A right pyramid with a square base has a base length of [tex]\( x \)[/tex] inches, and the height is two inches longer than the length of the base. Which expression represents the volume in terms of [tex]\( x \)[/tex]?

A. [tex]\(\frac{x^2(x+2)}{3}\)[/tex] cubic inches
B. [tex]\(\frac{x(x+2)}{3}\)[/tex] cubic inches
C. [tex]\(\frac{x^3}{3}+2\)[/tex] cubic inches
D. [tex]\(\frac{x^3+2}{3}\)[/tex] cubic inches



Answer :

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.