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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Question : The surface areas of the two spheres are in the ratio of 64 : 81. Find the ratio of their volumes, in the order given.
Option 1: 512 : 729
Option 2: 64 : 729
Option 3: 8 : 81
Option 4: 4 : 9
Question : If the ratio of the volumes of two cylinders is 7 : 5 and the ratio of their heights is 5 : 7, then what is the ratio of their radii?
Option 1: 5 : 7
Option 2: 3 : 4
Option 3: 5 : 9
Option 4: 4 : 5
Question : If the heights of two cylinders are in the ratio of 7 : 9 and their diameters are in the ratio of 6 : 7, then what is the ratio of the volumes of the cylinders?
Option 1: 6 : 7
Option 2: 7 : 9
Option 4: 4 : 7
Question : If the surface area of two spheres is in the ratio 81 : 25, then what is the ratio of their radius?
Option 1: 5 : 8
Option 3: 5 : 7
Option 4: 9 : 5
Question : If the ratio of areas of two circles is 4 : 9, then the ratio of their circumferences will be:
Option 1: 3 : 2
Option 2: 9 : 4
Option 3: 2 : 3
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