Question

The height and radius of a cylinder are both enlarged by the same factor.  The surface area of the new cylinder is 25 times the surface area of the original cylinder.  By what factor were the original height and the original radius enlarged?

Answer 1

$\bf \qquad \qquad \textit{ratio relations} \\\\ \begin{array}{ccccllll} &Sides&Area&Volume\\ &-----&-----&-----\\ \cfrac{\textit{similar shape}}{\textit{similar shape}}&\cfrac{s}{s}&\cfrac{s^2}{s^2}&\cfrac{s^3}{s^3} \end{array} \\\\ -----------------------------\\\\ \cfrac{\textit{similar shape}}{\textit{similar shape}}\qquad \cfrac{s}{s}=\cfrac{\sqrt{s^2}}{\sqrt{s^2}}=\cfrac{\sqrt[3]{s^3}}{\sqrt[3]{s^3}}\\\\ -------------------------------$

$\bf \textit{let's enlarge them both radius and height by "k"} \\\\\\ \cfrac{\textit{smaller cylinder}}{\textit{larger cylindder}}\qquad \cfrac{r}{kr}=\cfrac{h}{kh}=\cfrac{1}{k}\impliedby ratio \\\\\\ \cfrac{1^2}{k^2}=\cfrac{\textit{original area}}{\textit{25 times original}}\implies \cfrac{1}{k^2}=\cfrac{a}{25a}\implies \cfrac{1}{k^2}=\cfrac{1}{25} \\\\\\ 25=k^2\implies \sqrt{25}=k$