Question

The pressure in a water pipe in the basement of an apartment house is $4.10 \times 10^{5} \mathrm{Pa},\( but on the seventh floor it is only \)1.85 \times 10^{5} \mathrm{Pa} .$ What is the height between the basement and the seventh floor? Assume the water is not flowing; no faucets are opened.

Short answer

Answer: The height difference between the basement and the seventh floor of the apartment house is approximately \(22.93\,\mathrm{m}\).

Step-by-step solution

Identify the formula for hydrostatic pressure

The hydrostatic pressure (P) in a fluid is given by the formula: \(P = \rho g h,\) where \(\rho\) is the density of the fluid, \(g\) is the acceleration due to gravity, and \(h\) is the height difference between the two points in the fluid.

Identify the pressure difference

The pressure in the basement is \(4.10 \times 10^{5} \mathrm{Pa}\) and on the seventh floor, it is \(1.85 \times 10^{5} \mathrm{Pa}\). The difference in pressure is: \(\Delta P = P_\text{basement} - P_\text{7th floor} = (4.10 \times 10^{5}) - (1.85 \times 10^{5}) \mathrm{Pa}\).

Calculate the pressure difference

Now we will calculate the pressure difference: \(\Delta P = (4.10-1.85) \times 10^5 = 2.25 \times 10^5 \mathrm{Pa}\).

Determine the water density and gravitational acceleration

The density of water \(\rho\) is approximately \(1000 \,\mathrm{kg/m^{3}}\). The acceleration due to gravity \(g\) is \(9.81 \,\mathrm{m/s^{2}}\).

Solve for height difference (h)

Rearrange the hydrostatic pressure formula to solve for height difference: \(h = \frac{\Delta P}{\rho g}\). Now, plug in the values: \(h = \frac{2.25 \times 10^{5}\,\mathrm{Pa}}{1000\, \mathrm{kg/m^{3}} \cdot 9.81\, \mathrm{m/s^2}}\).

Calculate the height difference

Calculate the height difference: \(h = \frac{2.25 \times 10^{5}}{1000 \cdot 9.81} = \frac{2.25 \times 10^{5}}{9810} = 22.93\,\mathrm{m}\).

Interpret the result

The height difference between the basement and the seventh floor of the apartment house is approximately \(22.93\,\mathrm{m}\).