Question

a right triangular prism is constructed so that its height is equal to the leg length of the base. what expression represents the volume of the prism, in cubic units? options: \\(\frac{1}{2}x^3\\), \\(\frac{1}{2}x^2 + x\\), \\(2x^3\\), \\(2x^2 + x\\)

Explanation:

Step1: Find the area of the triangular base

The formula for the area of a triangle is \(A=\frac{1}{2}\times base\times height\). For the right - angled triangular base, if the legs (base and height of the triangle) are both \(x\), then \(A = \frac{1}{2}\times x\times x=\frac{1}{2}x^{2}\).

Step2: Use the volume formula for a prism

The volume formula for a prism is \(V=A\times h\), where \(A\) is the area of the base and \(h\) is the height of the prism. The height of the prism \(h = x\) (given that the height of the prism is equal to the leg length of the base). Substitute \(A=\frac{1}{2}x^{2}\) and \(h = x\) into the formula: \(V=\frac{1}{2}x^{2}\times x\).
Using the rule of exponents \(a^{m}\times a^{n}=a^{m + n}\) (here \(m = 2\) and \(n=1\)), we get \(V=\frac{1}{2}x^{3}\).

Answer:

\(\frac{1}{2}x^{3}\)