The atmospheric temperature above a lake is below `0^(@)C` and constant. It is found that a 2 cm layer of ice is formed in four days. In how many days will the thickness increase to 3 cm?

Answer

To solve the problem of how many days it will take for the thickness of ice on a lake to increase from 2 cm to 3 cm given that a 2 cm layer forms in 4 days, we can use the concept of heat transfer and the relationship between the thickness of the ice and time.

### Step-by-Step Solution:

1. **Understanding the Problem**: We know that a 2 cm thick layer of ice forms in 4 days. We need to find out how long it will take for the ice to increase to 3 cm.

2. **Using the Stefan's Law of Heat Transfer**: According to Stefan's law, the rate of heat transfer through a material is proportional to the temperature difference and the area, and inversely proportional to the thickness of the material. The heat transfer can be expressed as:
\[ dq = K \cdot A \cdot \frac{\Delta T}{x} \]
where \( K \) is the thermal conductivity, \( A \) is the area, \( \Delta T \) is the temperature difference, and \( x \) is the thickness of the ice.

3. **Relating Heat Transfer to Ice Formation**: The heat removed from the water to form ice can also be expressed in terms of the mass of ice formed:
\[ dq = A \cdot \rho \cdot dx \cdot L \]
where \( \rho \) is the density of ice, \( dx \) is the thickness increase, and \( L \) is the latent heat of fusion.

4. **Equating the Two Expressions**: Since both expressions represent the same amount of heat transfer, we can set them equal to each other:
\[ K \cdot A \cdot \frac{\Delta T}{x} = A \cdot \rho \cdot dx \cdot L \]

5. **Simplifying the Equation**: We can simplify this equation by canceling out \( A \) from both sides:
\[ K \cdot \frac{\Delta T}{x} = \rho \cdot dx \cdot L \]

6. **Finding the Relationship Between Thickness and Time**: The thickness of the ice is proportional to the square of the time taken for its formation:
\[ x \propto t^2 \]
Therefore, we can write:
\[ t \propto x^2 \]

7. **Calculating Time for Different Thickness**: We know that for 2 cm thickness, the time taken is 4 days. We can express this as:
\[ t_1 = 4 \text{ days for } x_1 = 2 \text{ cm} \]
For 3 cm thickness, let the time be \( t_2 \):
\[ t_2 \propto (3 \text{ cm})^2 \]

8. **Setting Up the Proportionality**:
\[ \frac{t_2}{t_1} = \frac{(3 \text{ cm})^2}{(2 \text{ cm})^2} \]
\[ \frac{t_2}{4} = \frac{9}{4} \]
\[ t_2 = 4 \cdot \frac{9}{4} = 9 \text{ days} \]

### Final Answer:
It will take **9 days** for the thickness of the ice to increase from 2 cm to 3 cm.