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 step by step, we will use the information provided about the heights of the mercury barometer at sea level and at the top of the hill, as well as the ratio of the densities of mercury and air.

### Step-by-Step Solution:

1. **Identify Given Values:**
- Height of mercury barometer at sea level (h1) = 75 cm
- Height of mercury barometer at the top of the hill (h2) = 50 cm
- Ratio of density of mercury (ρ_m) to density of air (ρ_a) = 10^4

2. **Convert Heights to Meters:**
- h1 = 75 cm = 0.75 m
- h2 = 50 cm = 0.50 m

3. **Calculate the Pressure Difference:**
- The pressure at sea level (P1) can be expressed as:
\[ P_1 = \rho_m \cdot g \cdot h_1 \]
- The pressure at the top of the hill (P2) can be expressed as:
\[ P_2 = \rho_m \cdot g \cdot h_2 \]
- The pressure difference (ΔP) is given by:
\[ \Delta P = P_1 - P_2 = \rho_m \cdot g \cdot (h_1 - h_2) \]

4. **Express the Pressure Difference in Terms of Air Density:**
- The pressure difference due to the height of the hill (h) can be expressed as:
\[ \Delta P = \rho_a \cdot g \cdot h \]

5. **Equate the Two Expressions for Pressure Difference:**
- Setting the two expressions for ΔP equal gives:
\[ \rho_m \cdot g \cdot (h_1 - h_2) = \rho_a \cdot g \cdot h \]
- We can cancel out g from both sides:
\[ \rho_m \cdot (h_1 - h_2) = \rho_a \cdot h \]

6. **Substitute the Density Ratio:**
- From the ratio of densities, we have:
\[ \frac{\rho_m}{\rho_a} = 10^4 \implies \rho_m = 10^4 \cdot \rho_a \]
- Substitute this into the equation:
\[ 10^4 \cdot \rho_a \cdot (h_1 - h_2) = \rho_a \cdot h \]
- Cancel ρ_a (assuming it is not zero):
\[ 10^4 \cdot (h_1 - h_2) = h \]

7. **Calculate the Height of the Hill:**
- Substitute the values of h1 and h2:
\[ h = 10^4 \cdot (0.75 - 0.50) = 10^4 \cdot 0.25 = 2500 \text{ m} \]
- Convert meters to kilometers:
\[ h = 2.5 \text{ km} \]

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