The pressure of air in a constant volume air thermometer is 80cm and 109.3cm at `0^(@)C` and `100^(@)C` respectively. When the bulb is placed in some hot water, the pressure is 100cm. Calculate the temperature of the hot water.

Explanation

To solve the problem, we will use the concept of the linear relationship between the pressure of air in a constant volume thermometer and temperature. We know the pressures at two known temperatures (0°C and 100°C) and we want to find the temperature corresponding to a pressure of 100 cm.

### Step-by-Step Solution:

1. **Identify the known values:**
- Pressure at 0°C, \( P_0 = 80 \, \text{cm} \)
- Pressure at 100°C, \( P_{100} = 109.3 \, \text{cm} \)
- Pressure at the unknown temperature, \( P = 100 \, \text{cm} \)

2. **Set up the formula for temperature:**
We can use the formula derived from the linear relationship:
\[ T = \frac{(P - P_0)}{(P_{100} - P_0)} \times 100 \]
where \( T \) is the temperature in degrees Celsius.

3. **Substitute the known values into the formula:**
\[ T = \frac{(100 - 80)}{(109.3 - 80)} \times 100 \]

4. **Calculate the numerator and denominator:**
- Numerator: \( 100 - 80 = 20 \)
- Denominator: \( 109.3 - 80 = 29.3 \)

5. **Plug the values back into the equation:**
\[ T = \frac{20}{29.3} \times 100 \]

6. **Perform the division:**
\[ \frac{20}{29.3} \approx 0.6825 \]

7. **Multiply by 100 to find the temperature:**
\[ T \approx 0.6825 \times 100 \approx 68.25 \, \text{°C} \]

8. **Final result:**
The temperature of the hot water is approximately \( 68.25 \, \text{°C} \).

### Summary:
The temperature of the hot water is \( 68.25 \, \text{°C} \).