Estimate the rate at which ice would melt in a wooden box 20 mm thick and of inside measurement 200 cm x 100 cm x 100 cm assumming that external temperature is `27^(@)C` and coefficient of thermal conductivity of wood is `0.0004 cal s^(-1) cm^(-1) .^(@)C^(-1)`.

Explanation

To estimate the rate at which ice would melt in a wooden box, we can follow these steps:

### Step 1: Understand the problem
We need to calculate the rate of heat transfer through the wooden box to determine how much ice will melt. The parameters provided include the dimensions of the box, the thickness of the wood, the external temperature, and the thermal conductivity of wood.

### Step 2: Convert units
- The thickness of the wooden box is given as 20 mm. We convert this to centimeters:
\[ \text{Thickness} = 20 \, \text{mm} = 2 \, \text{cm} \]

### Step 3: Calculate the surface area of the box
The box has the following dimensions:
- Inside dimensions: 200 cm x 100 cm x 100 cm
- The surface area consists of:
- 2 faces of 200 cm x 100 cm
- 2 faces of 100 cm x 100 cm
- Calculating the area:
\[ \text{Area} = 2 \times (200 \times 100) + 2 \times (100 \times 100) = 2 \times 20000 + 2 \times 10000 = 40000 + 20000 = 60000 \, \text{cm}^2 \]

### Step 4: Use the formula for heat transfer
The rate of heat transfer \( \frac{dq}{dt} \) through the box can be calculated using Fourier's law of heat conduction:
\[ \frac{dq}{dt} = \frac{K \cdot A \cdot \Delta T}{d} \]
Where:
- \( K \) = thermal conductivity of wood = \( 0.0004 \, \text{cal s}^{-1} \text{cm}^{-1} \degree C^{-1} \)
- \( A \) = surface area = \( 60000 \, \text{cm}^2 \)
- \( \Delta T \) = temperature difference = \( 27 \degree C - 0 \degree C = 27 \degree C \)
- \( d \) = thickness of the wood = \( 2 \, \text{cm} \)

### Step 5: Substitute the values into the equation
Substituting the values into the equation:
\[ \frac{dq}{dt} = \frac{0.0004 \cdot 60000 \cdot 27}{2} \]

### Step 6: Calculate the heat transfer rate
Calculating the above expression:
\[ \frac{dq}{dt} = \frac{0.0004 \cdot 60000 \cdot 27}{2} = \frac{648}{2} = 324 \, \text{cal/s} \]

### Step 7: Calculate the mass of ice melted per second
To find the mass of ice melted per second, we use the latent heat of fusion for ice, which is \( L = 80 \, \text{cal/g} \):
\[ \text{Mass melted per second} = \frac{\frac{dq}{dt}}{L} = \frac{324 \, \text{cal/s}}{80 \, \text{cal/g}} = 4.05 \, \text{g/s} \]

### Conclusion
The rate at which ice would melt in the wooden box is approximately \( 4.05 \, \text{g/s} \).

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