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Chapter 9: Problem 81

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

Expert verified
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

01

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.
02

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: \(\Delta P = P_\text{basement} - P_\text{7th floor} = (4.10 \times 10^{5}) - (1.85 \times 10^{5}) \mathrm{Pa}\).
03

Calculate the pressure difference

Now we will calculate the pressure difference: \(\Delta P = (4.10-1.85) \times 10^5 = 2.25 \times 10^5 \mathrm{Pa}\).
04

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 \(9.81 \,\mathrm{m/s^{2}}\).
05

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: \(h = \frac{2.25 \times 10^{5}\,\mathrm{Pa}}{1000\, \mathrm{kg/m^{3}} \cdot 9.81\, \mathrm{m/s^2}}\).
06

Calculate the height difference

Calculate the height difference: \(h = \frac{2.25 \times 10^{5}}{1000 \cdot 9.81} = \frac{2.25 \times 10^{5}}{9810} = 22.93\,\mathrm{m}\).
07

Interpret the result

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

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Most popular questions from this chapter

A flat-bottomed barge, loaded with coal, has a mass of $3.0 \times 10^{5} \mathrm{kg} .\( The barge is \)20.0 \mathrm{m}\( long and \)10.0 \mathrm{m}$ wide. It floats in fresh water. What is the depth of the barge below the waterline? (W) tutorial: boat)
A viscous liquid is flowing steadily through a pipe of diameter \(D .\) Suppose you replace it by two parallel pipes, each of diameter \(D / 2,\) but the same length as the original pipe. If the pressure difference between the ends of these two pipes is the same as for the original pipe, what is the total rate of flow in the two pipes compared to the original flow rate?
A cylindrical disk has volume \(8.97 \times 10^{-3} \mathrm{m}^{3}\) and mass \(8.16 \mathrm{kg} .\) The disk is floating on the surface of some water with its flat surfaces horizontal. The area of each flat surface is $0.640 \mathrm{m}^{2} .$ (a) What is the specific gravity of the disk? (b) How far below the water level is its bottom surface? (c) How far above the water level is its top surface?
The diameter of a certain artery has decreased by \(25 \%\) due to arteriosclerosis. (a) If the same amount of blood flows through it per unit time as when it was unobstructed, by what percentage has the blood pressure difference between its ends increased? (b) If, instead, the pressure drop across the artery stays the same, by what factor does the blood flow rate through it decrease? (In reality we are likely to see a combination of some pressure increase with some reduction in flow.)
The density of platinum is \(21500 \mathrm{kg} / \mathrm{m}^{3} .\) Find the ratio of the volume of \(1.00 \mathrm{kg}\) of platinum to the volume of $1.00 \mathrm{kg}$ of aluminum.
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