Respuesta :
[tex]\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{\sqrt{s^2}}{\sqrt{s^2}}=\cfrac{\sqrt[3]{s^3}}{\sqrt[3]{s^3}}\\\\
-------------------------------\\\\[/tex]
[tex]\bf \cfrac{\sqrt{s^2}}{\sqrt{s^2}}=\cfrac{\sqrt[3]{s^3}}{\sqrt[3]{s^3}}\implies \cfrac{\sqrt{450}}{\sqrt{578}}=\cfrac{\sqrt[3]{511}}{\sqrt[3]{v}}\implies \cfrac{15\sqrt{2}}{17\sqrt{2}}=\cfrac{\sqrt[3]{511}}{\sqrt[3]{v}} \\\\\\ \cfrac{15}{17}=\cfrac{\sqrt[3]{511}}{\sqrt[3]{v}}\implies \sqrt[3]{v}=\cfrac{17\sqrt[3]{511}}{15}\implies v=\left( \cfrac{17\sqrt[3]{511}}{15} \right)^3 \\\\\\ v=\cfrac{17^3\cdot 511}{15^3}[/tex]
[tex]\bf \cfrac{\sqrt{s^2}}{\sqrt{s^2}}=\cfrac{\sqrt[3]{s^3}}{\sqrt[3]{s^3}}\implies \cfrac{\sqrt{450}}{\sqrt{578}}=\cfrac{\sqrt[3]{511}}{\sqrt[3]{v}}\implies \cfrac{15\sqrt{2}}{17\sqrt{2}}=\cfrac{\sqrt[3]{511}}{\sqrt[3]{v}} \\\\\\ \cfrac{15}{17}=\cfrac{\sqrt[3]{511}}{\sqrt[3]{v}}\implies \sqrt[3]{v}=\cfrac{17\sqrt[3]{511}}{15}\implies v=\left( \cfrac{17\sqrt[3]{511}}{15} \right)^3 \\\\\\ v=\cfrac{17^3\cdot 511}{15^3}[/tex]
Answer:
The volume of the larger solid is 743.86 cubic mm.
Step-by-step explanation:
The surface areas of two similar solids are 450 mm squared and 578 mm squared.
Since the solids are similar, the ratio of the surface areas is equal to the square of the scale factor, k.
[tex]k^{2}= \frac{578}{450}[/tex] = [tex]\frac{289}{225}[/tex]
[tex]k=\sqrt{\frac{289}{225}}[/tex]
[tex]k=\frac{17}{15}[/tex]
The ratio of the volumes of the solids is equal to [tex]k^{3}[/tex], so further we can solve as:
[tex]\frac{V}{511} =(\frac{17}{15})^{3}[/tex]
[tex]\frac{V}{511} =\frac{4913}{3375}[/tex]
[tex]V =\frac{4913*511}{3375}[/tex]
V=743.86 cubic mm.
