Question

The cylinders are similar. The volume of the larger cylinder is 2264 cubic centimeters. What is the volume of the smaller cylinder?

Answer 1

$\bf ~\hspace{5em} \textit{ratio relations of two similar shapes} \\[2em] \begin{array}{ccccllll} &\stackrel{ratio~of~the}{Sides}&\stackrel{ratio~of~the}{Areas}&\stackrel{ratio~of~the}{Volumes}\\ \cline{2-4}&\\ \cfrac{\textit{similar shape}}{\textit{similar shape}}&\cfrac{s}{s}&\cfrac{s^2}{s^2}&\cfrac{s^3}{s^3} \end{array}\\\\[-0.35em] ~\dotfill$

$\bf \cfrac{\textit{similar shape}}{\textit{similar shape}}\qquad \cfrac{s}{s}=\cfrac{\sqrt{s^2}}{\sqrt{s^2}}=\cfrac{\sqrt[3]{s^3}}{\sqrt[3]{s^3}}\\\\[-0.35em] \rule{34em}{0.25pt}\\\\ \cfrac{s}{s}=\cfrac{4}{8}\implies \cfrac{s}{s}=\cfrac{1}{2}~\hspace{7em}\cfrac{1}{2}=\cfrac{\sqrt[3]{x}}{\sqrt[3]{2264}}\implies \cfrac{1}{2}=\sqrt[3]{\cfrac{x}{2264}} \\\\\\ \left( \cfrac{1}{2} \right)^3=\cfrac{x}{2264}\implies \cfrac{1}{8}=\cfrac{x}{2264}\implies \cfrac{2264}{8}=x\implies 283=x$