To find the expression that represents the volume of a right pyramid with a square base in terms of [tex]\( x \)[/tex], let's follow these steps:
1. Identify the given dimensions:
- The base length of the square is [tex]\( x \)[/tex] inches.
- The height of the pyramid is [tex]\( x + 2 \)[/tex] inches.
2. Formula for the volume of a pyramid:
The volume [tex]\( V \)[/tex] of a pyramid with a square base is given by:
[tex]\[
V = \frac{1}{3} \cdot \text{base area} \cdot \text{height}
\][/tex]
3. Calculate the base area:
Since the base is a square with side length [tex]\( x \)[/tex], the area of the base is:
[tex]\[
\text{Base area} = x^2
\][/tex]
4. Substitute the base area and the height into the volume formula:
The height of the pyramid is [tex]\( x + 2 \)[/tex]. Substituting the values, the volume becomes:
[tex]\[
V = \frac{1}{3} \cdot x^2 \cdot (x + 2)
\][/tex]
5. Simplify the expression:
Combine terms to express the volume:
[tex]\[
V = \frac{x^2 (x + 2)}{3}
\][/tex]
Therefore, the correct expression for the volume of the right pyramid with a square base in terms of [tex]\( x \)[/tex] is:
[tex]\[
\frac{x^2 (x + 2)}{3} \text{ cubic inches}
\][/tex]
The choice matching this expression is:
[tex]\(\frac{x^2(x+2)}{3}\)[/tex] cubic inches.
