Question

The rectangular prism shown below has a base area of x
+ 5x + 6 in
and a height of x+ 5 in. Write the polynomial expression for the volume of the rectangular prism.

Answer 1

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

\[
\text{Volume} = \text{Base Area} \times \text{Height}
\]

In this case, the base area is given as \(x^2 + 5x + 6\) square inches, and the height is \(x + 5\) inches.

To calculate the volume, we can multiply the base area by the height:

\[
\text{Volume} = (x^2 + 5x + 6)(x + 5)
\]

Now, we'll expand this expression using the distributive property (or the FOIL method):

1. Distribute \(x^2\) across \(x + 5\):
\[
x^2 \cdot x + x^2 \cdot 5 = x^3 + 5x^2
\]

2. Distribute \(5x\) across \(x + 5\):
\[
5x \cdot x + 5x \cdot 5 = 5x^2 + 25x
\]

3. Distribute \(6\) across \(x + 5\):
\[
6 \cdot x + 6 \cdot 5 = 6x + 30
\]

Now, combine all of these results to get the total:

\[
x^3 + 5x^2 + 5x^2 + 25x + 6x + 30
\]

Combine like terms:

\[
x^3 + (5x^2 + 5x^2) + (25x + 6x) + 30 = x^3 + 10x^2 + 31x + 30
\]

Thus, the polynomial expression for the volume of the rectangular prism is:

\[
\boxed{x^3 + 10x^2 + 31x + 30}
\]