Answer:
Part 1) [tex]\frac{a^2}{b^2}=\frac{4}{9}[/tex]
Part 2) [tex]\frac{a}{b}=\frac{2}{3}[/tex]
Part 3) [tex]\frac{a^3}{b^3}=\frac{8}{27}[/tex]
Step-by-step explanation:
Part 1) Find the ratio Area I/Area II
we know that
If two figures are similar, then the ratio of its areas is equal to the scale factor squared
The ratio of the areas is equal to divide the surface area cylinder I by the surface area cylinder II
Let
a^2 -----> the surface area cylinder I
b^2 ----> the surface area cylinder II
we have
[tex]a^2= 8\pi\ in^2[/tex]
[tex]b^2= 18\pi\ in^2[/tex]
Find the ratio
[tex]\frac{a^2}{b^2}=\frac{8\pi }{18\pi}[/tex]
Simplify
[tex]\frac{a^2}{b^2}=\frac{4}{9}[/tex]
That means
[tex]\frac{a^2}{b^2}=\frac{4}{9}[/tex] --->[tex]4b^2=9a^2[/tex]
Four times area cylinder II is equal to nine times the surface area of cylinder I.
Part 2) Find the ratio a/b
we know that
If two figures are similar, then the ratio of its corresponding sides is equal to the scale factor.
In this problem
[tex]\frac{r_1}{r_2}=\frac{h_1}{h_2}=\frac{a}{b}[/tex] ----> scale factor
we have
[tex]\frac{a^2}{b^2}=\frac{4}{9}[/tex]
so
square root both sides
[tex]\frac{a}{b}=\frac{2}{3}[/tex]
That means
[tex]\frac{r_1}{r_2}=\frac{2}{3}[/tex] --->[tex]2r_2=3r_1[/tex]
Two times radius cylinder II (r_2) is equal to three times radius cylinder I (r_1)
[tex]\frac{h_1}{h_2}=\frac{2}{3}[/tex] --->[tex]2h_2=3h_1[/tex]
Two times height cylinder II is equal to three times height cylinder I
Part 3) Find the ratio Volume I/Volume II
we know that
If two figures are similar, then the ratio of its volumes is equal to the scale factor elevated to the cube
we have
[tex]\frac{a}{b}=\frac{2}{3}[/tex] ----> scale factor
[tex]\frac{Volume\ I}{Volume\ II}=\frac{a^3}{b^3}[/tex]
substitute the values
[tex]\frac{Volume\ I}{Volume\ II}=\frac{2^3}{3^3}[/tex]
[tex]\frac{Volume\ I}{Volume\ II}=\frac{8}{27}[/tex]
[tex]8Volume\ II=27Volume\ I[/tex]
8 times volume cylinder II is equal to 27 times volume cylinder I
