A vessel contains oil (density = 0.8 `gm//cm^(2)`) over mercury (density = 13.6 `gm//cm^(2)`). A homogeneous sphere floats with half of its volume immersed in mercury and the other half in oil. The density of material of the sphere in `gm//cm^(3)` is

Options

A. `3.3`
B. `6.4`
C. `7.2`
D. `2.8`

Explanation

To find the density of the sphere that is floating in the oil and mercury, we will use the principle of buoyancy. The weight of the sphere is equal to the total buoyant force acting on it from both the oil and the mercury.

### Step-by-Step Solution:

1. **Identify the Variables:**
- Density of oil (\( \rho_{oil} \)) = 0.8 g/cm³
- Density of mercury (\( \rho_{mercury} \)) = 13.6 g/cm³
- Let the volume of the sphere be \( V \).

2. **Determine the Volume Immersed:**
- Half of the sphere's volume is immersed in mercury: \( V_{mercury} = \frac{V}{2} \)
- Half of the sphere's volume is immersed in oil: \( V_{oil} = \frac{V}{2} \)

3. **Calculate the Buoyant Force:**
- The buoyant force from mercury (\( F_{buoyant, mercury} \)):
\[ F_{buoyant, mercury} = \rho_{mercury} \cdot V_{mercury} \cdot g = 13.6 \cdot \frac{V}{2} \cdot g \]
- The buoyant force from oil (\( F_{buoyant, oil} \)):
\[ F_{buoyant, oil} = \rho_{oil} \cdot V_{oil} \cdot g = 0.8 \cdot \frac{V}{2} \cdot g \]

4. **Total Buoyant Force:**
- The total buoyant force acting on the sphere is the sum of the buoyant forces from both fluids:
\[ F_{buoyant, total} = F_{buoyant, mercury} + F_{buoyant, oil} \]
\[ F_{buoyant, total} = \left( 13.6 \cdot \frac{V}{2} \cdot g \right) + \left( 0.8 \cdot \frac{V}{2} \cdot g \right) \]
\[ F_{buoyant, total} = \left( 13.6 + 0.8 \right) \cdot \frac{V}{2} \cdot g = 14.4 \cdot \frac{V}{2} \cdot g \]

5. **Weight of the Sphere:**
- The weight of the sphere (\( W \)) is given by:
\[ W = \text{Density of sphere} \cdot V \cdot g \]
Let the density of the sphere be \( \rho_{sphere} \):
\[ W = \rho_{sphere} \cdot V \cdot g \]

6. **Equating Weight and Buoyant Force:**
- Since the sphere is floating, the weight of the sphere is equal to the total buoyant force:
\[ \rho_{sphere} \cdot V \cdot g = 14.4 \cdot \frac{V}{2} \cdot g \]

7. **Cancel Common Terms:**
- We can cancel \( V \) and \( g \) from both sides:
\[ \rho_{sphere} = 14.4 \cdot \frac{1}{2} \]
\[ \rho_{sphere} = 7.2 \text{ g/cm}^3 \]

### Conclusion:
The density of the material of the sphere is **7.2 g/cm³**.