Two vessels of different materials are similar in size in every respect. The same quantity of ice filled in them gets melted in 20 min and 35 min, respectively. The ratio of coefficients of thermal conduction of the metals is

Options

A. `4:7`
B. `7:4`
C. `25:16`
D. `16:25`

Explanation

To solve the problem, we need to find the ratio of the coefficients of thermal conduction (K1 and K2) of the two metals based on the time taken to melt the same quantity of ice in each vessel.

### Step-by-Step Solution:

1. **Understand the Problem**:
We have two vessels made of different materials (metals) that are identical in size. The same quantity of ice is placed in both vessels, and it melts in different times: 20 minutes for the first vessel and 35 minutes for the second vessel.

2. **Identify the Variables**:
- Let K1 be the thermal conductivity of the first metal.
- Let K2 be the thermal conductivity of the second metal.
- Let T1 = 20 minutes (time for the first vessel).
- Let T2 = 35 minutes (time for the second vessel).

3. **Use the Concept of Heat Transfer**:
The rate of heat transfer (Q/t) through a material is given by the formula:
\[ \frac{Q}{t} = \frac{K \cdot A \cdot \Delta T}{L} \]
where:
- Q is the heat transferred,
- t is the time,
- K is the thermal conductivity,
- A is the cross-sectional area,
- ΔT is the temperature difference,
- L is the thickness of the material.

4. **Relate Heat Transfer to Time**:
Since the same quantity of ice is melted in both vessels, the heat (Q) required to melt the ice is the same. Therefore, we can set up the equations for both vessels:
\[ \frac{K_1 \cdot A \cdot \Delta T}{L} \cdot T_1 = \frac{K_2 \cdot A \cdot \Delta T}{L} \cdot T_2 \]

5. **Simplify the Equation**:
The areas (A), temperature differences (ΔT), and thicknesses (L) are the same for both vessels, so they cancel out:
\[ K_1 \cdot T_1 = K_2 \cdot T_2 \]

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

7. **Substituting the Values**:
Substitute the values of T1 and T2:
\[ \frac{K_1}{K_2} = \frac{35}{20} = \frac{7}{4} \]

8. **Final Result**:
The ratio of the coefficients of thermal conduction of the two metals is:
\[ \frac{K_1}{K_2} = \frac{7}{4} \]