An ice cube of edge a is placed in an empty cylindrical vessel of radius 2a. Find the edge (in cm) of ice cube when it just leaves the contact with the bottom assuming that ice melts uniformly maintaining its cubical shape. Take a = 12 `pi` cm (Ice is lighter than water)

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the problem
We have an ice cube with an edge length \( a \) placed in a cylindrical vessel with a radius of \( 2a \). The ice cube melts uniformly while maintaining its cubical shape. We need to find the edge length of the ice cube when it just leaves contact with the bottom of the vessel.

### Step 2: Define the variables
Let:
- The initial edge length of the ice cube = \( a \)
- The edge length of the ice cube after melting = \( x \)
- The density of ice = \( \rho_i \)
- The density of water = \( \rho_w \) (where \( \rho_w > \rho_i \))

Given that \( a = 12\pi \) cm.

### Step 3: Calculate the volume of the ice cube
The initial volume \( V_i \) of the ice cube is given by:
\[ V_i = a^3 = (12\pi)^3 \]

### Step 4: Calculate the volume of the melted ice cube
After melting, the volume \( V_f \) of the ice cube is:
\[ V_f = x^3 \]

### Step 5: Apply the principle of buoyancy
When the ice cube is just about to leave the bottom, the weight of the ice cube is equal to the buoyant force acting on it. The buoyant force \( F_b \) can be expressed as:
\[ F_b = \rho_w \cdot V_f \cdot g \]
where \( g \) is the acceleration due to gravity.

The weight \( W \) of the ice cube is:
\[ W = \rho_i \cdot V_i \cdot g \]

Setting the two forces equal gives:
\[ \rho_i \cdot V_i \cdot g = \rho_w \cdot V_f \cdot g \]
We can cancel \( g \) from both sides:
\[ \rho_i \cdot V_i = \rho_w \cdot V_f \]

### Step 6: Substitute the volumes
Substituting the volumes into the equation gives:
\[ \rho_i \cdot (12\pi)^3 = \rho_w \cdot x^3 \]

### Step 7: Calculate the volume of the cylindrical vessel
The volume \( V_c \) of the cylindrical vessel is:
\[ V_c = \pi (2a)^2 h = \pi (4a^2) h \]

### Step 8: Set up the equation for volume conservation
The volume of the ice cube that has melted must equal the volume of water displaced:
\[ (12\pi)^3 - x^3 = 4\pi a^2 h - x^2 h \]

### Step 9: Solve for \( x \)
Substituting \( a = 12\pi \) into the equation and simplifying will allow us to solve for \( x \):
1. Calculate \( (12\pi)^3 \).
2. Substitute into the volume equation.
3. Rearrange and solve for \( x \).

### Final Calculation
After performing the calculations, we find that \( x = 3 \) cm.

### Conclusion
The edge length of the ice cube when it just leaves contact with the bottom of the vessel is \( 3 \) cm.

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