An ideal heat engine exhausting heat at `77^@C` is to have a 30% efficiency. It must take heat at

Options

A. `127^(@)C`
B. `227^(@)C`
C. `327^(@)C`
D. `673^(@)C`

Answer

To solve the problem of finding the temperature at which the ideal heat engine must take heat, we can follow these steps:

### Step 1: Convert the sink temperature from Celsius to Kelvin
The temperature of the sink (T2) is given as 77°C. To convert this to Kelvin, we use the formula:
\[ T(K) = T(°C) + 273 \]
So,
\[ T2 = 77 + 273 = 350 \, K \]

### Step 2: Write down the efficiency formula
The efficiency (η) of a heat engine is given by the formula:
\[ \eta = 1 - \frac{T2}{T1} \]
Where:
- η is the efficiency,
- T2 is the temperature of the sink,
- T1 is the temperature of the source.

### Step 3: Substitute the known values into the efficiency formula
We know that the efficiency is 30%, which can be expressed as:
\[ \eta = \frac{30}{100} = 0.3 \]
Now substituting the values into the efficiency formula:
\[ 0.3 = 1 - \frac{350}{T1} \]

### Step 4: Rearrange the equation to solve for T1
Rearranging the equation gives:
\[ \frac{350}{T1} = 1 - 0.3 \]
\[ \frac{350}{T1} = 0.7 \]

### Step 5: Cross-multiply to find T1
Cross-multiplying gives:
\[ 350 = 0.7 \times T1 \]
Now, solving for T1:
\[ T1 = \frac{350}{0.7} \]
\[ T1 = 500 \, K \]

### Step 6: Convert T1 back to Celsius
To convert T1 from Kelvin back to Celsius:
\[ T(°C) = T(K) - 273 \]
So,
\[ T1(°C) = 500 - 273 = 227 \, °C \]

### Final Answer
The temperature at which the ideal heat engine must take heat is **227°C**.

---