The main water line enters a house on the first floor . The line has a gauge pressure of `1.90 xx 10^(5) Pa`. A faucet on the second floor , 6.50 m above the first floor , is turned off . What is the gauge pressure at this faucet ?

Options

A. `1.45 xx 10^(5) Pa`
B. `1.26 xx 10^(5) Pa`
C. `2.29 xx 10^(5) Pa`
D. `2.55 xx 10^(5) Pa`

Answer

To solve the problem of finding the gauge pressure at the faucet on the second floor, we can follow these steps:

### Step 1: Understand the Problem
We need to find the gauge pressure at a faucet located 6.5 meters above the main water line, which has a gauge pressure of \(1.90 \times 10^5 \, \text{Pa}\).

### Step 2: Identify Given Values
- Gauge pressure at the first floor (\(P_1\)): \(1.90 \times 10^5 \, \text{Pa}\)
- Height difference (\(h\)): \(6.5 \, \text{m}\)
- Density of water (\(\rho\)): \(1000 \, \text{kg/m}^3\)
- Acceleration due to gravity (\(g\)): \(9.81 \, \text{m/s}^2\)

### Step 3: Use the Hydrostatic Pressure Equation
The pressure difference due to height can be calculated using the hydrostatic pressure equation:
\[ \Delta P = \rho g h \]
Where:
- \(\Delta P\) is the change in pressure,
- \(\rho\) is the density of the fluid,
- \(g\) is the acceleration due to gravity,
- \(h\) is the height difference.

### Step 4: Calculate the Pressure Difference
Substituting the values into the equation:
\[ \Delta P = (1000 \, \text{kg/m}^3)(9.81 \, \text{m/s}^2)(6.5 \, \text{m}) \]
Calculating this gives:
\[ \Delta P = 1000 \times 9.81 \times 6.5 = 63865 \, \text{Pa} \]

### Step 5: Calculate the Gauge Pressure at the Faucet
The gauge pressure at the faucet (\(P_2\)) can be found by subtracting the pressure difference from the gauge pressure at the first floor:
\[ P_2 = P_1 - \Delta P \]
Substituting the values:
\[ P_2 = 1.90 \times 10^5 \, \text{Pa} - 63865 \, \text{Pa} \]
Calculating this gives:
\[ P_2 = 190000 - 63865 = 126135 \, \text{Pa} \]

### Step 6: Present the Final Answer
Thus, the gauge pressure at the faucet on the second floor is:
\[ P_2 \approx 1.26 \times 10^5 \, \text{Pa} \]