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
Correct Answer: 512 : 729
Solution :
let $r_1$ and $r_2$ be the radius of the sphere.
Given: $\frac{\text{surface area of 1st sphere}}{\text{surface area of 2nd sphere}} = \frac{64}{81}$
⇒ $\frac{4\pi r_1^2}{4\pi r_2^2} = \frac{64}{81}$
⇒ $\frac{r_1}{r_2}=\frac{8}{9}$
Now, $\frac{\text{The volume of 1st sphere}}{\text{The volume of 2nd sphere}} = \frac{\frac{4}{3}\pi r_1^3}{\frac{4}{3}\pi r_2^3} = \frac{r_1^3}{r_2^3}$
Putting the value of $\frac{r_1}{r_2}=\frac{8}{9}$ in above equation, we get:
$\frac{\text{The volume of 1st sphere}}{\text{The volume of 2nd sphere}}= (\frac{8}{9})^3= \frac{512}{729}$
Hence, the correct answer is 512 : 729.
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