Same quantity of ice is filled in each of the two metal container P and Q having the same size, shape and will thickness but make of different materials. The containers are kept in identical surroundings, The ice in P melts completely in time `t_(1)`, whereas in Q takes a time `t_(2)`. The ratio of thermal conductivities of the materials of P and Q is:

Options

A. `t_(2):t_(1)`
B. `t_(1):t_(2)`
C. `t_(1)^(2):t_(2)^(2)`
D. `t_(2)^(2):t_(1)^(2)`

Explanation

To solve the problem, we need to analyze the heat transfer through the two metal containers P and Q, which have the same size, shape, and thickness but are made of different materials. The ice in container P melts in time \( t_1 \), while in container Q it melts in time \( t_2 \). We want to find the ratio of the thermal conductivities of the materials of containers P and Q.

### Step-by-Step Solution:

1. **Understanding Heat Transfer**:
The heat \( Q \) required to melt the ice in both containers is the same since the quantity of ice is the same. Therefore, we can set up the equation based on the principle of calorimetry, which states that the heat transferred through the containers is equal.

2. **Heat Transfer Formula**:
The heat transferred through a material can be expressed using the formula:
\[ Q = \frac{K \cdot A \cdot (T_1 - T_2) \cdot t}{d} \]
where:
- \( K \) is the thermal conductivity of the material,
- \( A \) is the area,
- \( (T_1 - T_2) \) is the temperature difference,
- \( t \) is the time,
- \( d \) is the thickness of the material.

3. **Applying the Formula to Containers P and Q**:
For container P:
\[ Q = \frac{K_1 \cdot A \cdot (T_1 - T_2) \cdot t_1}{d} \]
For container Q:
\[ Q = \frac{K_2 \cdot A \cdot (T_1 - T_2) \cdot t_2}{d} \]

4. **Setting the Heat Equations Equal**:
Since the heat \( Q \) is the same for both containers, we can equate the two equations:
\[ \frac{K_1 \cdot A \cdot (T_1 - T_2) \cdot t_1}{d} = \frac{K_2 \cdot A \cdot (T_1 - T_2) \cdot t_2}{d} \]

5. **Canceling Common Terms**:
We can cancel \( A \), \( (T_1 - T_2) \), and \( d \) from both sides (assuming they are the same for both containers):
\[ K_1 \cdot t_1 = K_2 \cdot t_2 \]

6. **Finding the Ratio of Thermal Conductivities**:
Rearranging the equation gives us:
\[ \frac{K_1}{K_2} = \frac{t_2}{t_1} \]

7. **Final Result**:
Therefore, the ratio of the thermal conductivities of the materials of containers P and Q is:
\[ K_1 : K_2 = t_2 : t_1 \]

### Conclusion:
The ratio of the thermal conductivities of the materials of containers P and Q is \( K_1 : K_2 = t_2 : t_1 \).

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