To determine which expression represents the volume of a right pyramid with a square base and a height that is two inches longer than the base length in terms of [tex]\( x \)[/tex], we can follow these steps:
1. Identify the base area:
- The base of the pyramid is a square with side length [tex]\( x \)[/tex] inches.
- The area of the square base, [tex]\( A \)[/tex], is calculated as:
[tex]\[ A = x^2 \][/tex]
2. Determine the height of the pyramid:
- The height of the pyramid is two inches longer than the base side length.
- Therefore, if the base side length is [tex]\( x \)[/tex], the height [tex]\( h \)[/tex] will be:
[tex]\[ h = x + 2 \][/tex]
3. Apply the volume formula for a pyramid:
- The volume [tex]\( V \)[/tex] of a pyramid with a square base is given by:
[tex]\[ V = \frac{1}{3} \times \text{base area} \times \text{height} \][/tex]
- Substituting the base area [tex]\( A = x^2 \)[/tex] and height [tex]\( h = x + 2 \)[/tex] into the formula:
[tex]\[ V = \frac{1}{3} \times x^2 \times (x + 2) \][/tex]
4. Simplify the expression:
- Simplifying the volume expression:
[tex]\[ V = \frac{1}{3} \times x^2 \times (x + 2) \][/tex]
- This gives:
[tex]\[ V = \frac{x^2 (x + 2)}{3} \][/tex]
Given the answer options:
1. [tex]\( \frac{x^2(x+2)}{3} \)[/tex] cubic inches
2. [tex]\( \frac{x(x+2)}{3} \)[/tex] cubic inches
3. [tex]\( \frac{x^3}{3} + 2 \)[/tex] cubic inches
4. [tex]\( \frac{x^3 + 2}{3} \)[/tex] cubic inches
The correct expression that represents the volume of the pyramid in terms of [tex]\( x \)[/tex] is:
[tex]\[ \boxed{\frac{x^2(x+2)}{3} \text{ cubic inches}} \][/tex]
