Question

the volume of the triangular prism is 54 cubic units. what is the value of x?
3
5
7
9

Explanation:

Step1: Find the formula for the volume of a triangular prism

The volume \(V\) of a triangular prism is \(V = (\text{Area of the triangular base})\times(\text{height of the prism})\). The area of a triangle \(A=\frac{1}{2}\times\text{base}\times\text{height}\). Here, the base of the triangle is \(4\) and the height of the triangle is \(x\), so \(A = \frac{1}{2}\times4\times x=2x\). The height of the prism (the length along the direction perpendicular to the triangular base) is \(3x\). Then \(V=A\times h=(2x)\times(3x)\).

Step2: Set up the equation and solve for \(x\)

Since \(V = 54\), we have the equation \((2x)\times(3x)=54\). Simplify the left - hand side: \(6x^{2}=54\). Divide both sides by \(6\): \(x^{2}=\frac{54}{6}=9\). Take the square root of both sides: \(x = 3\) (we consider the positive value since \(x\) represents a length).

Answer:

3