Mathematics

Question

welp! help! brainly deleted my last question so here's this!
welp! help! brainly deleted my last question so here's this!

1 Answer

  • 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

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