To determine the volume of a right triangular prism, we need to consider the formula for the volume of a prism, which is:
[tex]\[ \text{Volume} = \text{Base Area} \times \text{Height} \][/tex]
In this problem, the base of the prism is a right triangle with leg lengths \( x \). The height of the prism is given to be equal to the leg length of the base, which is also \( x \).
### Step-by-Step Solution:
1. Determine the base area of the right triangle:
- For a right triangle with legs of length \( x \), the area is given by:
[tex]\[ \text{Base Area} = \frac{1}{2} \times \text{leg} \times \text{leg} = \frac{1}{2} \times x \times x = \frac{1}{2} x^2 \][/tex]
2. Determine the height of the prism:
- The height of the prism is equal to the leg length, so \( \text{Height} = x \).
3. Calculate the volume of the prism:
- Using the formula for volume:
[tex]\[ \text{Volume} = \text{Base Area} \times \text{Height} \][/tex]
- Substitute the base area and the height we found:
[tex]\[ \text{Volume} = \left(\frac{1}{2} x^2\right) \times x = \frac{1}{2} x^2 \times x = \frac{1}{2} x^3 \][/tex]
Therefore, the expression that represents the volume of the prism is:
[tex]\[ \boxed{\frac{1}{2} x^3} \][/tex]
