Question

Two houses are served by a nearby water storage tank. House A is \(55 \mathrm{ft}\) above House \(\mathrm{B}\) which sits at \(725 \mathrm{ft}\) above sea level. The base of the tank sits at \(855 \mathrm{ft}\) above sea level. The low water level in the tank is at \(7.5 \mathrm{ft}\). At the low level, will House A meet the minimum pressure requirements of \(35 \mathrm{psi} ?\)

Short answer

At the low water level, House A will meet the minimum pressure requirement of 35 psi, as the pressure is 35.7225 psi.

Step-by-step solution

- Determine the height difference between the tank's water level and House A

First, find the total height of the water in the tank by adding the base height of the tank to the low water level: \( 855 \text{ ft} + 7.5 \text{ ft} = 862.5 \text{ ft} \). Then, calculate the height difference between the low water level in the tank and House A: \( 862.5 \text{ ft} - (725 \text{ ft} + 55 \text{ ft}) = 862.5 \text{ ft} - 780 \text{ ft} = 82.5 \text{ ft} \).

- Convert the height difference to pressure in psi

To find the pressure in psi that the water exerts, use the conversion factor for water height to pressure: 1 foot of water height = 0.433 psi. So, multiply the height difference by this factor: \( 82.5 \text{ ft} \times 0.433 \text{ psi/ft} = 35.7225 \text{ psi} \).

- Compare the calculated pressure to the minimum pressure requirement

The calculated pressure at House A is \( 35.7225 \text{ psi} \). The minimum pressure requirement for House A is \( 35 \text{ psi} \). Since \( 35.7225 \text{ psi} > 35 \text{ psi} \), House A meets the minimum pressure requirement.

Key concepts

hydrostatic pressure

To understand how we calculate water pressure, we first need to grasp the idea of hydrostatic pressure. This is the pressure exerted by a fluid due to the force of gravity. In simpler terms, it's how much weight the water comes to bear standing still in a tank or other container. The deeper you go beneath the water surface, the greater the hydrostatic pressure. Water molecules are pulled down by gravity and the weight of the water above. This pressure can be calculated using the equation:

\[ P = \rho gh \]
where: \({\rho}\) is the density of the fluid \({g}\) is the acceleration due to gravity and \({h}\) is the height or depth of the fluid.

In our example, knowing how to convert the height of the water column into pressure is crucial to determine if House A meets the minimum pressure requirements.

elevation difference

Now, let's delve into the concept of elevation difference. This is simply the vertical height difference between two points. In calculating water pressure at different elevations, it's vital to know how far one point is above or below another. For this exercise, we calculated the elevation difference between the tank's water level and House A. Here's how:

First, we determined the water height in the tank:

- Base height of the tank: 855 ft
- Low water level height: 7.5 ft
Adding these: 855 ft + 7.5 ft = 862.5 ft

Then, we found the elevation of House A:

- House B's elevation: 725 ft
- House A is 55 ft higher than House B
Adding these: 725 ft + 55 ft = 780 ft

Finally, we subtracted House A's elevation from the water level's height:

862.5 ft - 780 ft = 82.5 ft
This gives us the elevation difference from the water in the tank to House A.

psi conversion factor

Next, let's talk about the conversion factor. To find how much pressure water exerts, we use a specific conversion factor: 1 foot of water height equals 0.433 psi (pounds per square inch). This factor helps us translate a water column height to pressure. In our exercise:

\[ 82.5 \text{ ft} \times 0.433 \text{ psi/ft} = 35.7225 \text{ psi} \]
This calculation tells us that an 82.5-feet column of water exerts 35.7225 psi of pressure. Converting elevation height to psi is key to determining if water pressure meets specific needs, like those of House A.

minimum pressure requirements

Finally, let's discuss minimum pressure requirements, which are standards that ensure systems function properly. For residential water systems, a common minimum requirement is 35 psi. This ensures there’s enough pressure for all household needs, including showering, washing, and cooking.

In our case, we calculated that the pressure at House A is 35.7225 psi, which is above the required 35 psi. Therefore, House A meets the minimum pressure requirement, guaranteeing reliable water flow and functionality.

To sum up, by understanding hydrostatic pressure, elevation difference, the psi conversion factor, and minimum pressure requirements, you can calculate and ensure sufficient water pressure for any situation.