There are three thermometers one in contact with the skin of the man other in between the vest and the shirt and third in between the shirt and coat. The readings of the thermomenters are `30^@C`,`25^@C` and `22^@C`, respectively. If the vest and the shirt are of the same thickness, the ratio of their thermal conductivities is

Options

A. `9:25`
B. `25:9`
C. `5:3`
D. `3:5`

Explanation

To solve the problem, we need to determine the ratio of the thermal conductivities of the vest and shirt based on the temperature readings from the thermometers. Let's break it down step by step.

### Step 1: Understand the Temperature Readings
We have three temperature readings:
- \( T_1 = 30^\circ C \) (skin temperature)
- \( T_2 = 25^\circ C \) (temperature between the vest and shirt)
- \( T_3 = 22^\circ C \) (temperature between the shirt and coat)

### Step 2: Calculate the Temperature Differences
We need to find the temperature differences for the two layers:
- For the vest (between skin and vest):
\[ \Delta T_1 = T_1 - T_2 = 30^\circ C - 25^\circ C = 5^\circ C \]
- For the shirt (between vest and coat):
\[ \Delta T_2 = T_2 - T_3 = 25^\circ C - 22^\circ C = 3^\circ C \]

### Step 3: Apply the Heat Transfer Equation
In a steady state, the heat transfer through both layers must be equal. The heat transfer \( Q \) through a material can be expressed as:
\[ Q = K \cdot A \cdot \frac{\Delta T}{L} \]
Where:
- \( K \) is the thermal conductivity,
- \( A \) is the area,
- \( \Delta T \) is the temperature difference,
- \( L \) is the thickness of the material.

Since the thickness \( L \) and area \( A \) are the same for both layers, we can simplify the equation for both layers:
\[ K_1 \cdot \Delta T_1 = K_2 \cdot \Delta T_2 \]

### Step 4: Substitute the Values
Substituting the temperature differences we calculated:
\[ K_1 \cdot 5 = K_2 \cdot 3 \]

### Step 5: Rearranging for the Ratio
Rearranging the equation to find the ratio of thermal conductivities:
\[ \frac{K_1}{K_2} = \frac{3}{5} \]

### Conclusion
The ratio of the thermal conductivities of the vest and shirt is:
\[ \frac{K_1}{K_2} = \frac{3}{5} \]