Two vessels of different materials are similar in size in every respect. The same quantity of ice filled in them gets melted in 20 minutes and 40 minutes respectively. The ratio of thermal conductivities of the metals is

Options

A. `5 : 6`
B. `6 : 5`
C. `3 : 1`
D. `2 : 1`

Explanation

To solve the problem, we need to determine the ratio of thermal conductivities of two materials based on the time taken for the same quantity of ice to melt in each vessel.

### Step-by-Step Solution:

1. **Understand the Problem**: We have two vessels made of different materials, both filled with the same quantity of ice. The ice melts in different times: 20 minutes for the first vessel and 40 minutes for the second vessel.

2. **Identify Relevant Principles**: According to the law of thermal conduction, the rate of heat transfer (Q/T) is given by the formula:
\[ \frac{Q}{T} = K \cdot A \cdot \frac{\Delta \theta}{L} \]
where:
- \(Q\) = heat transferred,
- \(T\) = time,
- \(K\) = thermal conductivity,
- \(A\) = area,
- \(\Delta \theta\) = temperature difference,
- \(L\) = thickness of the material.

3. **Analyze Given Information**: Since both vessels are similar in size, the area \(A\) and length \(L\) are the same for both vessels. The quantity of ice melted (which corresponds to the heat \(Q\)) is also the same.

4. **Set Up the Ratios**: From the law of thermal conduction, we can express the heat transfer for both vessels:
- For vessel 1 (melting time \(T_1 = 20\) minutes):
\[ \frac{Q}{T_1} = K_1 \cdot A \cdot \frac{\Delta \theta}{L} \]
- For vessel 2 (melting time \(T_2 = 40\) minutes):
\[ \frac{Q}{T_2} = K_2 \cdot A \cdot \frac{\Delta \theta}{L} \]

5. **Equate the Heat Transfer**: Since the quantity of heat \(Q\) is the same for both vessels, we can set the two equations equal:
\[ K_1 \cdot A \cdot \frac{\Delta \theta}{L} \cdot T_1 = K_2 \cdot A \cdot \frac{\Delta \theta}{L} \cdot T_2 \]
The terms \(A\), \(\Delta \theta\), and \(L\) cancel out, leading to:
\[ K_1 \cdot T_1 = K_2 \cdot T_2 \]

6. **Rearranging for the Ratio**: We can rearrange this to find the ratio of thermal conductivities:
\[ \frac{K_1}{K_2} = \frac{T_2}{T_1} \]

7. **Substituting the Values**: Substitute \(T_1 = 20\) minutes and \(T_2 = 40\) minutes into the equation:
\[ \frac{K_1}{K_2} = \frac{40}{20} = 2 \]

8. **Final Result**: The ratio of thermal conductivities \(K_1 : K_2\) is:
\[ K_1 : K_2 = 2 : 1 \]

### Conclusion:
The ratio of thermal conductivities of the two materials is \(2:1\).