Question

The volume of a rectangular prism is represented by the function x3 + 9x2 + 6x − 16. The length of the box is x + 2, while the height is x + 8. Find the expression representing the width of the box.

x − 4
x − 1
x + 1
x + 4

Answer 1

To find the expression representing the width of the box, we can use the formula for the volume of a rectangular prism, which is given by:

\[
\text{Volume} = \text{Length} \times \text{Width} \times \text{Height}
\]

In this case, the volume is given by \( V(x) = x^3 + 9x^2 + 6x - 16 \), the length is \( L = x + 2 \), and the height is \( H = x + 8 \).

Let the width be represented by \( W \). Thus, we have:

\[
V(x) = (x + 2)(W)(x + 8)
\]

We can rearrange the equation to isolate \( W \):

\[
W = \frac{V(x)}{(x + 2)(x + 8)}
\]

Now, we need to first determine \( (x + 2)(x + 8) \):

\[
(x + 2)(x + 8) = x^2 + 8x + 2x + 16 = x^2 + 10x + 16
\]

Next, we will perform polynomial long division to divide \( V(x) = x^3 + 9x^2 + 6x - 16 \) by \( x^2 + 10x + 16 \).

1. Divide the leading term of \( V(x) \) by the leading term of \( x^2 + 10x + 16 \):
\[
\frac{x^3}{x^2} = x.
\]
2. Multiply \( x \) by the entire polynomial \( x^2 + 10x + 16 \):
\[
x(x^2 + 10x + 16) = x^3 + 10x^2 + 16x.
\]
3. Subtract this from \( V(x) \):
\[
(x^3 + 9x^2 + 6x - 16) - (x^3 + 10x^2 + 16x) = (9x^2 - 10x^2) + (6x - 16x) - 16 = -x^2 - 10x - 16.
\]
4. Now divide the leading term of the remainder \( -x^2 \) by the leading term of \( x^2 + 10x + 16 \):
\[
\frac{-x^2}{x^2} = -1.
\]
5. Multiply this result by \( x^2 + 10x + 16 \):
\[
-1(x^2 + 10x + 16) = -x^2 - 10x - 16.
\]
6. Subtract:
\[
(-x^2 - 10x - 16) - (-x^2 - 10x - 16) = 0.
\]

Thus, we find:

\[
W = x - 1.
\]

The width of the box is represented by the expression:

\[
\boxed{x - 1}.
\]