A sphere of solid material of relative density 9 has a concentric spherical cavity and sinks in water. If te radius of the sphere be R. Then the radius of the cavity (r) will be related to R as

Options

A. `r^(3)=(8)/(9)R^(3)`
B. `r^(3)=(2)/(3)R^(3)`
C. `r^(3)=(sqrt(8))/(3)R^(3)`
D. `r^(3)=sqrt((2)/(3))R^(3)`

Explanation

To solve the problem, we need to relate the radius of the cavity (r) to the radius of the sphere (R) given that the sphere sinks in water and has a relative density of 9. Here are the steps to derive the relationship:

### Step 1: Understand the Concept of Relative Density
Relative density (RD) is defined as the ratio of the density of a substance to the density of water. For this problem, the relative density of the sphere is given as 9. This means:
\[ \text{Relative Density} = \frac{\text{Density of Sphere (d)}}{\text{Density of Water} (\rho_w)} = 9 \]
Thus, we can express the density of the sphere as:
\[ d = 9 \cdot \rho_w \]

...