Question

the oblique prism below has an isosceles right triangle base. what expression represents the volume of the prism, in cubic units? o $\frac{1}{2}x^{3}+x^{2}$ o $\frac{1}{2}x^{3}+\frac{3}{2}x^{2}$ o $x^{3}+x^{2}$ o $x^{3}+3x^{2}$

Explanation:

Step1: Find the area of the base

The base is an isosceles right - triangle with legs of length $x$. The area formula for a triangle is $A=\frac{1}{2}bh$. Here, $b = x$ and $h = x$, so $A=\frac{1}{2}x\cdot x=\frac{1}{2}x^{2}$.

Step2: Find the volume of the prism

The volume formula for a prism is $V = Ah$, where $A$ is the area of the base and $h$ is the height of the prism. The height of the prism is $(x + 3)$. So $V=\frac{1}{2}x^{2}(x + 3)$.

Step3: Expand the expression

Using the distributive property $a(b + c)=ab+ac$, we have $V=\frac{1}{2}x^{2}\cdot x+\frac{1}{2}x^{2}\cdot3=\frac{1}{2}x^{3}+\frac{3}{2}x^{2}$.

Answer:

$\frac{1}{2}x^{3}+\frac{3}{2}x^{2}$