A cubic body floats on mercury with 0.25 fraction of its volume below the surface. What fraction of the volume of the body will be immeresed in the mercury if a layer of water poured on top of the mercury covers the body conpletely.

Explanation

To solve the problem, we will follow these steps:

### Step 1: Understand the initial condition
The cubic body floats on mercury with 0.25 of its volume submerged. This means that the buoyant force provided by the mercury equals the weight of the body.

### Step 2: Write the equation for the initial condition
Let:
- \( V \) = total volume of the body
- \( \rho_m \) = density of mercury
- \( g \) = acceleration due to gravity

The weight of the body is given by:
\[ W = V \cdot \rho_b \cdot g \]
where \( \rho_b \) is the density of the body.

The buoyant force (upthrust) from the mercury is:
\[ F_b = 0.25V \cdot \rho_m \cdot g \]

Setting the weight equal to the buoyant force:
\[ V \cdot \rho_b \cdot g = 0.25V \cdot \rho_m \cdot g \]

### Step 3: Simplify the equation
Cancel out \( V \) and \( g \) (assuming they are not zero):
\[ \rho_b = 0.25 \cdot \rho_m \]

### Step 4: Introduce the new condition with water
When water is poured on top of the mercury, we need to find the new fraction \( x \) of the body that is submerged in mercury. The body will now have part of its volume submerged in both mercury and water.

### Step 5: Write the equation for the new condition
The weight of the body remains the same:
\[ W = V \cdot \rho_b \cdot g \]

The buoyant force now consists of two parts:
1. The buoyant force from the mercury:
\[ F_{b,m} = xV \cdot \rho_m \cdot g \]
2. The buoyant force from the water:
\[ F_{b,w} = (1 - x)V \cdot \rho_w \cdot g \]

Setting the total buoyant force equal to the weight of the body:
\[ V \cdot \rho_b \cdot g = xV \cdot \rho_m \cdot g + (1 - x)V \cdot \rho_w \cdot g \]

### Step 6: Simplify the equation
Cancel \( V \) and \( g \):
\[ \rho_b = x \cdot \rho_m + (1 - x) \cdot \rho_w \]

### Step 7: Substitute known values
We know:
- \( \rho_m = 13.6 \, \text{g/cm}^3 \)
- \( \rho_w = 1 \, \text{g/cm}^3 \)
- \( \rho_b = 0.25 \cdot \rho_m = 0.25 \cdot 13.6 = 3.4 \, \text{g/cm}^3 \)

Substituting these values into the equation:
\[ 3.4 = x \cdot 13.6 + (1 - x) \cdot 1 \]

### Step 8: Solve for \( x \)
Expanding the equation:
\[ 3.4 = x \cdot 13.6 + 1 - x \]
\[ 3.4 - 1 = x \cdot 13.6 - x \]
\[ 2.4 = x \cdot (13.6 - 1) \]
\[ 2.4 = x \cdot 12.6 \]
\[ x = \frac{2.4}{12.6} \]
\[ x \approx 0.1905 \]

### Step 9: Conclusion
Thus, the fraction of the volume of the body that will be immersed in mercury when water is poured on top is approximately:
\[ x \approx 0.19 \]