Question

if the surface area of two smilar spheres is 256 feet and 576 feet, what is the ratio of the volume of the smaller sphere to the volume of the largee sphere

Answer 1

`\bf \qquad \qquad \textit{ratio relations} \\\\ \begin{array}{ccccllll} &\stackrel{ratio~of~the}{Sides}&\stackrel{ratio~of~the}{Areas}&\stackrel{ratio~of~the}{Volumes}\\ &-----&-----&-----\\ \cfrac{\textit{similar shape}}{\textit{similar shape}}&\cfrac{s}{s}&\cfrac{s^2}{s^2}&\cfrac{s^3}{s^3} \end{array}\\\\ -----------------------------`


`\bf \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}}\\\\ -------------------------------\\\\ \cfrac{smaller}{larger}\qquad \cfrac{s}{s}=\cfrac{√(256)}{√(576)}\implies \cfrac{s}{s}=\cfrac{16}{24}\implies \cfrac{s}{s}=\stackrel{simplified}{\cfrac{2}{3}}`