, width, and height.
1. Understand the given polynomial:
The volume of the rectangular prism is represented by the polynomial [tex]\( V(x) = x^3 + 11x^2 + 20x - 32 \)[/tex].
2. Identify the given dimensions:
- Width ([tex]\( W \)[/tex]) = [tex]\( x - 1 \)[/tex]
- Height ([tex]\( H \)[/tex]) = [tex]\( x + 8 \)[/tex]
3. Set up the equation:
We know that the volume [tex]\( V \)[/tex] of a rectangular prism is given by:
[tex]\[
V = \text{length} \times \text{width} \times \text{height}
\][/tex]
Substitute the given expressions:
[tex]\[
x^3 + 11x^2 + 20x - 32 = \text{length} \times (x - 1) \times (x + 8)
\][/tex]
4. Find the expression for the length:
To isolate the length, we need to divide the volume polynomial by the product of the width and height. Let's denote the length as [tex]\( L \)[/tex].
First, calculate the product of the width and height:
[tex]\[
(x - 1)(x + 8)
\][/tex]
Use the distributive property to expand:
[tex]\[
(x - 1)(x + 8) = x^2 + 8x - x - 8 = x^2 + 7x - 8
\][/tex]
5. Divide the volume by the result:
[tex]\[
L = \frac{V(x)}{(x - 1)(x + 8)} = \frac{x^3 + 11x^2 + 20x - 32}{x^2 + 7x - 8}
\][/tex]
6. Simplify the division:
Perform the polynomial division:
[tex]\[
\frac{x^3 + 11x^2 + 20x - 32}{x^2 + 7x - 8}
\][/tex]
The quotient of this division, which represents the length of the box, simplifies to:
[tex]\[
L = x + 4
\][/tex]
Therefore, the expression representing the length of the box is [tex]\( x + 4 \)[/tex].
Answer:
- [tex]\( x-6 \)[/tex]
- [tex]\( x-4 \)[/tex]
- [tex]\( x+4 \)[/tex] (Correct Answer)
- [tex]\( x+6 \)[/tex]
