Question

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

Explanation:

Step1: Calculate the area of the base

The base is an isosceles right - triangle. The area formula for a triangle is $A=\frac{1}{2}\times base\times height$. Here, base = height = $x$, so $A_{base}=\frac{1}{2}\times x\times x=\frac{1}{2}x^{2}$.

Step2: Use the volume formula for a prism

The volume formula for a prism is $V = A_{base}\times h$, where $h$ is the height (or length) 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 $V=\frac{1}{2}x^{2}\times x+\frac{1}{2}x^{2}\times3=\frac{1}{2}x^{3}+\frac{3}{2}x^{2}$.

Answer:

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