A Carnot reversible engine converts `1//6` of heat input into work. When the temperature of the sink is redused by 62 K, the efficiency of Carnot’s cycle becomes `1//3`. The sum of temperature (in kelvin) of the source and sink will be

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the efficiency of the Carnot engine
The efficiency \( \eta \) of a Carnot engine is given by the formula:
\[ \eta = 1 - \frac{T_2}{T_1} \]
where \( T_1 \) is the temperature of the heat source and \( T_2 \) is the temperature of the heat sink.

### Step 2: Set up the first condition
From the problem, we know that the engine initially converts \( \frac{1}{6} \) of the heat input into work. Therefore, the efficiency can be expressed as:
\[ \eta_1 = \frac{1}{6} \]
Using the efficiency formula:
\[ \frac{1}{6} = 1 - \frac{T_2}{T_1} \]
Rearranging gives:
\[ \frac{T_2}{T_1} = 1 - \frac{1}{6} = \frac{5}{6} \]
Thus, we have:
\[ T_2 = \frac{5}{6} T_1 \tag{1} \]

### Step 3: Set up the second condition
When the temperature of the sink is reduced by \( 62 \, K \), the new efficiency becomes \( \frac{1}{3} \):
\[ \eta_2 = \frac{1}{3} \]
Using the efficiency formula again:
\[ \frac{1}{3} = 1 - \frac{T_2 - 62}{T_1} \]
Rearranging gives:
\[ \frac{T_2 - 62}{T_1} = 1 - \frac{1}{3} = \frac{2}{3} \]
Thus, we have:
\[ T_2 - 62 = \frac{2}{3} T_1 \tag{2} \]

### Step 4: Substitute equation (1) into equation (2)
Substituting \( T_2 \) from equation (1) into equation (2):
\[ \frac{5}{6} T_1 - 62 = \frac{2}{3} T_1 \]

### Step 5: Solve for \( T_1 \)
To solve for \( T_1 \), first eliminate the fractions by multiplying through by 6:
\[ 5 T_1 - 372 = 4 T_1 \]
Rearranging gives:
\[ 5 T_1 - 4 T_1 = 372 \]
Thus:
\[ T_1 = 372 \, K \]

### Step 6: Find \( T_2 \)
Now, substitute \( T_1 \) back into equation (1) to find \( T_2 \):
\[ T_2 = \frac{5}{6} \times 372 = 310 \, K \]

### Step 7: Calculate the sum of \( T_1 \) and \( T_2 \)
Finally, we find the sum of the temperatures:
\[ T_1 + T_2 = 372 + 310 = 682 \, K \]

### Final Answer
The sum of the temperature of the source and sink is:
\[ \boxed{682 \, K} \]