To find the volume of a right pyramid with a square base in terms of [tex]\( x \)[/tex], follow these steps:
1. Identify the given dimensions:
- Base length: [tex]\( x \)[/tex] inches.
- Height: Since the height is two inches longer than the base length, it is [tex]\( x + 2 \)[/tex] inches.
2. Write the formula for the volume of a pyramid:
The volume [tex]\( V \)[/tex] of a pyramid is given by:
[tex]\[
V = \frac{1}{3} \times \text{Base Area} \times \text{Height}
\][/tex]
3. Calculate the base area:
The base is a square with a side length of [tex]\( x \)[/tex] inches. Therefore, the area of the base [tex]\( \text{Base Area} \)[/tex] is:
[tex]\[
\text{Base Area} = x \times x = x^2
\][/tex]
4. Substitute the base area and height into the volume formula:
[tex]\[
V = \frac{1}{3} \times x^2 \times (x + 2)
\][/tex]
5. Expand and simplify the expression:
To simplify the expression, multiply [tex]\( x^2 \)[/tex] by [tex]\( (x + 2) \)[/tex]:
[tex]\[
x^2 \times (x + 2) = x^3 + 2x^2
\][/tex]
6. Divide by 3:
[tex]\[
V = \frac{x^3 + 2x^2}{3}
\][/tex]
Therefore, the volume of the pyramid in terms of [tex]\( x \)[/tex] is:
[tex]\[
\frac{x^3 + 2x^2}{3}
\][/tex]
This corresponds to the algebraic expression:
[tex]\[
\frac{x^3}{3} + \frac{2x^2}{3}
\][/tex]
So among the options given, the correct expression representing the volume is:
[tex]\[
\frac{x^3}{3} + \frac{2x^2}{3}
\][/tex]
This confirms that the correct choice is:
[tex]\[
\frac{x^3}{3} + 2x^2/3 \text{ cubic inches}
\][/tex]
