[tex]\bf \qquad \qquad \textit{ratio relations}
\\\\
\begin{array}{ccccllll}
&Sides&Area&Volume\\
&-----&-----&-----\\
\cfrac{\textit{similar shape}}{\textit{similar shape}}&\cfrac{s}{s}&\cfrac{s^2}{s^2}&\cfrac{s^3}{s^3}
\end{array} \\\\
-----------------------------\\\\
\cfrac{\textit{similar shape}}{\textit{similar shape}}\qquad \cfrac{s}{s}=\cfrac{\sqrt{s^2}}{\sqrt{s^2}}=\cfrac{\sqrt[3]{s^3}}{\sqrt[3]{s^3}}\\\\
-------------------------------\\\\[/tex]
[tex]\bf \textit{let's enlarge them both radius and height by "k"}
\\\\\\
\cfrac{\textit{smaller cylinder}}{\textit{larger cylindder}}\qquad \cfrac{r}{kr}=\cfrac{h}{kh}=\cfrac{1}{k}\impliedby ratio
\\\\\\
\cfrac{1^2}{k^2}=\cfrac{\textit{original area}}{\textit{25 times original}}\implies \cfrac{1}{k^2}=\cfrac{a}{25a}\implies \cfrac{1}{k^2}=\cfrac{1}{25}
\\\\\\
25=k^2\implies \sqrt{25}=k[/tex]
