The ratio of the areas of cross section of two rods of different materials is 1:2 and the ratio of ther themal conductivities of their mateirals is 4: 3. On keeping equal temperature difference between the ends of theserods, the rate of conduction of heat are equal. The ratio of the lengths of the rods is

Options

A. `2:3`
B. `2:4`
C. `4:2`
D. `4:5`

Answer

To solve the problem, we need to find the ratio of the lengths of two rods made of different materials, given the ratios of their cross-sectional areas and thermal conductivities, while ensuring that the rate of heat conduction is equal for both rods.

### Step-by-Step Solution:

1. **Identify Given Ratios:**
- The ratio of the areas of cross-section of the two rods is given as:
\[ \frac{A_1}{A_2} = \frac{1}{2} \]
- The ratio of the thermal conductivities of the materials is given as:
\[ \frac{K_1}{K_2} = \frac{4}{3} \]

2. **Understand the Heat Conduction Equation:**
The rate of heat conduction \( Q \) through a rod is given by Fourier's law:
\[ Q = \frac{K \cdot A \cdot (T_1 - T_2)}{L} \]
where \( K \) is the thermal conductivity, \( A \) is the area of cross-section, \( T_1 - T_2 \) is the temperature difference, and \( L \) is the length of the rod.

3. **Set Up the Equation for Both Rods:**
For rod 1:
\[ Q = \frac{K_1 \cdot A_1 \cdot (T_1 - T_2)}{L_1} \]
For rod 2:
\[ Q = \frac{K_2 \cdot A_2 \cdot (T_2 - T_3)}{L_2} \]

4. **Since the Temperature Difference is Equal:**
Given that \( T_1 - T_2 = T_2 - T_3 \), we can denote this common temperature difference as \( \Delta T \). Thus, we can rewrite the equations:
\[ Q = \frac{K_1 \cdot A_1 \cdot \Delta T}{L_1} \]
\[ Q = \frac{K_2 \cdot A_2 \cdot \Delta T}{L_2} \]

5. **Equate the Two Expressions for Q:**
Since the rate of heat conduction is equal for both rods:
\[ \frac{K_1 \cdot A_1}{L_1} = \frac{K_2 \cdot A_2}{L_2} \]

6. **Rearranging the Equation:**
Rearranging gives us:
\[ \frac{L_1}{L_2} = \frac{K_1 \cdot A_1}{K_2 \cdot A_2} \]

7. **Substituting the Ratios:**
Substitute the given ratios:
\[ \frac{L_1}{L_2} = \frac{K_1}{K_2} \cdot \frac{A_1}{A_2} = \frac{4/3}{1/2} = \frac{4}{3} \cdot \frac{2}{1} = \frac{8}{3} \]

8. **Final Ratio of Lengths:**
Thus, the ratio of the lengths of the rods is:
\[ \frac{L_1}{L_2} = \frac{8}{3} \]

### Conclusion:
The ratio of the lengths of the two rods is \( \frac{8}{3} \).