the two triangular prisms shown are similar. what is the volume of the larger prism? 36 m² 36 m³ 144 m² 144 m³
the two triangular prisms shown are similar. what is the volume of the larger prism? 36 m² 36 m³ 144 m² 144 m³
Answer
Explanation:
Step1: Find the scale - factor
The ratio of the corresponding side lengths of the two similar prisms is $\frac{6}{1.5}=4$.
Step2: Recall the volume - ratio formula for similar solids
If the scale - factor of two similar solids is $k$, the ratio of their volumes is $k^{3}$. Here $k = 4$, so the ratio of the volume of the larger prism to the volume of the smaller prism is $k^{3}=4^{3}=64$. First, find the volume of the smaller prism. The base of the smaller triangular prism is a right - triangle with legs $a = 1.5$m and $b = 1.5$m, and the height of the prism $h = 2$m. The area of the base $B=\frac{1}{2}\times1.5\times1.5=\frac{2.25}{2}$m². The volume of the smaller prism $V_{s}=B\times h=\frac{2.25}{2}\times2 = 2.25$m³. Let the volume of the larger prism be $V_{l}$. Since $\frac{V_{l}}{V_{s}}=k^{3}=64$, then $V_{l}=64\times V_{s}$. Substituting $V_{s}=2.25$m³, we get $V_{l}=64\times2.25 = 144$m³.
Answer:
$144$ m³