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Step-by-step Solution
Step 1: Write the formula for Carnot engine efficiency
The efficiency $ \eta $ of a Carnot engine operating between a source temperature $T_1$ and sink temperature $T_2$ is given by:
$$ \eta = \frac{T_1 - T_2}{T_1} $$
Step 2: Express the given data for the initial efficiency
Initially, the efficiency is $40\%$ (i.e., $0.40$), and the sink temperature is $300\text{ K}$. Let the corresponding source temperature be $T_1$. Therefore,
$$ 0.40 = \frac{T_1 - 300}{T_1} .$$
Step 3: Solve for the initial source temperature
Rearranging the above equation:
$$ 0.40 \times T_1 = T_1 - 300, $$
$$ T_1 - 0.40\,T_1 = 300, $$
$$ 0.60\,T_1 = 300, $$
$$ T_1 = \frac{300}{0.60} = 500\text{ K}. $$
Thus, the initial source temperature is $500\text{ K}.$
Step 4: Determine the new efficiency
The efficiency is to be increased by $50\%$ of its original efficiency. The original efficiency is $0.40$, so the new efficiency is:
$$ \eta_{\text{new}} = 0.40 + 0.50 \times 0.40 = 0.40 + 0.20 = 0.60. $$
Step 5: Write the equation for new efficiency
Let the new required source temperature be $T_1^\prime$. Using the Carnot efficiency formula again with sink at $300\text{ K}$,
$$ 0.60 = \frac{T_1^\prime - 300}{T_1^\prime}. $$
Step 6: Solve for the new source temperature
Rearranging, we get:
$$ 0.60 \times T_1^\prime = T_1^\prime - 300, $$
$$ 0.40\,T_1^\prime = 300, $$
$$ T_1^\prime = \frac{300}{0.40} = 750\text{ K}. $$
Step 7: Calculate the increase in source temperature
The initial source temperature was $500\text{ K}$, and the new source temperature required is $750\text{ K}$. Therefore, the increase in temperature is:
$$ \Delta T = 750 - 500 = 250\text{ K}. $$
Final Answer:
The source temperature must be increased by $\boxed{250\text{ K}}$.
