A cube made of material having a density of `900 kgm^(-3)` floats between water of density `1000 kgm^(-3)` and a liquid of density `700kgm^(-3)`, which is immiscible with water. What part of the cube is inside the water?

Options

A. `(1)/(3)`
B. `(2)/(3)`
C. `(3)/(4)`
D. `(3)/(7)`

Answer

To solve the problem, we need to determine what part of the cube is submerged in water when it is floating between two liquids of different densities. Let's break down the solution step by step.

### Step 1: Understand the Problem
We have a cube with a density of \(900 \, \text{kg/m}^3\) floating between water (\(1000 \, \text{kg/m}^3\)) and another liquid (\(700 \, \text{kg/m}^3\)). We need to find the volume of the cube that is submerged in water.

### Step 2: Define Variables
Let:
- \(V\) = Total volume of the cube
- \(V_D\) = Volume of the cube submerged in water
- \(V_L\) = Volume of the cube submerged in the liquid (700 kg/m³)

Since the cube is floating, the total volume submerged in both liquids will equal the total volume of the cube:
\[ V_D + V_L = V \]

### Step 3: Apply the Principle of Buoyancy
According to the principle of buoyancy (Archimedes' principle), the weight of the cube is equal to the buoyant force acting on it. The buoyant force is the sum of the buoyant forces from both liquids.

The weight of the cube can be expressed as:
\[ W = \text{Density of cube} \times V \times g = 900 \times V \times g \]

The buoyant force from the water is:
\[ F_B(\text{water}) = \text{Density of water} \times V_D \times g = 1000 \times V_D \times g \]

The buoyant force from the liquid is:
\[ F_B(\text{liquid}) = \text{Density of liquid} \times V_L \times g = 700 \times V_L \times g \]

### Step 4: Set Up the Equation
According to the law of flotation:
\[ W = F_B(\text{water}) + F_B(\text{liquid}) \]

Substituting the expressions we derived:
\[ 900 \times V \times g = 1000 \times V_D \times g + 700 \times V_L \times g \]

We can cancel \(g\) from both sides:
\[ 900 \times V = 1000 \times V_D + 700 \times V_L \]

### Step 5: Substitute \(V_L\)
From the earlier equation \(V_L = V - V_D\). Substitute this into the buoyancy equation:
\[ 900 \times V = 1000 \times V_D + 700 \times (V - V_D) \]

### Step 6: Simplify the Equation
Expanding the equation gives:
\[ 900V = 1000V_D + 700V - 700V_D \]
\[ 900V = 300V_D + 700V \]

Now, isolate \(V_D\):
\[ 900V - 700V = 300V_D \]
\[ 200V = 300V_D \]
\[ V_D = \frac{200V}{300} = \frac{2}{3}V \]

### Conclusion
Thus, the volume of the cube that is submerged in water is:
\[ V_D = \frac{2}{3}V \]

### Final Answer
The part of the cube that is inside the water is \(\frac{2}{3}\) of the total volume of the cube.

---