A cube of iron of edge `5 cm` floats on the surface of mercury, contained in a tank. Water is then poured on top, so that the cube just gets immersed. Find the depth of water layer. (Specific gravities of iron-and mercury are `7.8` and `13.6`, respectively.)

Explanation

To solve the problem of finding the depth of the water layer above the iron cube that is floating on mercury, we will follow these steps:

### Step-by-Step Solution

1. **Understand the Problem**:
We have a cube of iron with an edge of 5 cm floating on mercury. When water is poured on top, the cube becomes fully immersed. We need to find the depth of the water layer (denoted as \( x \)).

2. **Calculate the Volume of the Iron Cube**:
The volume \( V \) of the cube can be calculated using the formula for the volume of a cube:
\[ V = \text{edge}^3 = (5 \, \text{cm})^3 = 125 \, \text{cm}^3 \]

3. **Calculate the Weight of the Iron Cube**:
The weight \( W \) of the iron cube can be calculated using the formula:
\[ W = \text{Volume} \times \text{Density} \times g \]
The density of iron can be found using its specific gravity:
\[ \text{Density of iron} = \text{Specific gravity} \times \text{Density of water} = 7.8 \times 1 \, \text{g/cm}^3 = 7.8 \, \text{g/cm}^3 \]
Therefore, the weight of the cube is:
\[ W = 125 \, \text{cm}^3 \times 7.8 \, \text{g/cm}^3 \times g = 975 \, \text{g} \times g \]

4. **Set Up the Buoyant Force Equation**:
The buoyant force \( F_b \) acting on the cube is equal to the weight of the fluid displaced by the cube. The total buoyant force is the sum of the buoyant forces from both the water and the mercury:
\[ F_b = F_{\text{water}} + F_{\text{mercury}} \]
Where:
- \( F_{\text{water}} = \rho_{\text{water}} \cdot V_{\text{submerged, water}} \cdot g = 1 \cdot (5^2 \cdot x) \cdot g = 25x \cdot g \)
- \( F_{\text{mercury}} = \rho_{\text{mercury}} \cdot V_{\text{submerged, mercury}} \cdot g = 13.6 \cdot (5^2 \cdot (5 - x)) \cdot g = 340(5 - x) \cdot g \)

5. **Equate the Weight and Buoyant Forces**:
Setting the total buoyant force equal to the weight of the cube:
\[ 975g = 25xg + 340(5 - x)g \]
Dividing through by \( g \):
\[ 975 = 25x + 340(5 - x) \]

6. **Solve for \( x \)**:
Expanding and rearranging the equation:
\[ 975 = 25x + 1700 - 340x \]
\[ 975 - 1700 = -315x \]
\[ -725 = -315x \]
\[ x = \frac{725}{315} \approx 2.3 \, \text{cm} \]

### Final Answer
The depth of the water layer \( x \) is approximately \( 2.3 \, \text{cm} \).