Question

the two triangular prisms shown are similar.
what is the volume of the larger prism?
2 m
1.5 m
1.5 m
6 m
36 m²
36 m³
144 m²
144 m³

Explanation:

Step1: Find the scale factor

The scale factor \(k\) of the side lengths is \(\frac{6}{1.5}=4\).

Step2: Use the volume - scale factor relationship

For similar solids, if the scale factor of side lengths is \(k\), the scale factor of volumes is \(k^{3}\).
First, find the volume of the smaller prism. The base of the smaller triangular prism is a right - triangle with base \(b = 1.5\)m and height \(h=1.5\)m. The area of the base \(B_{1}=\frac{1}{2}\times1.5\times1.5=\frac{2.25}{2}\text{ m}^2\), and the height of the prism \(H_{1}=2\)m. So the volume of the smaller prism \(V_{1}=B_{1}H_{1}=\frac{1}{2}\times1.5\times1.5\times2 = 2.25\text{ m}^3\).
Since \(k = 4\), the volume of the larger prism \(V_{2}=V_{1}\times k^{3}\).
Substitute \(V_{1}=2.25\) and \(k = 4\) into the formula: \(V_{2}=2.25\times4^{3}\).
Calculate \(4^{3}=64\), then \(V_{2}=2.25\times64 = 144\text{ m}^3\).

Answer:

\(144\text{ m}^3\) (the fourth option)