Question

the oblique prism below has an isosceles right triangle base.
what expression represents the volume of the prism, in 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 formula for the area of 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 \(V=\frac{1}{2}x^{2}\times(x + 3)\).
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:

\(\frac{1}{2}x^{3}+\frac{3}{2}x^{2}\) (the second option)