The height of a mercury barometer is 75 cm at sea level and 50 cm at the top of a hill. Ration of density of mercury to that of air is `10^(4)`. The height of the hill is

Options

A. 250 m
B. 2.5 km
C. 1.25 km
D. 750 m

Answer

To solve the problem of finding the height of the hill based on the given data about the mercury barometer, we can follow these steps:

### Step 1: Understand the Problem
We have two heights of mercury in a barometer:
- At sea level (H1) = 75 cm
- At the top of the hill (H2) = 50 cm

We also know the ratio of the density of mercury (ρ_M) to the density of air (ρ_air) is 10^4.

### Step 2: Write the Pressure Equations
The pressure at sea level (P1) can be expressed as:
\[ P_1 = \rho_M g H_1 \]
The pressure at the top of the hill (P2) can be expressed as:
\[ P_2 = \rho_M g H_2 \]

### Step 3: Calculate the Change in Pressure
The change in pressure (ΔP) between the two locations is:
\[ \Delta P = P_1 - P_2 = \rho_M g (H_1 - H_2) \]

### Step 4: Relate Change in Pressure to Height of the Hill
The change in pressure due to the height of the hill (h) can be expressed as:
\[ \Delta P = \rho_{air} g h \]

### Step 5: Equate the Two Pressure Changes
Setting the two expressions for ΔP equal gives us:
\[ \rho_M g (H_1 - H_2) = \rho_{air} g h \]
We can cancel g from both sides:
\[ \rho_M (H_1 - H_2) = \rho_{air} h \]

### Step 6: Substitute the Density Ratio
Using the given ratio of densities:
\[ \frac{\rho_M}{\rho_{air}} = 10^4 \]
We can rewrite the equation as:
\[ 10^4 (H_1 - H_2) = h \]

### Step 7: Calculate the Height of the Hill
Substituting the values of H1 and H2:
\[ h = 10^4 (75 \, \text{cm} - 50 \, \text{cm}) \]
\[ h = 10^4 (25 \, \text{cm}) \]
\[ h = 250000 \, \text{cm} \]

### Step 8: Convert to Kilometers
To convert centimeters to kilometers:
\[ h = \frac{250000 \, \text{cm}}{100000} = 2.5 \, \text{km} \]

### Final Answer
The height of the hill is **2.5 kilometers**.

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