There are two identical vessels filled with equal amounts of ice. The vessels are of different metals., If the ice melts in the two vessels in 20 and 35 minutes respectively, the ratio of the coefficients of thermal conductivity of the two metals is

Options

A. `4 : 7`
B. `7 : 4`
C. `16 : 49`
D. `49 : 16`

Answer

To solve the problem, we need to find the ratio of the coefficients of thermal conductivity of two metals based on the time taken for ice to melt in vessels made of these metals.

### Step-by-Step Solution:

1. **Understanding the Problem**:
We have two identical vessels filled with equal amounts of ice, made from two different metals. The ice melts in different times: 20 minutes for the first vessel and 35 minutes for the second vessel.

2. **Defining Variables**:
Let:
- \( k_1 \) = thermal conductivity of the first metal
- \( k_2 \) = thermal conductivity of the second metal
- \( t_1 = 20 \) minutes (time taken for the first vessel)
- \( t_2 = 35 \) minutes (time taken for the second vessel)

3. **Using the Heat Transfer Equation**:
The heat required to melt the ice in both vessels can be expressed using the formula for heat transfer:
\[ Q = k \cdot A \cdot \Delta T \cdot \frac{t}{l} \]
where:
- \( Q \) = heat required to melt the ice
- \( A \) = area of the vessel
- \( \Delta T \) = temperature difference
- \( t \) = time
- \( l \) = thickness of the vessel

Since the amount of ice is the same in both cases, we can equate the heat transferred in both vessels.

4. **Setting Up the Equation**:
For the first metal:
\[ Q = k_1 \cdot A \cdot \Delta T \cdot \frac{t_1}{l} \]
For the second metal:
\[ Q = k_2 \cdot A \cdot \Delta T \cdot \frac{t_2}{l} \]

Since both equations equal \( Q \), we can set them equal to each other:
\[ k_1 \cdot A \cdot \Delta T \cdot \frac{t_1}{l} = k_2 \cdot A \cdot \Delta T \cdot \frac{t_2}{l} \]

5. **Simplifying the Equation**:
We can cancel out \( A \), \( \Delta T \), and \( l \) from both sides:
\[ k_1 \cdot t_1 = k_2 \cdot t_2 \]

6. **Finding the Ratio**:
Rearranging the equation gives us:
\[ \frac{k_1}{k_2} = \frac{t_2}{t_1} \]

Substituting the values of \( t_1 \) and \( t_2 \):
\[ \frac{k_1}{k_2} = \frac{35}{20} \]

7. **Calculating the Ratio**:
Simplifying \( \frac{35}{20} \):
\[ \frac{35}{20} = \frac{7}{4} \]

8. **Final Answer**:
The ratio of the coefficients of thermal conductivity of the two metals is:
\[ \frac{k_1}{k_2} = \frac{7}{4} \]