Answer: The correct option is (B) 8 : 343.
Step-by-step explanation: We are given that the ratio for the radius of two similar spheres is 2 : 7.
We are to find the ratio of their volumes.
We know that
the VOLUME of a sphere with radius r units is given by
[tex]V=\dfrac{4}{3}\pi r^3.[/tex]
Let r and r' be the radii of the given similar spheres.
Then,
[tex]r:r'=2:7\\\\\Rightarrow \dfrac{r}{r'}=\dfrac{2}{7}~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(i)[/tex]
Now, if V and V' represents their corresponding volumes, then we get
[tex]\dfrac{V}{V'}\\\\\\=\dfrac{\dfrac{4}{3}\pi r^3}{\dfrac{4}{3}\pi r'^3}\\\\\\=\dfrac{r^3}{r'^3}\\\\\\=\left(\dfrac{r}{r'}\right)^3\\\\\\=\left(\dfrac{2}{7}\right)^3~~~~~~~~~~~~~~~~~~~~~~[\textup{Using equation (i)}]\\\\\\=\dfrac{8}{343}\\\\\\=8:343.[/tex]
Thus, the required ratio of the volumes is 8 : 343.
Option (B) is CORRECT.
