Answer:
Part 1)
The scale factor of the smaller prism to the larger prism is 3
or
The scale factor of the larger prism to the smaller prism is 1/3
Part 2) The surface area of the smaller prism is 66 square centimeters
Step-by-step explanation:
Part 1) What is the scale factor for the similar figures?
we know that
If two figures are similar, then the ratio of their volumes is equal to the scale factor raised to the cube
Let
z ----> the scale factor
x ---> volume of the larger prism
y ---> volume of the smaller prism
so
[tex]z^3=\frac{x}{y}[/tex]
we have
[tex]x=972\ cm^3\\y=36\ cm^3[/tex]
substitute
[tex]z^3=\frac{972}{36}[/tex]
Simplify
[tex]z^3=27\\z=3[/tex]
therefore
The scale factor of the smaller prism to the larger prism is 3
or
The scale factor of the larger prism to the smaller prism is 1/3
Part 2) What is the surface area of the smaller prism?
we know that
If two figures are similar, then the ratio of their surface areas is equal to the scale factor squared
Let
z ----> the scale factor
x ---> surface area of the larger prism
y ---> surface area of the smaller prism
so
[tex]z^2=\frac{x}{y}[/tex]
we have
[tex]x=594\ cm^2[/tex]
[tex]z=3[/tex]
substitute
[tex]3^2=\frac{594}{y}[/tex]
solve for y
[tex]y=\frac{594}{9}=66\ cm^2[/tex]
therefore
The surface area of the smaller prism is 66 square centimeters
