Question

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

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 $k^{3}=4^{3}=64$.

Step3: Find the volume of the smaller prism

The volume of a triangular prism $V=\text{Base Area}\times\text{Height}$. The base of the smaller prism is a right - triangle with legs $1.5$ m and $1.5$ m, so the base area $A=\frac{1}{2}\times1.5\times1.5 = 1.125$ m², and the height of the smaller prism is $2$ m. Then the volume of the smaller prism $V_{s}=1.125\times2 = 2.25$ m³.

Step4: Find the volume of the larger prism

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