Question

Two similar pyramids have lateral areas 20 ft2 and 45 ft2. The volume of the smaller pyramid is 8 ft3. What is the volume of the larger pyramid?

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{small}{large}\qquad \cfrac{√(20)}{√(45)}=\cfrac{\sqrt[3]{8}}{\sqrt[3]{v}}\implies \cfrac{2\underline{√(5)}}{3\underline{√(5)}}=\cfrac{2}{\sqrt[3]{v}}\implies \cfrac{2}{3}=\cfrac{2}{\sqrt[3]{v}} \\\\\\ \sqrt[3]{v}=3\implies v=3^3\implies v=27`