An ice box is built of wood of thickness 1.75 cm. The box has an inner lining of cork 2 cm thick. If the difference in temperature between the interior of the box and outside is 12°C, calculate the temperature of the interface between wood and cork. Thermal conductivity of wood and cork are respectively `0.25 and 0.05` in S.I. unit.

Explanation

To solve the problem, we will follow these steps:

### Step 1: Understand the Setup
We have a wooden ice box with cork lining. The thickness of the wood is 1.75 cm and the thickness of the cork is 2 cm. The temperature difference between the inside (0°C) and outside (12°C) is given as 12°C. We need to find the temperature at the interface between the wood and the cork.

### Step 2: Write Down the Known Values
- Thickness of wood, \( L_w = 1.75 \, \text{cm} = 0.0175 \, \text{m} \)
- Thickness of cork, \( L_c = 2 \, \text{cm} = 0.02 \, \text{m} \)
- Temperature difference, \( \Delta T = 12 \, \text{°C} \)
- Thermal conductivity of wood, \( K_w = 0.25 \, \text{W/m°C} \)
- Thermal conductivity of cork, \( K_c = 0.05 \, \text{W/m°C} \)

### Step 3: Set Up the Heat Conduction Equations
Since the heat flow through both materials is the same, we can use the formula for heat conduction:
\[ \frac{K_c \cdot A \cdot (T - 0)}{L_c} = \frac{K_w \cdot A \cdot (12 - T)}{L_w} \]
Here, \( A \) is the area, which cancels out on both sides.

### Step 4: Substitute Known Values
Substituting the known values into the equation:
\[ \frac{0.05 \cdot (T - 0)}{0.02} = \frac{0.25 \cdot (12 - T)}{0.0175} \]

### Step 5: Simplify the Equation
Now we simplify both sides:
\[ \frac{0.05T}{0.02} = \frac{0.25(12 - T)}{0.0175} \]
This simplifies to:
\[ 2.5T = \frac{0.25 \cdot 12 - 0.25T}{0.0175} \]

Calculating the right side:
\[ 2.5T = \frac{3 - 0.25T}{0.0175} \]

### Step 6: Cross Multiply to Eliminate the Fraction
Cross-multiplying gives:
\[ 2.5T \cdot 0.0175 = 3 - 0.25T \]
\[ 0.04375T = 3 - 0.25T \]

### Step 7: Combine Like Terms
Now, combine the terms involving \( T \):
\[ 0.04375T + 0.25T = 3 \]
\[ 0.29375T = 3 \]

### Step 8: Solve for \( T \)
Now, solve for \( T \):
\[ T = \frac{3}{0.29375} \approx 10.21 \, \text{°C} \]

### Final Answer
The temperature at the interface between the wood and cork is approximately \( 10.21 \, \text{°C} \).

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