To calculate the height of the water column in a reservoir based on the pressure exerted at the floor, you can use the hydrostatic pressure formula, which is:
\[ P = \rho \times g \times h \]
where:
- \( P \) is the pressure at a certain depth in pascals (Pa),
- \( \rho \) is the density of the fluid (in this case, water) in kilograms per cubic meter (kg/m³),
- \( g \) is the acceleration due to gravity in meters per second squared (m/s²), and
- \( h \) is the height of the fluid column above the point where the pressure is measured (in meters).
Given that the pressure at the floor is 12.5 psi, we'll first convert this to pascals:
1 psi can be converted to pascals using the conversion factor: \( 1 \text{ psi} = 6894.76 \text{ Pa} \).
So, \( P \) in pascals = \( 12.5 \text{ psi} \times 6894.76 \text{ Pa/psi} \).
Now, let's use the density of water. The problem with using the density in pounds per cubic inch (lb/in³) is that we need to convert it to the SI unit of kg/m³ to use it in our formula. The conversion factor from lb/in³ to kg/m³ is: \( 1 \text{ lb/in}^3 = 27679.9 \text{ kg/m}^3 \).
The acceleration due to gravity \( g \) is usually taken to be \( 9.80665 \text{ m/s}^2 \).
By rearranging the hydrostatic pressure formula, we can solve for \( h \), the height of the water column:
\[ h = \frac{P}{\rho \times g} \]
We have the value for \( P \) in pascals, and we can convert the given water density to kg/m³ and then plug in the value for \( g \). Finally, we solve for \( h \), which will give us the height of the water in meters.
Let's go through the calculations:
1. Convert pressure to pascals:
\[ P = 12.5 \text{ psi} \times 6894.76 \text{ Pa/psi} = 86209.5 \text{ Pa} \]
2. Convert water density to kg/m³:
\[ \rho = 0.0361 \text{ lb/in}^3 \times 27679.9 \text{ kg/m}^3/\text{lb/in}^3 = 999.11079 \text{ kg/m}^3 \]
3. Use the values for \( P \), \( \rho \), and \( g \) in the formula to solve for \( h \):
\[ h = \frac{86209.5 \text{ Pa}}{999.11079 \text{ kg/m}^3 \times 9.80665 \text{ m/s}^2} \]
\[ h \approx \frac{86209.5}{999.11079 \times 9.80665} \]
\[ h \approx \frac{86209.5}{9796.998782635} \]
\[ h \approx 8.8 \text{ m} \]
So the height of the water column, or the water level in the reservoir, is approximately 8.8 meters.
