The Coefficient of cubical expansion of mercury is `0.0018//^(@)C` and that of brass `0.000006//^(@)C`, find the true barmetric height at `0^(@)C`.The scale is supposed to be coreect at `15^(@)C`

Options

A. 74.122 cm
B. 79.125 cm
C. 42.161cm
D. 142.39 cm

Answer

To find the true barometric height at \(0^\circ C\) given the coefficients of cubical expansion for mercury and brass, we can follow these steps:

### Step-by-Step Solution:

1. **Identify Given Values**:
- Coefficient of cubical expansion of mercury (\(\gamma\)): \(0.0018 \, /^\circ C\)
- Coefficient of cubical expansion of brass (\(\alpha\)): \(0.000006 \, /^\circ C\)
- Initial barometric height at \(15^\circ C\) (\(h_0\)): \(760 \, mm\) (which is equivalent to \(76 \, cm\))
- Change in temperature (\(\Delta t\)): \(0^\circ C - 15^\circ C = -15^\circ C\)

2. **Use the Formula for True Barometric Height**:
The formula for the true barometric height (\(h\)) at a different temperature is given by:
\[ h = h_0 \left(1 - \left(\gamma - \alpha\right) \Delta t\right) \]

3. **Substitute the Values into the Formula**:
- Substitute \(h_0 = 76 \, cm\), \(\gamma = 0.0018 \, /^\circ C\), \(\alpha = 0.000006 \, /^\circ C\), and \(\Delta t = -15^\circ C\) into the formula:
\[ h = 76 \left(1 - \left(0.0018 - 0.000006\right)(-15)\right) \]

4. **Calculate the Difference in Coefficients**:
- Calculate \(\gamma - \alpha\):
\[ \gamma - \alpha = 0.0018 - 0.000006 = 0.001794 \]

5. **Calculate the Product with \(\Delta t\)**:
- Now calculate \((\gamma - \alpha) \Delta t\):
\[ (\gamma - \alpha)(-15) = 0.001794 \times (-15) = -0.02691 \]

6. **Substitute Back into the Height Equation**:
\[ h = 76 \left(1 + 0.02691\right) = 76 \times 1.02691 \]

7. **Final Calculation**:
- Now calculate \(h\):
\[ h \approx 76 \times 1.02691 \approx 77.04 \, cm \]

### Final Answer:
The true barometric height at \(0^\circ C\) is approximately \(77.04 \, cm\).