The efficiency of a carnot engine is `(1)/(6)`. If the temperature of the sink is reduced by 62 K, the efficiency becomes `(1)/(3)`. The temperature of the source and the sink in the first case are respectively.

Options

A. 372 K, 290 K
B. 372 K, 310 K
C. 744 K, 310 K
D. 744 K, 290 K

Explanation

To solve the problem, we need to find the temperatures of the source (T2) and sink (T1) of a Carnot engine given its efficiencies at two different states.

### Step-by-Step Solution:

1. **Understanding Efficiency of Carnot Engine**:
The efficiency (η) of a Carnot engine is given by the formula:
\[ \eta = 1 - \frac{T_1}{T_2} \]
where \( T_1 \) is the temperature of the sink and \( T_2 \) is the temperature of the source.

2. **Setting Up the First Equation**:
Given that the efficiency is \( \frac{1}{6} \):
\[ 1 - \frac{T_1}{T_2} = \frac{1}{6} \]
Rearranging gives:
\[ \frac{T_1}{T_2} = 1 - \frac{1}{6} = \frac{5}{6} \]
Thus, we can express \( T_1 \) in terms of \( T_2 \):
\[ T_1 = \frac{5}{6} T_2 \quad \text{(Equation 1)} \]

3. **Setting Up the Second Equation**:
When the sink temperature is reduced by 62 K, the new efficiency becomes \( \frac{1}{3} \):
\[ 1 - \frac{T_1 - 62}{T_2} = \frac{1}{3} \]
Rearranging gives:
\[ \frac{T_1 - 62}{T_2} = 1 - \frac{1}{3} = \frac{2}{3} \]
Thus, we can express \( T_1 - 62 \) in terms of \( T_2 \):
\[ T_1 - 62 = \frac{2}{3} T_2 \quad \text{(Equation 2)} \]

4. **Substituting Equation 1 into Equation 2**:
Substitute \( T_1 = \frac{5}{6} T_2 \) into Equation 2:
\[ \frac{5}{6} T_2 - 62 = \frac{2}{3} T_2 \]

5. **Solving for T2**:
To eliminate the fractions, multiply the entire equation by 6:
\[ 5 T_2 - 372 = 4 T_2 \]
Rearranging gives:
\[ 5 T_2 - 4 T_2 = 372 \]
Thus:
\[ T_2 = 372 \, \text{K} \]

6. **Finding T1**:
Now substitute \( T_2 = 372 \, \text{K} \) back into Equation 1:
\[ T_1 = \frac{5}{6} \times 372 = 310 \, \text{K} \]

### Final Answer:
The temperatures of the source and sink in the first case are:
- Source temperature \( T_2 = 372 \, \text{K} \)
- Sink temperature \( T_1 = 310 \, \text{K} \)