Question

in the right triangular prism below, \\( \overline{bg} \perp \overline{fc} \\) the volume of the prism is 900 cubic centimeters. the height of the prism is 15 centimeters and the height of the triangular base is 10 centimeters. what is the length of edge fc? \\( \square \\) cm

Explanation:

Step1: Recall the volume formula for a prism

The volume formula for a prism is \(V = B\times h\), where \(B\) is the area of the base and \(h\) is the height of the prism. Given \(V = 900\space cm^{3}\) and \(h=15\space cm\). First, find the area of the triangular base.
From \(V = B\times h\), we can solve for \(B\): \(B=\frac{V}{h}\).
Substitute \(V = 900\) and \(h = 15\) into the formula: \(B=\frac{900}{15}=60\space cm^{2}\).

Step2: Use the area formula for a triangle

The area formula for a triangle is \(B=\frac{1}{2}\times base\times height\). For the triangular base, let the base be \(FC\) (denoted as \(x\)) and the height of the triangle \(BG = 10\space cm\).
We know \(B = 60\space cm^{2}\), and \(B=\frac{1}{2}\times x\times10\).
Set up the equation: \(\frac{1}{2}\times x\times10=60\).
Simplify the left - hand side: \(5x = 60\).
Solve for \(x\): \(x=\frac{60}{5}=12\).

Answer:

\(12\)