1. Use the formula for linear expansion

For a rod of initial length $L$ at initial temperature $30^\circ\text{C}$, the final length after heating is

$L' = L\left(1 + \alpha \Delta T\right)$

where $\alpha$ is the coefficient of linear expansion and $\Delta T$ is the rise in temperature.

2. Write expressions for the two rods

Let the initial length of each rod be $L$.

• For rod $A$:
  • Initial temperature $= 30^\circ\text{C}$
  • Final temperature $= 180^\circ\text{C}$
  • So,

$\Delta T_A = 180 - 30 = 150^\circ\text{C}$

Thus,

$L_A = L(1 + \alpha_A \cdot 150)$

• For rod $B$:
  • Initial temperature $= 30^\circ\text{C}$
  • Final temperature $= T^\circ\text{C}$
  • So,

$\Delta T_B = T - 30$

Thus,

$L_B = L(1 + \alpha_B (T-30))$

3. Given that the new lengths are the same

So,

$L_A = L_B$

$L(1 + \alpha_A \cdot 150) = L(1 + \alpha_B (T-30))$

Cancelling $L$,

$1 + 150\alpha_A = 1 + \alpha_B (T-30)$

$150\alpha_A = \alpha_B (T-30)$

4. Use the ratio of coefficients

Given,

$\alpha_A : \alpha_B = 4 : 3$

So,

$\frac{\alpha_A}{\alpha_B} = \frac{4}{3}$

Substitute into the equation:

$150\left(\frac{\alpha_A}{\alpha_B}\right) = T - 30$

$150 \cdot \frac{4}{3} = T - 30$

$200 = T - 30$

$T = 230^\circ\text{C}$

5. Check options

The correct option is:

$\boxed{\text{C: }230^\circ\text{C}}$