A concrete sphere of radius `R` has cavity of radius `r` which is packed with sawdust. The specific gravities of concrete and sawdust are respectively `2.4 and 0.3` for this sphere to float with its entire volume submerged under water. Ratio of mass of concrete to mass of swadust will be

Options

A. 8
B. 4
C. 3
D. Zero

Explanation

To solve the problem, we need to find the ratio of the mass of concrete to the mass of sawdust in a concrete sphere with a cavity filled with sawdust, given that the sphere floats with its entire volume submerged in water.

### Step-by-Step Solution:

1. **Understanding the Problem**:
- We have a concrete sphere of radius \( R \) with a cavity of radius \( r \) filled with sawdust.
- The specific gravities of concrete and sawdust are given as \( 2.4 \) and \( 0.3 \) respectively.
- We need to find the ratio of the mass of concrete to the mass of sawdust.

2. **Calculate the Volume of Concrete and Sawdust**:
- The volume of the entire sphere (concrete + cavity) is:
\[ V_{\text{total}} = \frac{4}{3} \pi R^3 \]
- The volume of the cavity (sawdust) is:
\[ V_{\text{cavity}} = \frac{4}{3} \pi r^3 \]
- The volume of concrete is:
\[ V_{\text{concrete}} = V_{\text{total}} - V_{\text{cavity}} = \frac{4}{3} \pi R^3 - \frac{4}{3} \pi r^3 = \frac{4}{3} \pi (R^3 - r^3) \]

3. **Calculate the Mass of Concrete and Sawdust**:
- The density of concrete (\( \rho_c \)) and sawdust (\( \rho_s \)) can be calculated using their specific gravities:
\[ \rho_c = 2.4 \times \rho_{\text{water}} \quad \text{and} \quad \rho_s = 0.3 \times \rho_{\text{water}} \]
- The mass of concrete (\( m_c \)) is:
\[ m_c = V_{\text{concrete}} \times \rho_c = \frac{4}{3} \pi (R^3 - r^3) \times (2.4 \times \rho_{\text{water}}) \]
- The mass of sawdust (\( m_s \)) is:
\[ m_s = V_{\text{cavity}} \times \rho_s = \frac{4}{3} \pi r^3 \times (0.3 \times \rho_{\text{water}}) \]

4. **Finding the Ratio of Mass of Concrete to Mass of Sawdust**:
- The ratio of the mass of concrete to the mass of sawdust is given by:
\[ \frac{m_c}{m_s} = \frac{\frac{4}{3} \pi (R^3 - r^3) \times (2.4 \times \rho_{\text{water}})}{\frac{4}{3} \pi r^3 \times (0.3 \times \rho_{\text{water}})} \]
- Simplifying this expression:
\[ \frac{m_c}{m_s} = \frac{(R^3 - r^3) \times 2.4}{r^3 \times 0.3} \]
- Further simplifying gives:
\[ \frac{m_c}{m_s} = \frac{2.4}{0.3} \times \frac{R^3 - r^3}{r^3} = 8 \times \frac{R^3 - r^3}{r^3} \]

5. **Final Expression**:
- Thus, the ratio of the mass of concrete to the mass of sawdust is:
\[ \frac{m_c}{m_s} = 8 \times \frac{R^3 - r^3}{r^3} \]