A glass thermometer is unmarked and has length of colum `L_(100^(@))=60 cm L_(0^(@))=10 cm.` If L=50 cm, then temperature thermometer will be :-

Options

A. `70^(@)C`
B. `80^(@)C`
C. `90^(@)C`
D. `100^(@)C`

Answer

To solve the problem, we need to find the temperature corresponding to the length of the mercury column at 50 cm in the unmarked glass thermometer. We know the lengths of the mercury column at two reference temperatures:

- At 0°C, the length of the column \( L_{0°C} = 10 \, \text{cm} \)
- At 100°C, the length of the column \( L_{100°C} = 60 \, \text{cm} \)

We can use the linear relationship between temperature and the length of the mercury column. The formula to find the temperature \( T \) corresponding to a length \( L \) can be derived from the two reference points:

1. **Identify the lengths and temperatures**:
- \( L_{0°C} = 10 \, \text{cm} \) corresponds to \( T_0 = 0°C \)
- \( L_{100°C} = 60 \, \text{cm} \) corresponds to \( T_{100} = 100°C \)

2. **Determine the temperature per cm**:
- The change in temperature from 0°C to 100°C is \( 100°C - 0°C = 100°C \).
- The change in length from 10 cm to 60 cm is \( 60 \, \text{cm} - 10 \, \text{cm} = 50 \, \text{cm} \).
- Therefore, the temperature change per cm of the column is:
\[ \frac{100°C}{50 \, \text{cm}} = 2°C/\text{cm} \]

3. **Calculate the length difference from the reference point**:
- The length \( L = 50 \, \text{cm} \) is between the two reference points (10 cm and 60 cm).
- The difference in length from the 0°C reference point is:
\[ L - L_{0°C} = 50 \, \text{cm} - 10 \, \text{cm} = 40 \, \text{cm} \]

4. **Calculate the temperature corresponding to 50 cm**:
- Using the temperature change per cm, we can find the temperature:
\[ T = T_0 + \left(\text{Length difference}\right) \times \left(\frac{100°C}{50 \, \text{cm}}\right) \]
\[ T = 0°C + 40 \, \text{cm} \times 2°C/\text{cm} = 80°C \]

Thus, the temperature indicated by the thermometer when the length of the mercury column is 50 cm is **80°C**.