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
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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