Determine the minimum gauge pressure needed in the water pipe leading into a building if water is to come out of a faucet on the twelfth floor, 38 m above that pipe.

P > (density) g H = (1000)(9.8)*38= 3.72*10^5 Pascals

= 3.72 atmospheres

To find the minimum gauge pressure needed in the water pipe, we can use the principles of fluid mechanics and Bernoulli's equation.

Bernoulli's equation states that the total pressure in a fluid is constant along a streamline. It can be written as:

P + 1/2 ρv^2 + ρgh = constant

Where:
P is the pressure in the fluid
ρ is the density of the fluid
v is the velocity of the fluid
g is the acceleration due to gravity
h is the height above a reference point

We can use this equation to determine the minimum gauge pressure needed at the base of the building, taking the 12th floor into account.

Given:
Height of the 12th floor = 38 m

To find the pressure at the base of the building, we need to take into account the height difference between the faucet on the 12th floor and the water pipe.

The pressure due to height difference can be calculated using the formula:

P = ρgh

Let's calculate the pressure due to the height difference:

P = ρgh
P = (1000 kg/m^3) * (9.81 m/s^2) * 38 m
P = 373580 Pa

So, the pressure due to the height difference is 373580 Pa.

Now, let's determine the minimum gauge pressure in the water pipe. We can assume the velocity of the water in the pipe is negligible compared to the velocity at the faucet on the 12th floor. Therefore, we can neglect the term 1/2 ρv^2 in Bernoulli's equation.

Using Bernoulli's equation, we can express the pressure at the base of the building as:

P_base + ρgh_base = P_faucet + ρgh_faucet

Where:
P_base is the pressure at the base of the building (what we want to find)
P_faucet is the pressure at the faucet on the 12th floor (which is atmospheric pressure)
h_base is the height of the water pipe above the reference point
h_faucet is the height of the faucet above the reference point

Since the pressure at the faucet, P_faucet, is atmospheric pressure (101325 Pa), and h_faucet is 38 m, the equation becomes:

P_base + ρgh_base = 101325 Pa + (1000 kg/m^3) * (9.81 m/s^2) * 38 m

Simplifying the equation, we can solve for P_base:

P_base = 101325 Pa + (1000 kg/m^3) * (9.81 m/s^2) * 38 m
P_base = 101325 Pa + 372041.4 Pa
P_base = 473,366.4 ≈ 473,400 Pa

So, the minimum gauge pressure needed in the water pipe leading into the building is approximately 473,400 Pa.

To determine the minimum gauge pressure needed in the water pipe, we need to take into account the height difference between the faucet on the twelfth floor and the location of the water pipe.

Here's how you can calculate it:

Step 1: Convert the height difference to meters
Since the height difference is given as 38 m, no conversion is necessary.

Step 2: Convert the height difference to pressure
The pressure needed at the faucet on the twelfth floor is dependent on the difference in height between the faucet and the water pipe. The pressure in a fluid due to its height is given by the equation P = ρgh, where P is the pressure, ρ is the density of the fluid, g is the acceleration due to gravity, and h is the height.

Step 3: Determine the density of water
The density of water is approximately 1000 kg/m³.

Step 4: Determine the acceleration due to gravity
The acceleration due to gravity is approximately 9.8 m/s².

Step 5: Calculate the pressure
Multiply the height difference by the density of water and the acceleration due to gravity to calculate the minimum gauge pressure needed in the water pipe.

P = (density) x (acceleration due to gravity) x (height difference)
P = 1000 kg/m³ x 9.8 m/s² x 38 m
P ≈ 372,400 Pa

Therefore, the minimum gauge pressure needed in the water pipe leading into the building is approximately 372,400 Pa.