Question

The dimensions of a certain right rectangular prism are described in terms of its width x. The length of the prism is (x – 4), and the height is represented by (x + 6). Which of the following expresses the volume of the prism in terms of x?

Answer 1

To find the volume of a right rectangular prism, you can use the formula:

\[
\text{Volume} = \text{length} \times \text{width} \times \text{height}
\]

In this case, the dimensions of the prism are given as follows:
- Width = \( x \)
- Length = \( x - 4 \)
- Height = \( x + 6 \)

Now plug these dimensions into the volume formula:

\[
\text{Volume} = (x - 4) \times x \times (x + 6)
\]

Next, we need to simplify this expression. We can start by multiplying the first two terms:

\[
(x - 4) \times x = x^2 - 4x
\]

Now, we can multiply this result by the height term \((x + 6)\):

\[
\text{Volume} = (x^2 - 4x) \times (x + 6)
\]

To perform this multiplication, we can use the distributive property (also known as the FOIL method for binomials):

\[
\begin{align*}
\text{Volume} & = (x^2 - 4x)(x) + (x^2 - 4x)(6) \\
& = x^3 - 4x^2 + 6x^2 - 24x \\
& = x^3 + (6x^2 - 4x^2) - 24x \\
& = x^3 + 2x^2 - 24x
\end{align*}
\]

Therefore, the volume of the prism in terms of \( x \) is:

\[
\text{Volume} = x^3 + 2x^2 - 24x
\]