Question

What is the similarity ratio of the smaller to the larger similar cones?

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{√(s^2)}{√(s^2)}=\cfrac{\sqrt[3]{s^3}}{\sqrt[3]{s^3}}\\\\ -------------------------------\\\\`


`\bf \cfrac{smaller}{larger}\qquad \cfrac{s}{s}=\cfrac{\sqrt[3]{s}}{\sqrt[3]{s}}\implies \cfrac{s}{s}=\cfrac{\sqrt[3]{250}}{\sqrt[3]{1024}}\implies \cfrac{s}{s}=\cfrac{\sqrt[3]{2\cdot 5^3}}{\sqrt[3]{2^9\cdot 2}} \\\\\\ \cfrac{s}{s}=\cfrac{5\sqrt[3]{2}}{\sqrt[3]{(2^3)^3\cdot 2}}\implies \cfrac{s}{s}=\cfrac{5\sqrt[3]{2}}{8\sqrt[3]{2}}\implies \cfrac{s}{s}=\cfrac{5}{8}`