Question

A design on the surface of a balloon is 9 cm wide when the balloon holds 62 cm3 of air. How much air does the balloon hold when the design is 18 cm wide?

Answer 1

`\bf \qquad \qquad \textit{ratio relations} \\\\ \begin{array}{cccllll} &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^2)}{√(s^2)}=\cfrac{\sqrt[3]{s^3}}{\sqrt[3]{s^3}}\implies \cfrac{√(9)}{√(18)}=\cfrac{\sqrt[3]{62}}{\sqrt[3]{x}}`


`\bf \\\\\\ \cfrac{3}{3√(2)}=\cfrac{\sqrt[3]{62}}{\sqrt[3]{x}}\implies \cfrac{1}{√(2)}=\cfrac{\sqrt[3]{62}}{\sqrt[3]{x}}\implies \sqrt[3]{x}=√(2)\cdot \sqrt[3]{62} \\\\\\ x=\left( √(2)\cdot \sqrt[3]{62} \right)^3\implies x=√(2^3)\cdot \sqrt[3]{62^3}\implies x=2√(2)\cdot 62 \\\\\\ \boxed{x=124√(2)}`