The efficiency ($\eta$) of a Carnot engine is given by the formula: $\eta = 1 - \frac{T_2}{T_1}$, where $T_1$ is the source temperature and $T_2$ is the sink temperature.

Step 1: Calculate the initial source temperature ($T_1$). Given: $\eta_1 = 40\% = 0.4$, $T_2 = 300 \text{ K}$. $0.4 = 1 - \frac{300}{T_1}$ $\frac{300}{T_1} = 1 - 0.4 = 0.6$ $T_1 = \frac{300}{0.6} = 500 \text{ K}$

Step 2: Calculate the new efficiency ($\eta_2$). The efficiency is increased by $50\%$ of its original value. Increase $= 50\% \text{ of } 0.4 = 0.5 \times 0.4 = 0.2$. New Efficiency $\eta_2 = 0.4 + 0.2 = 0.6$ (or $60\%$).

Step 3: Calculate the new source temperature ($T_1'$). Keeping the sink temperature ($T_2$) constant at $300 \text{ K}$: $0.6 = 1 - \frac{300}{T_1'}$ $\frac{300}{T_1'} = 1 - 0.6 = 0.4$ $T_1' = \frac{300}{0.4} = 750 \text{ K}$

Step 4: Calculate the increase in temperature. $\Delta T = T_1' - T_1 = 750 \text{ K} - 500 \text{ K} = 250 \text{ K}$