Question

the volume of the triangular prism is 54 cubic units. what is the value of x?
○ 3
○ 5
○ 7
○ 9
(there is a diagram of a triangular prism with a right triangle base where one leg is 4, the height of the triangle is x, and the length of the prism is 3x)

Explanation:

Step1: Recall the volume formula for a triangular prism

The volume formula for a triangular prism is \(V = B\times h\), where \(B\) is the area of the triangular base and \(h\) is the height (length) of the prism. The area of a triangle \(B=\frac{1}{2}\times base\times height\). Here, the base of the triangular base is \(4\) and its height is \(x\), so \(B = \frac{1}{2}\times4\times x=2x\). The height (length) of the prism \(h = 3x\).

Step2: Substitute into the volume formula

Substitute \(B = 2x\) and \(h=3x\) into \(V=B\times h\). We get \(V=(2x)\times(3x)=6x^{2}\).

Step3: Solve for \(x\)

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

Answer:

A. 3