Brass scale of a Barometer gives correct reading at `0^(@)C`. Coefficient of linear expansion of brass is `18 xx 10^(-6 xx @)C`. If the barometer reads `76cm` at `20^(@)C`, the correct reading is
`(gamma_(Hg) = 18 xx 10^(-5 //@)C)`

Options

A. `76.426 cm`
B. `75.7 cm`
C. `76.642 cm`
D. `76.264 cm`

Explanation

To find the correct reading of the barometer at 20°C, we need to account for the linear expansion of the brass scale due to the increase in temperature. Here’s a step-by-step solution:

### Step 1: Understand the problem
The brass scale gives a correct reading at 0°C, and we need to find the correct reading when the temperature is 20°C. The coefficient of linear expansion for brass is given as \( \alpha = 18 \times 10^{-6} \, \text{°C}^{-1} \).

### Step 2: Identify the parameters
- Scale reading at 20°C, \( L = 76 \, \text{cm} \)
- Coefficient of linear expansion of brass, \( \alpha = 18 \times 10^{-6} \, \text{°C}^{-1} \)
- Coefficient of linear expansion of mercury, \( \gamma = 18 \times 10^{-5} \, \text{°C}^{-1} \)
- Change in temperature, \( \Delta T = 20 - 0 = 20 \, \text{°C} \)

### Step 3: Calculate the effective change in the scale reading
The effective change in the scale reading due to the expansion can be calculated using the formula:
\[ \text{True Value} = \text{Scale Reading} \times \left(1 - (\gamma - \alpha) \Delta T\right) \]

### Step 4: Calculate \( \gamma - \alpha \)
First, we need to calculate \( \gamma - \alpha \):
\[ \gamma - \alpha = 18 \times 10^{-5} - 1.8 \times 10^{-5} = 16.2 \times 10^{-5} \, \text{°C}^{-1} \]

### Step 5: Substitute the values into the formula
Now substitute the values into the formula:
\[ \text{True Value} = 76 \, \text{cm} \times \left(1 - (16.2 \times 10^{-5}) \times 20\right) \]

### Step 6: Calculate the term inside the parentheses
Calculate \( (16.2 \times 10^{-5}) \times 20 \):
\[ 16.2 \times 10^{-5} \times 20 = 0.000324 \]
Thus,
\[ 1 - 0.000324 = 0.999676 \]

### Step 7: Calculate the True Value
Now calculate the true value:
\[ \text{True Value} = 76 \, \text{cm} \times 0.999676 \approx 75.782 \, \text{cm} \]

### Step 8: Round the answer
Rounding to one decimal place gives:
\[ \text{True Value} \approx 75.7 \, \text{cm} \]

### Final Answer
The correct reading of the barometer at 20°C is approximately **75.7 cm**.

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