Question : The ratio of the total surface areas of the two cubes is 49 : 81. What is the ratio of their volumes?
Option 1: 343 : 729
Option 2: 294 : 486
Option 3: 7 : 9
Option 4: 49 : 121
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Correct Answer: 343 : 729
Solution : Given: The ratio of the total surface areas of the two cubes is 49 : 81. The volume of the cube = $x^3$, The total surface area of the cube = $6x^2$ where $x$ is the side of the cube. Let the sides of the cubes be $y$ and $z$, respectively. According to the question, $\frac{6y^2}{6z^2}=\frac{49}{81}$ ⇒ $\frac{6y^2}{6z^2}=\frac{49}{81}$ ⇒ $\frac{y}{z}=\frac{7}{9}$ Let $y=7k$ and $z=9k$ The ratio of their volumes $=\frac{y^3}{z^3}=\frac{7k^3}{9k^3}=\frac{343}{729}=343: 729$ Hence, the correct answer is 343 : 729.
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