To determine how the volumes of two similar right triangular prisms compare given a scale factor of [tex]\(\frac{3}{2}\)[/tex], we need to understand a few important concepts about similar solids:
1. Scale Factor: This is the ratio of corresponding linear dimensions (such as height, width, or length) of two similar figures. Here, the scale factor from the smaller prism to the larger prism is [tex]\(\frac{3}{2}\)[/tex].
2. Volume Ratio: For similar three-dimensional solids, the volumes change in proportion to the cube of the scale factor. If the scale factor between the linear dimensions is [tex]\(\frac{a}{b}\)[/tex], the ratio of the volumes is [tex]\((\frac{a}{b})^3\)[/tex].
Given the scale factor [tex]\(\frac{3}{2}\)[/tex], we calculate the volume ratio as follows:
[tex]\[
\left(\frac{3}{2}\right)^3
\][/tex]
Let's break this down step-by-step:
1. Cubing the Numerator and Denominator: We compute the cube of 3 and the cube of 2 separately.
[tex]\[
3^3 = 27
\][/tex]
[tex]\[
2^3 = 8
\][/tex]
2. Form the Volume Ratio: Now, we form the ratio of these cubed values.
[tex]\[
\left(\frac{3}{2}\right)^3 = \frac{27}{8}
\][/tex]
Therefore, the volume of the larger prism is [tex]\(\frac{27}{8}\)[/tex] times the volume of the smaller prism. In simplified form, the fraction [tex]\(\frac{27}{8}\)[/tex] can also be expressed as the decimal 3.375.
So, the correct way the volume changes with the given scale factor is:
[tex]\[
\frac{3^3}{2^3} = \frac{27}{8} = 3.375
\][/tex]
This means the volume of the larger prism is 3.375 times the volume of the smaller prism.
