Answer:
A) ![\frac{1}{2}x^3](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7B2%7Dx%5E3)
Step-by-step explanation:
The volume of a triangular prism can be found using the formula
(1)
where
A is the area of the base
h is the height of the prism
Here, the base is a right triangle; the area of a triangle is
![A=\frac{1}{2}bh'](https://tex.z-dn.net/?f=A%3D%5Cfrac%7B1%7D%7B2%7Dbh%27)
where
b is the base
h' is the height of the triangle
Here, the triangle has the two sides equal, and it is a right triangle, so we have
![b=h=x](https://tex.z-dn.net/?f=b%3Dh%3Dx)
So the area of the base is
![A=\frac{1}{2}x\cdot x = \frac{1}{2}x^2](https://tex.z-dn.net/?f=A%3D%5Cfrac%7B1%7D%7B2%7Dx%5Ccdot%20x%20%3D%20%5Cfrac%7B1%7D%7B2%7Dx%5E2)
We also know that the height of the prism is equal to the leg length of the base, so
![h=x](https://tex.z-dn.net/?f=h%3Dx)
Therefore substituting into (1) we find:
![V=\frac{1}{2}x^2 \cdot x = \frac{1}{2}x^3](https://tex.z-dn.net/?f=V%3D%5Cfrac%7B1%7D%7B2%7Dx%5E2%20%5Ccdot%20x%20%3D%20%5Cfrac%7B1%7D%7B2%7Dx%5E3)